Optimal. Leaf size=215 \[ -\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^2 (5 A b-8 a B)}{6 b^3}+\frac{x^8 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.139865, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {457, 302, 292, 31, 634, 617, 204, 628} \[ -\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^2 (5 A b-8 a B)}{6 b^3}+\frac{x^8 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 302
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{(-5 A b+8 a B) \int \frac{x^7}{a+b x^3} \, dx}{3 a b}\\ &=\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{(-5 A b+8 a B) \int \left (-\frac{a x}{b^2}+\frac{x^4}{b}+\frac{a^2 x}{b^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}-\frac{(a (5 A b-8 a B)) \int \frac{x}{a+b x^3} \, dx}{3 b^3}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^{10/3}}-\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \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}{9 b^{10/3}}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}-\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \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}{18 b^{11/3}}-\frac{(a (5 A b-8 a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{10/3}}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}-\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}-\frac{\left (a^{2/3} (5 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 b^{11/3}}\\ &=\frac{(5 A b-8 a B) x^2}{6 b^3}-\frac{(5 A b-8 a B) x^5}{15 a b^2}+\frac{(A b-a B) x^8}{3 a b \left (a+b x^3\right )}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}-\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}\\ \end{align*}
Mathematica [A] time = 0.124517, size = 185, normalized size = 0.86 \[ \frac{5 a^{2/3} (8 a B-5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a^{2/3} (8 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} a^{2/3} (8 a B-5 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+45 b^{2/3} x^2 (A b-2 a B)+\frac{30 a b^{2/3} x^2 (A b-a B)}{a+b x^3}+18 b^{5/3} B x^5}{90 b^{11/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 266, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,{b}^{2}}}+{\frac{A{x}^{2}}{2\,{b}^{2}}}-{\frac{B{x}^{2}a}{{b}^{3}}}+{\frac{aA{x}^{2}}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{B{x}^{2}{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,aA}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,aA}{18\,{b}^{3}}\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{5\,aA\sqrt{3}}{9\,{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{8\,{a}^{2}B}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{4\,{a}^{2}B}{9\,{b}^{4}}\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{8\,{a}^{2}B\sqrt{3}}{9\,{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}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.65707, size = 599, normalized size = 2.79 \begin{align*} \frac{18 \, B b^{2} x^{8} - 9 \,{\left (8 \, B a b - 5 \, A b^{2}\right )} x^{5} - 15 \,{\left (8 \, B a^{2} - 5 \, A a b\right )} x^{2} + 10 \, \sqrt{3}{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + 5 \,{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \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 ) - 10 \,{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \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 )}{90 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.51983, size = 151, normalized size = 0.7 \begin{align*} \frac{B x^{5}}{5 b^{2}} - \frac{x^{2} \left (- A a b + B a^{2}\right )}{3 a b^{3} + 3 b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{11} - 125 A^{3} a^{2} b^{3} + 600 A^{2} B a^{3} b^{2} - 960 A B^{2} a^{4} b + 512 B^{3} a^{5}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} b^{7}}{25 A^{2} a b^{2} - 80 A B a^{2} b + 64 B^{2} a^{3}} + x \right )} \right )\right )} - \frac{x^{2} \left (- A b + 2 B a\right )}{2 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10413, size = 319, normalized size = 1.48 \begin{align*} -\frac{{\left (8 \, B a^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, A a b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{3}} - \frac{\sqrt{3}{\left (8 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \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 )}{9 \, b^{5}} - \frac{B a^{2} x^{2} - A a b x^{2}}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (8 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{5}} + \frac{2 \, B b^{8} x^{5} - 10 \, B a b^{7} x^{2} + 5 \, A b^{8} x^{2}}{10 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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