Optimal. Leaf size=246 \[ -\frac{2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{14/3}}+\frac{x^8 (5 A b-11 a B)}{18 a b^2 \left (a+b x^3\right )}-\frac{4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac{2 x^2 (5 A b-11 a B)}{9 b^4}+\frac{x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.155731, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {457, 288, 302, 292, 31, 634, 617, 204, 628} \[ -\frac{2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{14/3}}+\frac{x^8 (5 A b-11 a B)}{18 a b^2 \left (a+b x^3\right )}-\frac{4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac{2 x^2 (5 A b-11 a B)}{9 b^4}+\frac{x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 288
Rule 302
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(-5 A b+11 a B) \int \frac{x^{10}}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}-\frac{(4 (5 A b-11 a B)) \int \frac{x^7}{a+b x^3} \, dx}{9 a b^2}\\ &=\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}-\frac{(4 (5 A b-11 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}{9 a b^2}\\ &=\frac{2 (5 A b-11 a B) x^2}{9 b^4}-\frac{4 (5 A b-11 a B) x^5}{45 a b^3}+\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}-\frac{(4 a (5 A b-11 a B)) \int \frac{x}{a+b x^3} \, dx}{9 b^4}\\ &=\frac{2 (5 A b-11 a B) x^2}{9 b^4}-\frac{4 (5 A b-11 a B) x^5}{45 a b^3}+\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac{\left (4 a^{2/3} (5 A b-11 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^{13/3}}-\frac{\left (4 a^{2/3} (5 A b-11 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}{27 b^{13/3}}\\ &=\frac{2 (5 A b-11 a B) x^2}{9 b^4}-\frac{4 (5 A b-11 a B) x^5}{45 a b^3}+\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac{\left (2 a^{2/3} (5 A b-11 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}{27 b^{14/3}}-\frac{(2 a (5 A b-11 a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^{13/3}}\\ &=\frac{2 (5 A b-11 a B) x^2}{9 b^4}-\frac{4 (5 A b-11 a B) x^5}{45 a b^3}+\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac{2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}-\frac{\left (4 a^{2/3} (5 A b-11 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 )}{9 b^{14/3}}\\ &=\frac{2 (5 A b-11 a B) x^2}{9 b^4}-\frac{4 (5 A b-11 a B) x^5}{45 a b^3}+\frac{(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac{(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac{4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac{2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.166405, size = 216, normalized size = 0.88 \[ \frac{\frac{45 a^2 b^{2/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+20 a^{2/3} (11 a B-5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-40 a^{2/3} (11 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-40 \sqrt{3} a^{2/3} (11 a B-5 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+135 b^{2/3} x^2 (A b-3 a B)+\frac{30 a b^{2/3} x^2 (7 A b-10 a B)}{a+b x^3}+54 b^{5/3} B x^5}{270 b^{14/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 308, normalized size = 1.3 \begin{align*}{\frac{B{x}^{5}}{5\,{b}^{3}}}+{\frac{A{x}^{2}}{2\,{b}^{3}}}-{\frac{3\,B{x}^{2}a}{2\,{b}^{4}}}+{\frac{7\,aA{x}^{5}}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{10\,{a}^{2}B{x}^{5}}{9\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{11\,{a}^{2}A{x}^{2}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{17\,B{a}^{3}{x}^{2}}{18\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{20\,aA}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{10\,aA}{27\,{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{20\,aA\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}}}}}}-{\frac{44\,{a}^{2}B}{27\,{b}^{5}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{22\,{a}^{2}B}{27\,{b}^{5}}\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{44\,{a}^{2}B\sqrt{3}}{27\,{b}^{5}}\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 = 1.4955, size = 838, normalized size = 3.41 \begin{align*} \frac{54 \, B b^{3} x^{11} - 27 \,{\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{8} - 96 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{5} - 60 \,{\left (11 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2} + 40 \, \sqrt{3}{\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\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 ) + 20 \,{\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\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 ) - 40 \,{\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\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 )}{270 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.72164, size = 189, normalized size = 0.77 \begin{align*} \frac{B x^{5}}{5 b^{3}} - \frac{x^{5} \left (- 14 A a b^{2} + 20 B a^{2} b\right ) + x^{2} \left (- 11 A a^{2} b + 17 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} b^{14} - 8000 A^{3} a^{2} b^{3} + 52800 A^{2} B a^{3} b^{2} - 116160 A B^{2} a^{4} b + 85184 B^{3} a^{5}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} b^{9}}{400 A^{2} a b^{2} - 1760 A B a^{2} b + 1936 B^{2} a^{3}} + x \right )} \right )\right )} - \frac{x^{2} \left (- A b + 3 B a\right )}{2 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12666, size = 350, normalized size = 1.42 \begin{align*} -\frac{4 \,{\left (11 \, 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 )}{27 \, a b^{4}} - \frac{4 \, \sqrt{3}{\left (11 \, \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 )}{27 \, b^{6}} + \frac{2 \,{\left (11 \, \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 )}{27 \, b^{6}} - \frac{20 \, B a^{2} b x^{5} - 14 \, A a b^{2} x^{5} + 17 \, B a^{3} x^{2} - 11 \, A a^{2} b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} b^{4}} + \frac{2 \, B b^{12} x^{5} - 15 \, B a b^{11} x^{2} + 5 \, A b^{12} x^{2}}{10 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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