Optimal. Leaf size=246 \[ -\frac{2 b^{2/3} (11 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{14/3}}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{4 b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{14/3}}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.141659, 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, 290, 325, 200, 31, 634, 617, 204, 628} \[ -\frac{2 b^{2/3} (11 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{14/3}}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{4 b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{14/3}}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 290
Rule 325
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^6 \left (a+b x^3\right )^3} \, dx &=\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{(11 A b-5 a B) \int \frac{1}{x^6 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{(4 (11 A b-5 a B)) \int \frac{1}{x^6 \left (a+b x^3\right )} \, dx}{9 a^2 b}\\ &=-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}-\frac{(4 (11 A b-5 a B)) \int \frac{1}{x^3 \left (a+b x^3\right )} \, dx}{9 a^3}\\ &=-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{(4 b (11 A b-5 a B)) \int \frac{1}{a+b x^3} \, dx}{9 a^4}\\ &=-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{(4 b (11 A b-5 a B)) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{14/3}}+\frac{(4 b (11 A b-5 a B)) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{14/3}}\\ &=-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{\left (2 b^{2/3} (11 A b-5 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 a^{14/3}}+\frac{(2 b (11 A b-5 a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{13/3}}\\ &=-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{2 b^{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 )}{27 a^{14/3}}+\frac{\left (4 b^{2/3} (11 A b-5 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 a^{14/3}}\\ &=-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}-\frac{4 b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{14/3}}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{2 b^{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 )}{27 a^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.170137, size = 210, normalized size = 0.85 \[ \frac{20 b^{2/3} (5 a B-11 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{45 a^{5/3} b x (a B-A b)}{\left (a+b x^3\right )^2}-\frac{15 a^{2/3} b x (11 a B-17 A b)}{a+b x^3}-\frac{135 a^{2/3} (a B-3 A b)}{x^2}-\frac{54 a^{5/3} A}{x^5}+40 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-40 \sqrt{3} b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{270 a^{14/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 295, normalized size = 1.2 \begin{align*}{\frac{17\,{b}^{3}A{x}^{4}}{18\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}B{x}^{4}}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{10\,{b}^{2}Ax}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,bBx}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{44\,Ab}{27\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{22\,Ab}{27\,{a}^{4}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{44\,Ab\sqrt{3}}{27\,{a}^{4}}\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{20\,B}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,B}{27\,{a}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,B\sqrt{3}}{27\,{a}^{3}}\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{A}{5\,{a}^{3}{x}^{5}}}+{\frac{3\,Ab}{2\,{a}^{4}{x}^{2}}}-{\frac{B}{2\,{a}^{3}{x}^{2}}} \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 = 1.51152, size = 876, normalized size = 3.56 \begin{align*} -\frac{60 \,{\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{9} + 96 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{6} + 54 \, A a^{3} + 27 \,{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{3} + 40 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{11} + 2 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{8} +{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - 20 \,{\left ({\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{11} + 2 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{8} +{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{5}\right )} \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 ) + 40 \,{\left ({\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{11} + 2 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{8} +{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{5}\right )} \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 )}{270 \,{\left (a^{4} b^{2} x^{11} + 2 \, a^{5} b x^{8} + a^{6} x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.6843, size = 173, normalized size = 0.7 \begin{align*} \operatorname{RootSum}{\left (19683 t^{3} a^{14} - 85184 A^{3} b^{5} + 116160 A^{2} B a b^{4} - 52800 A B^{2} a^{2} b^{3} + 8000 B^{3} a^{3} b^{2}, \left ( t \mapsto t \log{\left (- \frac{27 t a^{5}}{- 44 A b^{2} + 20 B a b} + x \right )} \right )\right )} - \frac{18 A a^{3} + x^{9} \left (- 220 A b^{3} + 100 B a b^{2}\right ) + x^{6} \left (- 352 A a b^{2} + 160 B a^{2} b\right ) + x^{3} \left (- 99 A a^{2} b + 45 B a^{3}\right )}{90 a^{6} x^{5} + 180 a^{5} b x^{8} + 90 a^{4} b^{2} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12628, size = 309, normalized size = 1.26 \begin{align*} -\frac{4 \, \sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 11 \, \left (-a b^{2}\right )^{\frac{1}{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^{5}} + \frac{4 \,{\left (5 \, B a b - 11 \, A b^{2}\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^{5}} - \frac{2 \,{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 11 \, \left (-a b^{2}\right )^{\frac{1}{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 \, a^{5}} - \frac{11 \, B a b^{2} x^{4} - 17 \, A b^{3} x^{4} + 14 \, B a^{2} b x - 20 \, A a b^{2} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4}} - \frac{5 \, B a x^{3} - 15 \, A b x^{3} + 2 \, A a}{10 \, a^{4} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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