∫ x7∕2 _________________________

3.5.62 \(\int \frac {x^{7/2}}{(a+b x^2) (c+d x^2)^3} \, dx\)

Optimal. Leaf size=631 \[ -\frac {a^{5/4} b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\sqrt {x} (b c-9 a d)}{16 d \left (c+d x^2\right ) (b c-a d)^2}-\frac {c \sqrt {x}}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

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Rubi [A] time = 0.81, antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 470, 527, 522, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac {a^{5/4} b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}+\frac {\sqrt {x} (b c-9 a d)}{16 d \left (c+d x^2\right ) (b c-a d)^2}-\frac {c \sqrt {x}}{4 d \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x^(7/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-(c*Sqrt[x])/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 9*a*d)*Sqrt[x])/(16*d*(b*c - a*d)^2*(c + d*x^2)) - (a^(

5/4)*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) + (a^(5/4)*b^(3/4)*ArcTan[

1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*ArcTan

[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d

- 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3) - (a^(

5/4)*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) + (a^(5/4)*

b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) - ((3*b^2*c^2 -

30*a*b*c*d - 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4

)*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[

d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di

st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[

{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b

]]))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno

minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*

x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege

rQ[p]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2

*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2

*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +

(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &

& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f

)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a

, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[

((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d

)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)

*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub

st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;

FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S

imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},

x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[

(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /

; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{7/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^8}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {a c+(b c-8 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 d (b c-a d)}\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-9 a d) \sqrt {x}}{16 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {a c (3 b c+5 a d)+3 b c (b c-9 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c d (b c-a d)^2}\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-9 a d) \sqrt {x}}{16 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (2 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 d (b c-a d)^3}\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-9 a d) \sqrt {x}}{16 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a^{3/2} b\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (a^{3/2} b\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} d (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} d (b c-a d)^3}\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-9 a d) \sqrt {x}}{16 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a^{3/2} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}+\frac {\left (a^{3/2} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}-\frac {\left (a^{5/4} b^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (a^{5/4} b^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} d^{3/2} (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} d^{3/2} (b c-a d)^3}-\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-9 a d) \sqrt {x}}{16 d (b c-a d)^2 \left (c+d x^2\right )}-\frac {a^{5/4} b^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (a^{5/4} b^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (a^{5/4} b^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}\\ &=-\frac {c \sqrt {x}}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-9 a d) \sqrt {x}}{16 d (b c-a d)^2 \left (c+d x^2\right )}-\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac {a^{5/4} b^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 0.54, size = 640, normalized size = 1.01 \begin {gather*} \frac {-32 \sqrt {2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+32 \sqrt {2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-64 \sqrt {2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+64 \sqrt {2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-\sqrt {2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+\sqrt {2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-2 \sqrt {2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )+2 \sqrt {2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )+8 c^{3/4} \sqrt [4]{d} \sqrt {x} \left (c+d x^2\right ) (b c-9 a d) (b c-a d)-32 c^{7/4} \sqrt [4]{d} \sqrt {x} (b c-a d)^2}{128 c^{3/4} d^{5/4} \left (c+d x^2\right )^2 (b c-a d)^3} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(7/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

(-32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x] + 8*c^(3/4)*d^(1/4)*(b*c - 9*a*d)*(b*c - a*d)*Sqrt[x]*(c + d*x^2) -

64*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 64*S

qrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]

*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*

(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 32*Sqrt[2]*

a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 32*

Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*

x] - Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]

+ Sqrt[d]*x] + Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4

)*Sqrt[x] + Sqrt[d]*x])/(128*c^(3/4)*d^(5/4)*(b*c - a*d)^3*(c + d*x^2)^2)

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IntegrateAlgebraic [A] time = 1.99, size = 378, normalized size = 0.60 \begin {gather*} -\frac {a^{5/4} b^{3/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} (b c-a d)^3}+\frac {a^{5/4} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{32 \sqrt {2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac {-5 a c d \sqrt {x}-9 a d^2 x^{5/2}-3 b c^2 \sqrt {x}+b c d x^{5/2}}{16 d \left (c+d x^2\right )^2 (a d-b c)^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[x^(7/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

(-3*b*c^2*Sqrt[x] - 5*a*c*d*Sqrt[x] + b*c*d*x^(5/2) - 9*a*d^2*x^(5/2))/(16*d*(-(b*c) + a*d)^2*(c + d*x^2)^2) -

(a^(5/4)*b^(3/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*(b*c -

a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]

)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3) + (a^(5/4)*b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(S

qrt[a] + Sqrt[b]*x)])/(Sqrt[2]*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)

*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3)

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [A] time = 1.57, size = 944, normalized size = 1.50

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

(a*b^3)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)

*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) + (a*b^3)^(1/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(

1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*

d^3) + 1/2*(a*b^3)^(1/4)*a*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^2

*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) - 1/2*(a*b^3)^(1/4)*a*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x +

sqrt(a/b))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) + 1/32*(3*(c*d

^3)^(1/4)*b^2*c^2 - 30*(c*d^3)^(1/4)*a*b*c*d - 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4

) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^4*d^2 - 3*sqrt(2)*a*b^2*c^3*d^3 + 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)

*a^3*c*d^5) + 1/32*(3*(c*d^3)^(1/4)*b^2*c^2 - 30*(c*d^3)^(1/4)*a*b*c*d - 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*

sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^4*d^2 - 3*sqrt(2)*a*b^2*c^3*d^3 + 3*sqrt

(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) + 1/64*(3*(c*d^3)^(1/4)*b^2*c^2 - 30*(c*d^3)^(1/4)*a*b*c*d - 5*(c*d^3)^

(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^4*d^2 - 3*sqrt(2)*a*b^2*c^3*d^3

+ 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) - 1/64*(3*(c*d^3)^(1/4)*b^2*c^2 - 30*(c*d^3)^(1/4)*a*b*c*d - 5

*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^4*d^2 - 3*sqrt(2)*a*b

^2*c^3*d^3 + 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) + 1/16*(b*c*d*x^(5/2) - 9*a*d^2*x^(5/2) - 3*b*c^2*sq

rt(x) - 5*a*c*d*sqrt(x))/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(d*x^2 + c)^2)

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maple [A] time = 0.02, size = 839, normalized size = 1.33

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

-1/4*a*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)

*x^(1/2)+(a/b)^(1/2)))-1/2*a*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2*a*b/(

a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-9/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*a^2*

d^2+5/8/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*a*b*c*d-1/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*b^2*c^2-5/16/(a*d-b*c)^3/

(d*x^2+c)^2*c*d*x^(1/2)*a^2+1/8/(a*d-b*c)^3/(d*x^2+c)^2*c^2*x^(1/2)*a*b+3/16/(a*d-b*c)^3/(d*x^2+c)^2*c^3/d*x^(

1/2)*b^2+5/64/(a*d-b*c)^3*d*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+15/32/(a*d-b*c)^3*

(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-3/64/(a*d-b*c)^3/d*(c/d)^(1/4)*c*2^(1/2)*arctan(

2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+5/64/(a*d-b*c)^3*d*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)

-1)*a^2+15/32/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-3/64/(a*d-b*c)^3/d*(c/

d)^(1/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128/(a*d-b*c)^3*d*(c/d)^(1/4)/c*2^(1/2)*ln((x+(

c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2+15/64/(a*d-b*c)^3*(c/

d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a

*b-3/128/(a*d-b*c)^3/d*(c/d)^(1/4)*c*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(

1/2)*x^(1/2)+(c/d)^(1/2)))*b^2

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maxima [A] time = 2.64, size = 653, normalized size = 1.03 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )} a^{2}}{4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (b c d - 9 \, a d^{2}\right )} x^{\frac {5}{2}} - {\left (3 \, b c^{2} + 5 \, a c d\right )} \sqrt {x}}{16 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} - 30 \, a b c d - 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} - 30 \, a b c d - 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (3 \, b^{2} c^{2} - 30 \, a b c d - 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b^{2} c^{2} - 30 \, a b c d - 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt

(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqr

t(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(3/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sq

rt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(3/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/

4))*a^2/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1/16*((b*c*d - 9*a*d^2)*x^(5/2) - (3*b*c^2 + 5*a

*c*d)*sqrt(x))/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^4 + 2*(b^2*c

^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2) + 1/128*(2*sqrt(2)*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*arctan(1/2*sq

rt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2

*sqrt(2)*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x)

)/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*log(sq

rt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(3*b^2*c^2 - 30*a*b*c*d - 5*a

^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c^3*d - 3*a*b^2*c^

2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)

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mupad [B] time = 4.22, size = 35251, normalized size = 55.87

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/((a + b*x^2)*(c + d*x^2)^3),x)

[Out]

atan((((((81*a^3*b^13*c^7)/2048 - (625*a^10*b^6*d^7)/2048 - (3159*a^4*b^12*c^6*d)/2048 + (148215*a^9*b^7*c*d^6

)/2048 + (44901*a^5*b^11*c^5*d^2)/2048 - (262899*a^6*b^10*c^4*d^3)/2048 + (386451*a^7*b^9*c^3*d^4)/2048 + (997

755*a^8*b^8*c^2*d^5)/2048)/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 +

70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) + (((-(a^5*b^3)/(16*a^12*d^12 +

16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 +

14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^

2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(1280*a^16*b^4*c*d^18 + 8960*a^3*b^17*c^14*d^5 - 106240

*a^4*b^16*c^13*d^6 + 576000*a^5*b^15*c^12*d^7 - 1886720*a^6*b^14*c^11*d^8 + 4153600*a^7*b^13*c^10*d^9 - 646272

0*a^8*b^12*c^9*d^10 + 7265280*a^9*b^11*c^8*d^11 - 5913600*a^10*b^10*c^7*d^12 + 3421440*a^11*b^9*c^6*d^13 - 133

7600*a^12*b^8*c^5*d^14 + 309760*a^13*b^7*c^4*d^15 - 23040*a^14*b^6*c^3*d^16 - 6400*a^15*b^5*c^2*d^17))/(a^8*d^

9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^

3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) - (x^(1/2)*(409600*a^19*b^4*d^20 + 147456*a^3*b^20*c^16*d^4 + 1205

8624*a^4*b^19*c^15*d^5 - 141950976*a^5*b^18*c^14*d^6 + 714080256*a^6*b^17*c^13*d^7 - 2086993920*a^7*b^16*c^12*

d^8 + 3911712768*a^8*b^15*c^11*d^9 - 4814143488*a^9*b^14*c^10*d^10 + 3714056192*a^10*b^13*c^9*d^11 - 139817779

2*a^11*b^12*c^8*d^12 - 259522560*a^12*b^11*c^7*d^13 + 508952576*a^13*b^10*c^6*d^14 - 116391936*a^14*b^9*c^5*d^

15 - 103612416*a^15*b^8*c^4*d^16 + 77070336*a^16*b^7*c^3*d^17 - 17694720*a^17*b^6*c^2*d^18))/(4096*(a^12*d^13

+ b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^

5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^

10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520

*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^

7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^

11))^(3/4))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^

4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 -

3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*1i - (x^(1/2)*(

81*a^4*b^15*c^8 + 26225*a^12*b^7*d^8 - 3240*a^5*b^14*c^7*d + 322200*a^11*b^8*c*d^7 + 48060*a^6*b^13*c^6*d^2 -

307800*a^7*b^12*c^5*d^3 + 658566*a^8*b^11*c^4*d^4 + 328680*a^9*b^10*c^3*d^5 + 1024380*a^10*b^9*c^2*d^6)*1i)/(4

096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c

^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c

^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*

c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672

*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d -

192*a^11*b*c*d^11))^(1/4) - ((((81*a^3*b^13*c^7)/2048 - (625*a^10*b^6*d^7)/2048 - (3159*a^4*b^12*c^6*d)/2048 +

(148215*a^9*b^7*c*d^6)/2048 + (44901*a^5*b^11*c^5*d^2)/2048 - (262899*a^6*b^10*c^4*d^3)/2048 + (386451*a^7*b^

9*c^3*d^4)/2048 + (997755*a^8*b^8*c^2*d^5)/2048)/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 -

56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) + (((-(a^5

*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12

672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*

d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(1280*a^16*b^4*c*d^18 + 8960*a^3*

b^17*c^14*d^5 - 106240*a^4*b^16*c^13*d^6 + 576000*a^5*b^15*c^12*d^7 - 1886720*a^6*b^14*c^11*d^8 + 4153600*a^7*

b^13*c^10*d^9 - 6462720*a^8*b^12*c^9*d^10 + 7265280*a^9*b^11*c^8*d^11 - 5913600*a^10*b^10*c^7*d^12 + 3421440*a

^11*b^9*c^6*d^13 - 1337600*a^12*b^8*c^5*d^14 + 309760*a^13*b^7*c^4*d^15 - 23040*a^14*b^6*c^3*d^16 - 6400*a^15*

b^5*c^2*d^17))/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c

^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) + (x^(1/2)*(409600*a^19*b^4*d^20 + 147456*a^

3*b^20*c^16*d^4 + 12058624*a^4*b^19*c^15*d^5 - 141950976*a^5*b^18*c^14*d^6 + 714080256*a^6*b^17*c^13*d^7 - 208

6993920*a^7*b^16*c^12*d^8 + 3911712768*a^8*b^15*c^11*d^9 - 4814143488*a^9*b^14*c^10*d^10 + 3714056192*a^10*b^1

3*c^9*d^11 - 1398177792*a^11*b^12*c^8*d^12 - 259522560*a^12*b^11*c^7*d^13 + 508952576*a^13*b^10*c^6*d^14 - 116

391936*a^14*b^9*c^5*d^15 - 103612416*a^15*b^8*c^4*d^16 + 77070336*a^16*b^7*c^3*d^17 - 17694720*a^17*b^6*c^2*d^

18))/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^

4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^

9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^

2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6

- 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^

11*d - 192*a^11*b*c*d^11))^(3/4))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3

*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 +

7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))

^(1/4)*1i + (x^(1/2)*(81*a^4*b^15*c^8 + 26225*a^12*b^7*d^8 - 3240*a^5*b^14*c^7*d + 322200*a^11*b^8*c*d^7 + 480

60*a^6*b^13*c^6*d^2 - 307800*a^7*b^12*c^5*d^3 + 658566*a^8*b^11*c^4*d^4 + 328680*a^9*b^10*c^3*d^5 + 1024380*a^

10*b^9*c^2*d^6)*1i)/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c

^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c

^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12

*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a

^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10

- 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4))/(((((81*a^3*b^13*c^7)/2048 - (625*a^10*b^6*d^7)/2048 - (3159*

a^4*b^12*c^6*d)/2048 + (148215*a^9*b^7*c*d^6)/2048 + (44901*a^5*b^11*c^5*d^2)/2048 - (262899*a^6*b^10*c^4*d^3)

/2048 + (386451*a^7*b^9*c^3*d^4)/2048 + (997755*a^8*b^8*c^2*d^5)/2048)/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2

+ 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a

^7*b*c*d^8) + (((-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 792

0*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d

^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(1280*a^16*

b^4*c*d^18 + 8960*a^3*b^17*c^14*d^5 - 106240*a^4*b^16*c^13*d^6 + 576000*a^5*b^15*c^12*d^7 - 1886720*a^6*b^14*c

^11*d^8 + 4153600*a^7*b^13*c^10*d^9 - 6462720*a^8*b^12*c^9*d^10 + 7265280*a^9*b^11*c^8*d^11 - 5913600*a^10*b^1

0*c^7*d^12 + 3421440*a^11*b^9*c^6*d^13 - 1337600*a^12*b^8*c^5*d^14 + 309760*a^13*b^7*c^4*d^15 - 23040*a^14*b^6

*c^3*d^16 - 6400*a^15*b^5*c^2*d^17))/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*

c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) - (x^(1/2)*(409600*a^1

9*b^4*d^20 + 147456*a^3*b^20*c^16*d^4 + 12058624*a^4*b^19*c^15*d^5 - 141950976*a^5*b^18*c^14*d^6 + 714080256*a

^6*b^17*c^13*d^7 - 2086993920*a^7*b^16*c^12*d^8 + 3911712768*a^8*b^15*c^11*d^9 - 4814143488*a^9*b^14*c^10*d^10

+ 3714056192*a^10*b^13*c^9*d^11 - 1398177792*a^11*b^12*c^8*d^12 - 259522560*a^12*b^11*c^7*d^13 + 508952576*a^

13*b^10*c^6*d^14 - 116391936*a^14*b^9*c^5*d^15 - 103612416*a^15*b^8*c^4*d^16 + 77070336*a^16*b^7*c^3*d^17 - 17

694720*a^17*b^6*c^2*d^18))/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^

3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^

8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 +

16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 +

14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^

2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^

10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12

672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d

- 192*a^11*b*c*d^11))^(1/4) - (x^(1/2)*(81*a^4*b^15*c^8 + 26225*a^12*b^7*d^8 - 3240*a^5*b^14*c^7*d + 322200*a

^11*b^8*c*d^7 + 48060*a^6*b^13*c^6*d^2 - 307800*a^7*b^12*c^5*d^3 + 658566*a^8*b^11*c^4*d^4 + 328680*a^9*b^10*c

^3*d^5 + 1024380*a^10*b^9*c^2*d^6))/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3

- 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8

+ 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^1

2*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c

^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^

10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + ((((81*a^3*b^13*c^7)/2048 - (625*a^10*b^6*d^

7)/2048 - (3159*a^4*b^12*c^6*d)/2048 + (148215*a^9*b^7*c*d^6)/2048 + (44901*a^5*b^11*c^5*d^2)/2048 - (262899*a

^6*b^10*c^4*d^3)/2048 + (386451*a^7*b^9*c^3*d^4)/2048 + (997755*a^8*b^8*c^2*d^5)/2048)/(a^8*d^9 + b^8*c^8*d -

8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b

^2*c^2*d^7 - 8*a^7*b*c*d^8) + (((-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b

^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 79

20*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(

1/4)*(1280*a^16*b^4*c*d^18 + 8960*a^3*b^17*c^14*d^5 - 106240*a^4*b^16*c^13*d^6 + 576000*a^5*b^15*c^12*d^7 - 18

86720*a^6*b^14*c^11*d^8 + 4153600*a^7*b^13*c^10*d^9 - 6462720*a^8*b^12*c^9*d^10 + 7265280*a^9*b^11*c^8*d^11 -

5913600*a^10*b^10*c^7*d^12 + 3421440*a^11*b^9*c^6*d^13 - 1337600*a^12*b^8*c^5*d^14 + 309760*a^13*b^7*c^4*d^15

- 23040*a^14*b^6*c^3*d^16 - 6400*a^15*b^5*c^2*d^17))/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d

^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) + (x^(

1/2)*(409600*a^19*b^4*d^20 + 147456*a^3*b^20*c^16*d^4 + 12058624*a^4*b^19*c^15*d^5 - 141950976*a^5*b^18*c^14*d

^6 + 714080256*a^6*b^17*c^13*d^7 - 2086993920*a^7*b^16*c^12*d^8 + 3911712768*a^8*b^15*c^11*d^9 - 4814143488*a^

9*b^14*c^10*d^10 + 3714056192*a^10*b^13*c^9*d^11 - 1398177792*a^11*b^12*c^8*d^12 - 259522560*a^12*b^11*c^7*d^1

3 + 508952576*a^13*b^10*c^6*d^14 - 116391936*a^14*b^9*c^5*d^15 - 103612416*a^15*b^8*c^4*d^16 + 77070336*a^16*b

^7*c^3*d^17 - 17694720*a^17*b^6*c^2*d^18))/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c

^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*

c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/

(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^

5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 +

1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^

12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*

b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 1

92*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + (x^(1/2)*(81*a^4*b^15*c^8 + 26225*a^12*b^7*d^8 - 3240*a^5*b^14*

c^7*d + 322200*a^11*b^8*c*d^7 + 48060*a^6*b^13*c^6*d^2 - 307800*a^7*b^12*c^5*d^3 + 658566*a^8*b^11*c^4*d^4 + 3

28680*a^9*b^10*c^3*d^5 + 1024380*a^10*b^9*c^2*d^6))/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a

^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792

*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(

a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 -

12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c

^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)))*(-(a^5*b^3)/(16*a^12*d^12 +

16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 +

14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^

2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*2i - ((x^(5/2)*(9*a*d - b*c))/(16*(a^2*d^2 + b^2*c^2 -

2*a*b*c*d)) + (x^(1/2)*(3*b*c^2 + 5*a*c*d))/(16*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(c^2 + d^2*x^4 + 2*c*d*x^2

) - 2*atan(((((((81*a^3*b^13*c^7)/2048 - (625*a^10*b^6*d^7)/2048 - (3159*a^4*b^12*c^6*d)/2048 + (148215*a^9*b^

7*c*d^6)/2048 + (44901*a^5*b^11*c^5*d^2)/2048 - (262899*a^6*b^10*c^4*d^3)/2048 + (386451*a^7*b^9*c^3*d^4)/2048

+ (997755*a^8*b^8*c^2*d^5)/2048)*1i)/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5

*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) + (((-(a^5*b^3)/(16*a

^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7

*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*

a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(1280*a^16*b^4*c*d^18 + 8960*a^3*b^17*c^14*d

^5 - 106240*a^4*b^16*c^13*d^6 + 576000*a^5*b^15*c^12*d^7 - 1886720*a^6*b^14*c^11*d^8 + 4153600*a^7*b^13*c^10*d

^9 - 6462720*a^8*b^12*c^9*d^10 + 7265280*a^9*b^11*c^8*d^11 - 5913600*a^10*b^10*c^7*d^12 + 3421440*a^11*b^9*c^6

*d^13 - 1337600*a^12*b^8*c^5*d^14 + 309760*a^13*b^7*c^4*d^15 - 23040*a^14*b^6*c^3*d^16 - 6400*a^15*b^5*c^2*d^1

7))/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56

*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) - (x^(1/2)*(409600*a^19*b^4*d^20 + 147456*a^3*b^20*c^16

*d^4 + 12058624*a^4*b^19*c^15*d^5 - 141950976*a^5*b^18*c^14*d^6 + 714080256*a^6*b^17*c^13*d^7 - 2086993920*a^7

*b^16*c^12*d^8 + 3911712768*a^8*b^15*c^11*d^9 - 4814143488*a^9*b^14*c^10*d^10 + 3714056192*a^10*b^13*c^9*d^11

- 1398177792*a^11*b^12*c^8*d^12 - 259522560*a^12*b^11*c^7*d^13 + 508952576*a^13*b^10*c^6*d^14 - 116391936*a^14

*b^9*c^5*d^15 - 103612416*a^15*b^8*c^4*d^16 + 77070336*a^16*b^7*c^3*d^17 - 17694720*a^17*b^6*c^2*d^18)*1i)/(40

96*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^

8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^

3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c

^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*

a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 1

92*a^11*b*c*d^11))^(3/4)*1i)*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*

c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*

a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4

) + (x^(1/2)*(81*a^4*b^15*c^8 + 26225*a^12*b^7*d^8 - 3240*a^5*b^14*c^7*d + 322200*a^11*b^8*c*d^7 + 48060*a^6*b

^13*c^6*d^2 - 307800*a^7*b^12*c^5*d^3 + 658566*a^8*b^11*c^4*d^4 + 328680*a^9*b^10*c^3*d^5 + 1024380*a^10*b^9*c

^2*d^6))/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 49

5*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 22

0*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 105

6*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*

d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^1

1*c^11*d - 192*a^11*b*c*d^11))^(1/4) - (((((81*a^3*b^13*c^7)/2048 - (625*a^10*b^6*d^7)/2048 - (3159*a^4*b^12*c

^6*d)/2048 + (148215*a^9*b^7*c*d^6)/2048 + (44901*a^5*b^11*c^5*d^2)/2048 - (262899*a^6*b^10*c^4*d^3)/2048 + (3

86451*a^7*b^9*c^3*d^4)/2048 + (997755*a^8*b^8*c^2*d^5)/2048)*1i)/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a

^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c

*d^8) + (((-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*

b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3

520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(1280*a^16*b^4*c*

d^18 + 8960*a^3*b^17*c^14*d^5 - 106240*a^4*b^16*c^13*d^6 + 576000*a^5*b^15*c^12*d^7 - 1886720*a^6*b^14*c^11*d^

8 + 4153600*a^7*b^13*c^10*d^9 - 6462720*a^8*b^12*c^9*d^10 + 7265280*a^9*b^11*c^8*d^11 - 5913600*a^10*b^10*c^7*

d^12 + 3421440*a^11*b^9*c^6*d^13 - 1337600*a^12*b^8*c^5*d^14 + 309760*a^13*b^7*c^4*d^15 - 23040*a^14*b^6*c^3*d

^16 - 6400*a^15*b^5*c^2*d^17))/(a^8*d^9 + b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^

4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8) + (x^(1/2)*(409600*a^19*b^4*

d^20 + 147456*a^3*b^20*c^16*d^4 + 12058624*a^4*b^19*c^15*d^5 - 141950976*a^5*b^18*c^14*d^6 + 714080256*a^6*b^1

7*c^13*d^7 - 2086993920*a^7*b^16*c^12*d^8 + 3911712768*a^8*b^15*c^11*d^9 - 4814143488*a^9*b^14*c^10*d^10 + 371

4056192*a^10*b^13*c^9*d^11 - 1398177792*a^11*b^12*c^8*d^12 - 259522560*a^12*b^11*c^7*d^13 + 508952576*a^13*b^1

0*c^6*d^14 - 116391936*a^14*b^9*c^5*d^15 - 103612416*a^15*b^8*c^4*d^16 + 77070336*a^16*b^7*c^3*d^17 - 17694720

*a^17*b^6*c^2*d^18)*1i)/(4096*(a^12*d^13 + b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b

^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b

^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12)))*(-(a^5*b^3)/(16*a^12*d^12 + 16*

b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 147

84*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d

^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4)*1i)*(-(a^5*b^3)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^