3.123 \(\int \frac{1}{x^8 (a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=352 \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac{b^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3} (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}+\frac{d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{d^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{10/3} (b c-a d)}-\frac{1}{7 a c x^7} \]

[Out]

-1/(7*a*c*x^7) + (b*c + a*d)/(4*a^2*c^2*x^4) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) + (b^(10/3)*ArcTan[(a
^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(10/3)*(b*c - a*d)) - (d^(10/3)*ArcTan[(c^(1/3) - 2*d^(1/
3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(10/3)*(b*c - a*d)) + (b^(10/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)*(b*
c - a*d)) - (d^(10/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(10/3)*(b*c - a*d)) - (b^(10/3)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(6*a^(10/3)*(b*c - a*d)) + (d^(10/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6
*c^(10/3)*(b*c - a*d))

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Rubi [A]  time = 0.500987, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {480, 583, 584, 292, 31, 634, 617, 204, 628} \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac{b^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3} (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}+\frac{d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{d^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{10/3} (b c-a d)}-\frac{1}{7 a c x^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(7*a*c*x^7) + (b*c + a*d)/(4*a^2*c^2*x^4) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) + (b^(10/3)*ArcTan[(a
^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(10/3)*(b*c - a*d)) - (d^(10/3)*ArcTan[(c^(1/3) - 2*d^(1/
3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(10/3)*(b*c - a*d)) + (b^(10/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)*(b*
c - a*d)) - (d^(10/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(10/3)*(b*c - a*d)) - (b^(10/3)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(6*a^(10/3)*(b*c - a*d)) + (d^(10/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6
*c^(10/3)*(b*c - a*d))

Rule 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac{1}{7 a c x^7}+\frac{\int \frac{-7 (b c+a d)-7 b d x^3}{x^5 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{7 a c}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{\int \frac{-28 \left (b^2 c^2+a b c d+a^2 d^2\right )-28 b d (b c+a d) x^3}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{28 a^2 c^2}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{\int \frac{x \left (-28 (b c+a d) \left (b^2 c^2+a^2 d^2\right )-28 b d \left (b^2 c^2+a b c d+a^2 d^2\right ) x^3\right )}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{28 a^3 c^3}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{\int \left (-\frac{28 b^4 c^3 x}{(b c-a d) \left (a+b x^3\right )}-\frac{28 a^3 d^4 x}{(-b c+a d) \left (c+d x^3\right )}\right ) \, dx}{28 a^3 c^3}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}-\frac{b^4 \int \frac{x}{a+b x^3} \, dx}{a^3 (b c-a d)}+\frac{d^4 \int \frac{x}{c+d x^3} \, dx}{c^3 (b c-a d)}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{b^{11/3} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{10/3} (b c-a d)}-\frac{b^{11/3} \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{10/3} (b c-a d)}-\frac{d^{11/3} \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{10/3} (b c-a d)}+\frac{d^{11/3} \int \frac{\sqrt [3]{c}+\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{10/3} (b c-a d)}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{b^{10/3} \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{10/3} (b c-a d)}-\frac{b^{11/3} \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^3 (b c-a d)}+\frac{d^{10/3} \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{10/3} (b c-a d)}+\frac{d^{11/3} \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^3 (b c-a d)}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac{d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac{b^{10/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{10/3} (b c-a d)}+\frac{d^{10/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{10/3} (b c-a d)}\\ &=-\frac{1}{7 a c x^7}+\frac{b c+a d}{4 a^2 c^2 x^4}-\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{b^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3} (b c-a d)}-\frac{d^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{10/3} (b c-a d)}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac{d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.194873, size = 304, normalized size = 0.86 \[ \frac{\frac{84 b^3 x^6}{a^3}-\frac{21 b^2 x^3}{a^2}-\frac{28 b^{10/3} x^7 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{10/3}}+\frac{14 b^{10/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{10/3}}-\frac{28 \sqrt{3} b^{10/3} x^7 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{10/3}}+\frac{12 b}{a}-\frac{84 d^3 x^6}{c^3}+\frac{21 d^2 x^3}{c^2}+\frac{28 d^{10/3} x^7 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{10/3}}-\frac{14 d^{10/3} x^7 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{10/3}}+\frac{28 \sqrt{3} d^{10/3} x^7 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{10/3}}-\frac{12 d}{c}}{84 x^7 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((12*b)/a - (12*d)/c - (21*b^2*x^3)/a^2 + (21*d^2*x^3)/c^2 + (84*b^3*x^6)/a^3 - (84*d^3*x^6)/c^3 - (28*Sqrt[3]
*b^(10/3)*x^7*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(10/3) + (28*Sqrt[3]*d^(10/3)*x^7*ArcTan[(1 - (2*
d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(10/3) - (28*b^(10/3)*x^7*Log[a^(1/3) + b^(1/3)*x])/a^(10/3) + (28*d^(10/3)*x^
7*Log[c^(1/3) + d^(1/3)*x])/c^(10/3) + (14*b^(10/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(10/
3) - (14*d^(10/3)*x^7*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(10/3))/(84*(-(b*c) + a*d)*x^7)

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Maple [A]  time = 0.014, size = 334, normalized size = 1. \begin{align*}{\frac{{d}^{3}}{3\,{c}^{3} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{{d}^{3}}{6\,{c}^{3} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{{d}^{3}\sqrt{3}}{3\,{c}^{3} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{{b}^{3}}{3\,{a}^{3} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{b}^{3}}{6\,{a}^{3} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{b}^{3}\sqrt{3}}{3\,{a}^{3} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{7\,ac{x}^{7}}}+{\frac{d}{4\,a{c}^{2}{x}^{4}}}+{\frac{b}{4\,{a}^{2}c{x}^{4}}}-{\frac{{d}^{2}}{a{c}^{3}x}}-{\frac{bd}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}cx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^3+a)/(d*x^3+c),x)

[Out]

1/3*d^3/c^3/(a*d-b*c)/(1/d*c)^(1/3)*ln(x+(1/d*c)^(1/3))-1/6*d^3/c^3/(a*d-b*c)/(1/d*c)^(1/3)*ln(x^2-(1/d*c)^(1/
3)*x+(1/d*c)^(2/3))-1/3*d^3/c^3/(a*d-b*c)*3^(1/2)/(1/d*c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/d*c)^(1/3)*x-1))-1/3*
b^3/a^3/(a*d-b*c)/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6*b^3/a^3/(a*d-b*c)/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(
2/3))+1/3*b^3/a^3/(a*d-b*c)*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/7/a/c/x^7+1/4/a/c^2/
x^4*d+1/4/a^2/c/x^4*b-1/a/c^3/x*d^2-1/a^2/c^2/x*b*d-1/a^3/c/x*b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.63884, size = 759, normalized size = 2.16 \begin{align*} -\frac{28 \, \sqrt{3} b^{3} c^{3} x^{7} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 28 \, \sqrt{3} a^{3} d^{3} x^{7} \left (\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{d}{c}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) - 14 \, b^{3} c^{3} x^{7} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 14 \, a^{3} d^{3} x^{7} \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (\frac{d}{c}\right )^{\frac{2}{3}} + c \left (\frac{d}{c}\right )^{\frac{1}{3}}\right ) + 28 \, b^{3} c^{3} x^{7} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 28 \, a^{3} d^{3} x^{7} \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d}{c}\right )^{\frac{2}{3}}\right ) + 84 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{6} + 12 \, a^{2} b c^{3} - 12 \, a^{3} c^{2} d - 21 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{3}}{84 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/84*(28*sqrt(3)*b^3*c^3*x^7*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 28*sqrt(3)*a^3*d
^3*x^7*(d/c)^(1/3)*arctan(2/3*sqrt(3)*x*(d/c)^(1/3) - 1/3*sqrt(3)) - 14*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x^2 - a
*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) - 14*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x^2 - c*x*(d/c)^(2/3) + c*(d/c)^(1/3)) +
28*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) + 28*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x + c*(d/c)^(2/3)) +
84*(b^3*c^3 - a^3*d^3)*x^6 + 12*a^2*b*c^3 - 12*a^3*c^2*d - 21*(a*b^2*c^3 - a^3*c*d^2)*x^3)/((a^3*b*c^4 - a^4*c
^3*d)*x^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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Giac [A]  time = 1.13945, size = 509, normalized size = 1.45 \begin{align*} \frac{b^{4} \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{4} b c - a^{5} d\right )}} - \frac{d^{4} \left (-\frac{c}{d}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{5} - a c^{4} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{4} b c - \sqrt{3} a^{5} d} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} d^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{5} - \sqrt{3} a c^{4} d} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} b^{2} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{4} b c - a^{5} d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} d^{2} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{5} - a c^{4} d\right )}} - \frac{28 \, b^{2} c^{2} x^{6} + 28 \, a b c d x^{6} + 28 \, a^{2} d^{2} x^{6} - 7 \, a b c^{2} x^{3} - 7 \, a^{2} c d x^{3} + 4 \, a^{2} c^{2}}{28 \, a^{3} c^{3} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")

[Out]

1/3*b^4*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b*c - a^5*d) - 1/3*d^4*(-c/d)^(2/3)*log(abs(x - (-c/d)^(1
/3)))/(b*c^5 - a*c^4*d) + (-a*b^2)^(2/3)*b^2*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^
4*b*c - sqrt(3)*a^5*d) - (-c*d^2)^(2/3)*d^2*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c
^5 - sqrt(3)*a*c^4*d) - 1/6*(-a*b^2)^(2/3)*b^2*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b*c - a^5*d) + 1/
6*(-c*d^2)^(2/3)*d^2*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^5 - a*c^4*d) - 1/28*(28*b^2*c^2*x^6 + 28*a*
b*c*d*x^6 + 28*a^2*d^2*x^6 - 7*a*b*c^2*x^3 - 7*a^2*c*d*x^3 + 4*a^2*c^2)/(a^3*c^3*x^7)