Optimal. Leaf size=222 \[ \frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.39914, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^7*(A + B*x^3))/(a + b*x^3)^3,x]
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Rubi in Sympy [A] time = 47.4581, size = 209, normalized size = 0.94 \[ \frac{x^{8} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{5} \left (A b - 4 B a\right )}{9 a b^{2} \left (a + b x^{3}\right )} - \frac{5 x^{2} \left (A b - 4 B a\right )}{18 a b^{3}} - \frac{5 \left (A b - 4 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 \sqrt [3]{a} b^{\frac{11}{3}}} + \frac{5 \left (A b - 4 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 \sqrt [3]{a} b^{\frac{11}{3}}} - \frac{5 \sqrt{3} \left (A b - 4 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 \sqrt [3]{a} b^{\frac{11}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7*(B*x**3+A)/(b*x**3+a)**3,x)
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Mathematica [A] time = 0.342959, size = 194, normalized size = 0.87 \[ \frac{\frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{6 b^{2/3} x^2 (4 A b-7 a B)}{a+b x^3}+\frac{9 a b^{2/3} x^2 (A b-a B)}{\left (a+b x^3\right )^2}+\frac{10 (4 a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{10 \sqrt{3} (4 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+27 b^{2/3} B x^2}{54 b^{11/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^3,x]
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Maple [A] time = 0.014, size = 275, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{3}}}-{\frac{4\,A{x}^{5}}{9\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{7\,B{x}^{5}a}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,aA{x}^{2}}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{11\,B{x}^{2}{a}^{2}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,A}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,A}{54\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,A\sqrt{3}}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{20\,Ba}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{10\,Ba}{27\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{20\,Ba\sqrt{3}}{27\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7*(B*x^3+A)/(b*x^3+a)^3,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((B*x^3 + A)*x^7/(b*x^3 + a)^3,x, algorithm="maxima")
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Fricas [A] time = 0.234363, size = 458, normalized size = 2.06 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \,{\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 10 \, \sqrt{3}{\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \,{\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 30 \,{\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \,{\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (9 \, B b^{2} x^{8} + 8 \,{\left (4 \, B a b - A b^{2}\right )} x^{5} + 5 \,{\left (4 \, B a^{2} - A a b\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^7/(b*x^3 + a)^3,x, algorithm="fricas")
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Sympy [A] time = 11.4758, size = 162, normalized size = 0.73 \[ \frac{B x^{2}}{2 b^{3}} + \frac{x^{5} \left (- 8 A b^{2} + 14 B a b\right ) + x^{2} \left (- 5 A a b + 11 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a b^{11} + 125 A^{3} b^{3} - 1500 A^{2} B a b^{2} + 6000 A B^{2} a^{2} b - 8000 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a b^{7}}{25 A^{2} b^{2} - 200 A B a b + 400 B^{2} a^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7*(B*x**3+A)/(b*x**3+a)**3,x)
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GIAC/XCAS [A] time = 0.222259, size = 313, normalized size = 1.41 \[ \frac{B x^{2}}{2 \, b^{3}} + \frac{5 \,{\left (4 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{3}} + \frac{5 \, \sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{5}} + \frac{14 \, B a b x^{5} - 8 \, A b^{2} x^{5} + 11 \, B a^{2} x^{2} - 5 \, A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} b^{3}} - \frac{5 \,{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^7/(b*x^3 + a)^3,x, algorithm="giac")
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