Optimal. Leaf size=301 \[ \frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d} \]
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Rubi [A] time = 0.728367, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d} \]
Antiderivative was successfully verified.
[In] Int[x^7/((a + b*x^3)*(c + d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 114.541, size = 269, normalized size = 0.89 \[ \frac{a^{\frac{5}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{5}{3}} \left (a d - b c\right )} - \frac{a^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{5}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} a^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}} \left (a d - b c\right )} - \frac{c^{\frac{5}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 d^{\frac{5}{3}} \left (a d - b c\right )} + \frac{c^{\frac{5}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{5}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} c^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 d^{\frac{5}{3}} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7/(b*x**3+a)/(d*x**3+c),x)
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Mathematica [A] time = 0.278244, size = 242, normalized size = 0.8 \[ \frac{\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-\frac{2 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}-\frac{2 \sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{5/3}}-\frac{3 a x^2}{b}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{5/3}}+\frac{2 c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{5/3}}+\frac{2 \sqrt{3} c^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{5/3}}+\frac{3 c x^2}{d}}{6 b c-6 a d} \]
Antiderivative was successfully verified.
[In] Integrate[x^7/((a + b*x^3)*(c + d*x^3)),x]
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Maple [A] time = 0.011, size = 269, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,bd}}+{\frac{{a}^{2}}{ \left ( 3\,ad-3\,bc \right ){b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}}{ \left ( 6\,ad-6\,bc \right ){b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ){b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}}{ \left ( 3\,ad-3\,bc \right ){d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{c}^{2}}{ \left ( 6\,ad-6\,bc \right ){d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{c}^{2}\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ){d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7/(b*x^3+a)/(d*x^3+c),x)
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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(x^7/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.486651, size = 412, normalized size = 1.37 \[ \frac{\sqrt{3}{\left (\sqrt{3} a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + \sqrt{3} b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x^{2} - d x \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}} - c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x + d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}\right ) + 3 \, \sqrt{3}{\left (b c - a d\right )} x^{2} - 6 \, a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) - 6 \, b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} c x - \sqrt{3} d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}}{3 \, d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \,{\left (b^{2} c d - a b d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/((b*x^3 + a)*(d*x^3 + c)),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(x**7/(b*x**3+a)/(d*x**3+c),x)
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GIAC/XCAS [A] time = 0.228266, size = 436, normalized size = 1.45 \[ \frac{\left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b^{5} c - a^{2} b^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} c^{2} d{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{2} d^{4} - a c d^{5}\right )}} - \frac{a^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{2} c - a^{2} b d\right )}} + \frac{c^{2} \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} d - a c d^{2}\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} a \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{4} c - \sqrt{3} a b^{3} d} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} c \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{3} - \sqrt{3} a d^{4}} + \frac{x^{2}}{2 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")
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