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

Optimal. Leaf size=476 \[ -\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}+\frac {c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{5/4} (b c-a d)}+\frac {2 \sqrt {x}}{b d} \]

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Rubi [A]  time = 0.49, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 479, 522, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}+\frac {c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{5/4} (b c-a d)}+\frac {2 \sqrt {x}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(2*n
- 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q) + 1)), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && 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 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 )} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^8}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x}}{b d}-\frac {2 \operatorname {Subst}\left (\int \frac {a c+(b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{b d}\\ &=\frac {2 \sqrt {x}}{b d}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)}-\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d (b c-a d)}\\ &=\frac {2 \sqrt {x}}{b d}+\frac {a^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)}+\frac {a^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)}-\frac {c^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d (b c-a d)}-\frac {c^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d (b c-a d)}\\ &=\frac {2 \sqrt {x}}{b d}+\frac {a^{3/2} \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^{3/2} (b c-a d)}+\frac {a^{3/2} \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^{3/2} (b c-a d)}-\frac {a^{5/4} \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^{5/4} (b c-a d)}-\frac {a^{5/4} \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^{5/4} (b c-a d)}-\frac {c^{3/2} \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 )}{2 d^{3/2} (b c-a d)}-\frac {c^{3/2} \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 )}{2 d^{3/2} (b c-a d)}+\frac {c^{5/4} \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 )}{2 \sqrt {2} d^{5/4} (b c-a d)}+\frac {c^{5/4} \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 )}{2 \sqrt {2} d^{5/4} (b c-a d)}\\ &=\frac {2 \sqrt {x}}{b d}-\frac {a^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}+\frac {a^{5/4} \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^{5/4} (b c-a d)}-\frac {a^{5/4} \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^{5/4} (b c-a d)}-\frac {c^{5/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{5/4} (b c-a d)}+\frac {c^{5/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{5/4} (b c-a d)}\\ &=\frac {2 \sqrt {x}}{b d}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{5/4} (b c-a d)}-\frac {a^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {a^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{5/4} (b c-a d)}+\frac {c^{5/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{5/4} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 409, normalized size = 0.86 \begin {gather*} \frac {-\frac {\sqrt {2} a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{b^{5/4}}+\frac {\sqrt {2} a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{b^{5/4}}-\frac {2 \sqrt {2} a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{b^{5/4}}+\frac {2 \sqrt {2} a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{b^{5/4}}-\frac {8 a \sqrt {x}}{b}+\frac {\sqrt {2} c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{d^{5/4}}-\frac {\sqrt {2} c^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{d^{5/4}}+\frac {2 \sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{d^{5/4}}-\frac {2 \sqrt {2} c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{d^{5/4}}+\frac {8 c \sqrt {x}}{d}}{4 b c-4 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a*Sqrt[x])/b + (8*c*Sqrt[x])/d - (2*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(5/4
) + (2*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(5/4) + (2*Sqrt[2]*c^(5/4)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) - (2*Sqrt[2]*c^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]
)/d^(5/4) - (Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(5/4) + (Sqrt[2]*a^
(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(5/4) + (Sqrt[2]*c^(5/4)*Log[Sqrt[c] - Sqr
t[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4) - (Sqrt[2]*c^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sq
rt[x] + Sqrt[d]*x])/d^(5/4))/(4*b*c - 4*a*d)

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IntegrateAlgebraic [A]  time = 0.57, size = 278, normalized size = 0.58 \begin {gather*} -\frac {a^{5/4} \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^{5/4} (b c-a d)}+\frac {a^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{5/4} (b c-a d)}-\frac {c^{5/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{c}}{\sqrt {2} \sqrt [4]{d}}-\frac {\sqrt [4]{d} x}{\sqrt {2} \sqrt [4]{c}}}{\sqrt {x}}\right )}{\sqrt {2} d^{5/4} (a d-b c)}+\frac {c^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} d^{5/4} (a d-b c)}+\frac {2 \sqrt {x}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.12, size = 1388, normalized size = 2.92

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),x, algorithm="fricas")

[Out]

-1/2*(4*(-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4))^(1/4)*b*d*arctan(
-((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*(-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2
- 4*a^3*b^6*c*d^3 + a^4*b^5*d^4))^(3/4)*sqrt(a^2*x + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*sqrt(-a^5/(b^9*c^4
- 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4))) - (a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3
*b^5*c*d^2 - a^4*b^4*d^3)*(-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4))
^(3/4)*sqrt(x))/a^5) - 4*(-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c*d^8 + a^4*d^9))^
(1/4)*b*d*arctan(-((b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*(-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*
d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c*d^8 + a^4*d^9))^(3/4)*sqrt(c^2*x + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*s
qrt(-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c*d^8 + a^4*d^9))) - (b^3*c^4*d^4 - 3*a*
b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3*c*d^7)*(-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*
c*d^8 + a^4*d^9))^(3/4)*sqrt(x))/c^5) - (-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 +
a^4*b^5*d^4))^(1/4)*b*d*log(a*sqrt(x) + (-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3
+ a^4*b^5*d^4))^(1/4)*(b^2*c - a*b*d)) + (-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3
+ a^4*b^5*d^4))^(1/4)*b*d*log(a*sqrt(x) - (-a^5/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3
+ a^4*b^5*d^4))^(1/4)*(b^2*c - a*b*d)) + (-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c
*d^8 + a^4*d^9))^(1/4)*b*d*log(c*sqrt(x) + (-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*
c*d^8 + a^4*d^9))^(1/4)*(b*c*d - a*d^2)) - (-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*
c*d^8 + a^4*d^9))^(1/4)*b*d*log(c*sqrt(x) - (-c^5/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b
*c*d^8 + a^4*d^9))^(1/4)*(b*c*d - a*d^2)) - 4*sqrt(x))/(b*d)

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giac [A]  time = 0.76, size = 476, normalized size = 1.00 \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 - \sqrt {2} a b^{2} d} + \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 - \sqrt {2} a b^{2} d} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \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 )}{\sqrt {2} b c d^{2} - \sqrt {2} a d^{3}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \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 )}{\sqrt {2} b c d^{2} - \sqrt {2} a 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 - \sqrt {2} a b^{2} d\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 - \sqrt {2} a b^{2} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c d^{2} - \sqrt {2} a d^{3}\right )}} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} c \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c d^{2} - \sqrt {2} a d^{3}\right )}} + \frac {2 \, \sqrt {x}}{b d} \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),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 - sqrt(2)*a*b
^2*d) + (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 - sq
rt(2)*a*b^2*d) - (c*d^3)^(1/4)*c*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b*
c*d^2 - sqrt(2)*a*d^3) - (c*d^3)^(1/4)*c*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(s
qrt(2)*b*c*d^2 - sqrt(2)*a*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 - sqrt(2)*a*b^2*d) - 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 - sqrt(2)*a*b^2*d) - 1/2*(c*d^3)^(1/4)*c*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c*d^2
- sqrt(2)*a*d^3) + 1/2*(c*d^3)^(1/4)*c*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c*d^2 - s
qrt(2)*a*d^3) + 2*sqrt(x)/(b*d)

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maple [A]  time = 0.02, size = 339, normalized size = 0.71 \begin {gather*} -\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right ) b}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 \left (a d -b c \right ) d}+\frac {2 \sqrt {x}}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.59, size = 384, normalized size = 0.81 \begin {gather*} \frac {\frac {2 \, \sqrt {2} a^{\frac {3}{2}} \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 {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} a^{\frac {3}{2}} \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 {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}}{4 \, {\left (b^{2} c - a b d\right )}} - \frac {\frac {2 \, \sqrt {2} c^{\frac {3}{2}} \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 {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} c^{\frac {3}{2}} \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 {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} c^{\frac {5}{4}} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{d^{\frac {1}{4}}} - \frac {\sqrt {2} c^{\frac {5}{4}} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{d^{\frac {1}{4}}}}{4 \, {\left (b c d - a d^{2}\right )}} + \frac {2 \, \sqrt {x}}{b d} \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),x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))
/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/s
qrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + sqrt(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/b^(1/4) - sqrt(2)*a^(5/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))/(b^2
*c - a*b*d) - 1/4*(2*sqrt(2)*c^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqr
t(c)*sqrt(d)))/sqrt(sqrt(c)*sqrt(d)) + 2*sqrt(2)*c^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt
(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/sqrt(sqrt(c)*sqrt(d)) + sqrt(2)*c^(5/4)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x
) + sqrt(d)*x + sqrt(c))/d^(1/4) - sqrt(2)*c^(5/4)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))
/d^(1/4))/(b*c*d - a*d^2) + 2*sqrt(x)/(b*d)

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mupad [B]  time = 1.60, size = 6428, normalized size = 13.50

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)),x)

[Out]

atan(((((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (256*x^(1/2)*(-a^5/(16*b^9*c
^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*
a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9))/(b*d))*(
-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4) - (256*x^(1
/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2
*d^2 - 64*a*b^8*c^3*d))^(1/4)*1i - (((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) +
(256*x^(1/2)*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(3
/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 +
16*a^8*b^4*c^3*d^9))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*
b^8*c^3*d))^(1/4) + (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a
^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4)*1i)/((((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^
8*d - a^8*b*c^4*d^5))/(b*d) - (256*x^(1/2)*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*
c^2*d^2 - 64*a*b^8*c^3*d))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^
5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3
+ 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4) - (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a^5/(16*b^
9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4) + (((512*(a^3*b^6*c^9
+ a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (256*x^(1/2)*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a
^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b
^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*
b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4) + (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4
*d^4))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1
/4)))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4)*2i -
2*atan(((((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (x^(1/2)*(-a^5/(16*b^9*c^
4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a
^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9)*256i)/(b*d
))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4)*1i + (2
56*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*
b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4) - (((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*
d) + (x^(1/2)*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(3
/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 +
16*a^8*b^4*c^3*d^9)*256i)/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 -
64*a*b^8*c^3*d))^(1/4)*1i - (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^
4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4))/((((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b
^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (x^(1/2)*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7
*c^2*d^2 - 64*a*b^8*c^3*d))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c
^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9)*256i)/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6
*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4)*1i + (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a
^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4)*1i + (((512*(
a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (x^(1/2)*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^
4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 +
32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9)*256i)/(b*d))*(-a^5/(16*b^9*
c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8*c^3*d))^(1/4)*1i - (256*x^(1/2)*(a^4*b
^4*c^8 + a^8*c^4*d^4))/(b*d))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*
a*b^8*c^3*d))^(1/4)*1i))*(-a^5/(16*b^9*c^4 + 16*a^4*b^5*d^4 - 64*a^3*b^6*c*d^3 + 96*a^2*b^7*c^2*d^2 - 64*a*b^8
*c^3*d))^(1/4) + atan(((((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (256*x^(1/2
)*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(3/4)*(16*a^3*
b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c
^3*d^9))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^
(1/4) - (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6
+ 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4)*1i - (((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*
c^4*d^5))/(b*d) + (256*x^(1/2)*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64
*a^3*b*c*d^8))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a
^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^
2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4) + (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b
^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4)*1i)/((((512*(a^3*b^6*c^9 + a^9*c^3
*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (256*x^(1/2)*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*
d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^
6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 -
64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4) - (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b
*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4) + (((
512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (256*x^(1/2)*(-c^5/(16*a^4*d^9 + 16*b
^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c
^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9))/(b*d))*(-c^5/(16*
a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4) + (256*x^(1/2)*(a^4*
b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64
*a^3*b*c*d^8))^(1/4)))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c
*d^8))^(1/4)*2i - 2*atan(((((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (x^(1/2)
*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(3/4)*(16*a^3*b
^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^
3*d^9)*256i)/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^
8))^(1/4)*1i + (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*
c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4) - (((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^
8*b*c^4*d^5))/(b*d) + (x^(1/2)*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64
*a^3*b*c*d^8))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a
^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9)*256i)/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a
^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4)*1i - (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-c^5/(16*a^4*d^
9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4))/((((512*(a^3*b^6*c^9 + a^
9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) - (x^(1/2)*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3
*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d
^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9)*256i)/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4
*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4)*1i + (256*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4
*d^4))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1
/4)*1i + (((512*(a^3*b^6*c^9 + a^9*c^3*d^6 - a^4*b^5*c^8*d - a^8*b*c^4*d^5))/(b*d) + (x^(1/2)*(-c^5/(16*a^4*d^
9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(3/4)*(16*a^3*b^9*c^8*d^4 - 48*a
^4*b^8*c^7*d^5 + 32*a^5*b^7*c^6*d^6 + 32*a^6*b^6*c^5*d^7 - 48*a^7*b^5*c^4*d^8 + 16*a^8*b^4*c^3*d^9)*256i)/(b*d
))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4)*1i - (2
56*x^(1/2)*(a^4*b^4*c^8 + a^8*c^4*d^4))/(b*d))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*
b^2*c^2*d^7 - 64*a^3*b*c*d^8))^(1/4)*1i))*(-c^5/(16*a^4*d^9 + 16*b^4*c^4*d^5 - 64*a*b^3*c^3*d^6 + 96*a^2*b^2*c
^2*d^7 - 64*a^3*b*c*d^8))^(1/4) + (2*x^(1/2))/(b*d)

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sympy [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),x)

[Out]

Timed out

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