Optimal. Leaf size=321 \[ -\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5} \]
[Out]
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Rubi [A] time = 1.10016, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 157.701, size = 289, normalized size = 0.9 \[ \frac{d^{\frac{8}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{8}{3}} \left (a d - b c\right )} - \frac{d^{\frac{8}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{8}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} d^{\frac{8}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{8}{3}} \left (a d - b c\right )} - \frac{1}{5 a c x^{5}} + \frac{a d + b c}{2 a^{2} c^{2} x^{2}} - \frac{b^{\frac{8}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{8}{3}} \left (a d - b c\right )} + \frac{b^{\frac{8}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{8}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} b^{\frac{8}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(b*x**3+a)/(d*x**3+c),x)
[Out]
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Mathematica [A] time = 0.417279, size = 282, normalized size = 0.88 \[ \frac{-\frac{10 b^{8/3} x^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}+\frac{10 \sqrt{3} b^{8/3} x^5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{8/3}}+\frac{5 b^{8/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}-\frac{15 b^2 x^3}{a^2}+\frac{6 b}{a}+\frac{10 d^{8/3} x^5 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{8/3}}-\frac{10 \sqrt{3} d^{8/3} x^5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{8/3}}-\frac{5 d^{8/3} x^5 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{8/3}}+\frac{15 d^2 x^3}{c^2}-\frac{6 d}{c}}{30 x^5 (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]
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Maple [A] time = 0.017, size = 293, normalized size = 0.9 \[ -{\frac{1}{5\,ac{x}^{5}}}+{\frac{d}{2\,a{c}^{2}{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}}{6\,{a}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{d}^{2}}{3\,{c}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{d}^{2}}{6\,{c}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{d}^{2}\sqrt{3}}{3\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(b*x^3+a)/(d*x^3+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.35578, size = 525, normalized size = 1.64 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3} b^{2} c^{2} x^{5} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + 5 \, \sqrt{3} a^{2} d^{2} x^{5} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3} b^{2} c^{2} x^{5} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3} a^{2} d^{2} x^{5} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 30 \, b^{2} c^{2} x^{5} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 30 \, a^{2} d^{2} x^{5} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (2 \, a b c^{2} - 2 \, a^{2} c d - 5 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}\right )}\right )}}{90 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^6),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(b*x**3+a)/(d*x**3+c),x)
[Out]
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GIAC/XCAS [A] time = 0.227652, size = 454, normalized size = 1.41 \[ -\frac{b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{3} b c - \sqrt{3} a^{4} d} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{4} - \sqrt{3} a c^{3} d} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b^{2}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{3} b c - a^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d^{2}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{5 \, b c x^{3} + 5 \, a d x^{3} - 2 \, a c}{10 \, a^{2} c^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^6),x, algorithm="giac")
[Out]