3.5.80 \(\int \frac {x^{7/2}}{(a+b x^2) (c+d x^2)^3} \, dx\) [480]

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

[Out]

-1/2*a^(5/4)*b^(3/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)+1/2*a^(5/4)*b^(3/4)*arctan
(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)-1/64*(-5*a^2*d^2-30*a*b*c*d+3*b^2*c^2)*arctan(1-d^(1/
4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(5/4)/(-a*d+b*c)^3*2^(1/2)+1/64*(-5*a^2*d^2-30*a*b*c*d+3*b^2*c^2)*arctan
(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(5/4)/(-a*d+b*c)^3*2^(1/2)-1/4*a^(5/4)*b^(3/4)*ln(a^(1/2)+x*b^(1
/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/(-a*d+b*c)^3*2^(1/2)+1/4*a^(5/4)*b^(3/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(
1/4)*2^(1/2)*x^(1/2))/(-a*d+b*c)^3*2^(1/2)-1/128*(-5*a^2*d^2-30*a*b*c*d+3*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4
)*d^(1/4)*2^(1/2)*x^(1/2))/c^(3/4)/d^(5/4)/(-a*d+b*c)^3*2^(1/2)+1/128*(-5*a^2*d^2-30*a*b*c*d+3*b^2*c^2)*ln(c^(
1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(3/4)/d^(5/4)/(-a*d+b*c)^3*2^(1/2)-1/4*c*x^(1/2)/d/(-a*d+b*c
)/(d*x^2+c)^2+1/16*(-9*a*d+b*c)*x^(1/2)/d/(-a*d+b*c)^2/(d*x^2+c)

________________________________________________________________________________________

Rubi [A]
time = 0.55, 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 = {477, 481, 541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {a^{5/4} b^{3/4} \text {ArcTan}\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} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\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} \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 {\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \text {ArcTan}\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 ) \text {ArcTan}\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 {\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 {\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]

-1/4*(c*Sqrt[x])/(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)*ArcTa
n[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)*ArcT
an[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] + Sqr
t[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

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 477

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 481

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 536

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 541

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 631

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 642

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 1176

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 1179

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 \text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 1.61, size = 328, normalized size = 0.52 \begin {gather*} \frac {\frac {4 (-b c+a d) \sqrt {x} \left (b c \left (3 c-d x^2\right )+a d \left (5 c+9 d x^2\right )\right )}{d \left (c+d x^2\right )^2}-32 \sqrt {2} a^{5/4} b^{3/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} \left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4} d^{5/4}}+32 \sqrt {2} 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 )+\frac {\sqrt {2} \left (3 b^2 c^2-30 a b c d-5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4} d^{5/4}}}{64 (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]

((4*(-(b*c) + a*d)*Sqrt[x]*(b*c*(3*c - d*x^2) + a*d*(5*c + 9*d*x^2)))/(d*(c + d*x^2)^2) - 32*Sqrt[2]*a^(5/4)*b
^(3/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - (Sqrt[2]*(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])])/(c^(3/4)*d^(5/4)) + 32*Sqrt[2]*a^(5/
4)*b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + (Sqrt[2]*(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)])/(c^(3/4)*d^(5/4)))/(64*(b*c -
a*d)^3)

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Maple [A]
time = 0.09, size = 330, normalized size = 0.52

method	result	size
derivativedivides	\(-\frac {a b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{2}+\frac {5}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) \sqrt {x}}{32 d}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+30 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d c}}{\left (a d -b c \right )^{3}}\)	\(330\)
default	\(-\frac {a b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{2}+\frac {5}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) \sqrt {x}}{32 d}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+30 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d c}}{\left (a d -b c \right )^{3}}\)	\(330\)

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,method=_RETURNVERBOSE)

[Out]

-1/4*a*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2
)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))+2/(a*
d-b*c)^3*(((-9/32*a^2*d^2+5/16*a*b*c*d-1/32*b^2*c^2)*x^(5/2)-1/32*c*(5*a^2*d^2-2*a*b*c*d-3*b^2*c^2)/d*x^(1/2))
/(d*x^2+c)^2+1/256*(5*a^2*d^2+30*a*b*c*d-3*b^2*c^2)/d*(c/d)^(1/4)/c*2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)
+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^
(1/2)/(c/d)^(1/4)*x^(1/2)-1)))

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Maxima [A]
time = 0.54, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5496 vs. \(2 (484) = 968\).
time = 100.80, size = 5496, normalized size = 8.71 \begin {gather*} \text {Too large to display} \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]

1/64*(4*(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)*(-(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^
5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8
*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9
- 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^
6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a^12*c^3*d^17))^(1/4)*arctan(-((b^9*c^11*d^4 - 9*a*b^8*c^
10*d^5 + 36*a^2*b^7*c^9*d^6 - 84*a^3*b^6*c^8*d^7 + 126*a^4*b^5*c^7*d^8 - 126*a^5*b^4*c^6*d^9 + 84*a^6*b^3*c^5*
d^10 - 36*a^7*b^2*c^4*d^11 + 9*a^8*b*c^3*d^12 - a^9*c^2*d^13)*sqrt((9*b^4*c^4 - 180*a*b^3*c^3*d + 870*a^2*b^2*
c^2*d^2 + 300*a^3*b*c*d^3 + 25*a^4*d^4)*x + (b^6*c^8*d^2 - 6*a*b^5*c^7*d^3 + 15*a^2*b^4*c^6*d^4 - 20*a^3*b^3*c
^5*d^5 + 15*a^4*b^2*c^4*d^6 - 6*a^5*b*c^3*d^7 + a^6*c^2*d^8)*sqrt(-(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*
b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^
6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*
c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^
8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a^12*c^3*d^17)))*(-(81*b^8
*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5
*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 +
66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*
d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c
^4*d^16 + a^12*c^3*d^17))^(3/4) + (3*b^11*c^13*d^4 - 57*a*b^10*c^12*d^5 + 373*a^2*b^9*c^11*d^6 - 1287*a^3*b^8*
c^10*d^7 + 2718*a^4*b^7*c^9*d^8 - 3738*a^5*b^6*c^8*d^9 + 3402*a^6*b^5*c^7*d^10 - 1998*a^7*b^4*c^6*d^11 + 687*a
^8*b^3*c^5*d^12 - 93*a^9*b^2*c^4*d^13 - 15*a^10*b*c^3*d^14 + 5*a^11*c^2*d^15)*sqrt(x)*(-(81*b^8*c^8 - 3240*a*b
^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 +
133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^
13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792*a^7*
b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a^12*
c^3*d^17))^(3/4))/(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4
*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)) - 128*(-a^5
*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^
5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^1
0*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(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)*arctan(-((b^9*c^9 - 9*a*b^8*c^
8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6
- 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^
2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^
7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(3/4)*sq
rt(a^2*b^2*x + (b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5
*b*c*d^5 + a^6*d^6)*sqrt(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 -
220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))) - (a*b^10*c^9 - 9*a^2*b^9*c^8*d
+ 36*a^3*b^8*c^7*d^2 - 84*a^4*b^7*c^6*d^3 + 126*a^5*b^6*c^5*d^4 - 126*a^6*b^5*c^4*d^5 + 84*a^7*b^4*c^3*d^6 - 3
6*a^8*b^3*c^2*d^7 + 9*a^9*b^2*c*d^8 - a^10*b*d^9)*(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d
^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d
^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(3/4)*s
qrt(x))/(a^5*b^3)) + 32*(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 -
220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 1...

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Sympy [F(-1)] Timed out
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)

[Out]

Timed out

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Giac [A]
time = 2.29, size = 944, normalized size = 1.50 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\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}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\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}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\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}\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\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}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\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}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\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}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\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}\right )}} - \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\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}\right )}} + \frac {b c d x^{\frac {5}{2}} - 9 \, a d^{2} x^{\frac {5}{2}} - 3 \, b c^{2} \sqrt {x} - 5 \, a c d \sqrt {x}}{16 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \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="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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Mupad [B]
time = 2.32, size = 2500, normalized size = 3.96 \begin {gather*} \text {Too large to display} \end {gather*}

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 - 25...

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