Optimal. Leaf size=215 \[ -\frac{b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}+\frac{8 A b-5 a B}{6 a^3 x^2}-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.127009, 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, 325, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}+\frac{8 A b-5 a B}{6 a^3 x^2}-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
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 )^2} \, dx &=\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac{(8 A b-5 a B) \int \frac{1}{x^6 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}-\frac{(8 A b-5 a B) \int \frac{1}{x^3 \left (a+b x^3\right )} \, dx}{3 a^2}\\ &=-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{8 A b-5 a B}{6 a^3 x^2}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac{(b (8 A b-5 a B)) \int \frac{1}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{8 A b-5 a B}{6 a^3 x^2}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac{(b (8 A b-5 a B)) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{11/3}}+\frac{(b (8 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}{9 a^{11/3}}\\ &=-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{8 A b-5 a B}{6 a^3 x^2}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{\left (b^{2/3} (8 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}{18 a^{11/3}}+\frac{(b (8 A b-5 a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{10/3}}\\ &=-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{8 A b-5 a B}{6 a^3 x^2}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{b^{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 )}{18 a^{11/3}}+\frac{\left (b^{2/3} (8 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 )}{3 a^{11/3}}\\ &=-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{8 A b-5 a B}{6 a^3 x^2}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )}-\frac{b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{b^{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 )}{18 a^{11/3}}\\ \end{align*}
Mathematica [A] time = 0.142564, size = 183, normalized size = 0.85 \[ \frac{5 b^{2/3} (5 a B-8 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{30 a^{2/3} b x (a B-A b)}{a+b x^3}-\frac{45 a^{2/3} (a B-2 A b)}{x^2}-\frac{18 a^{5/3} A}{x^5}+10 b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{90 a^{11/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 252, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}Ax}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{bBx}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{8\,Ab}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,Ab}{9\,{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{8\,Ab\sqrt{3}}{9\,{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{5\,B}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,B}{18\,{a}^{2}}\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{5\,B\sqrt{3}}{9\,{a}^{2}}\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}^{2}{x}^{5}}}+{\frac{Ab}{{a}^{3}{x}^{2}}}-{\frac{B}{2\,{a}^{2}{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 = 1.65282, size = 633, normalized size = 2.94 \begin{align*} -\frac{15 \,{\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 9 \,{\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 18 \, A a^{2} + 10 \, \sqrt{3}{\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} +{\left (5 \, B a^{2} - 8 \, A a 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 ) - 5 \,{\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} +{\left (5 \, B a^{2} - 8 \, A a 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 ) + 10 \,{\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} +{\left (5 \, B a^{2} - 8 \, A a 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 )}{90 \,{\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.60649, size = 138, normalized size = 0.64 \begin{align*} \operatorname{RootSum}{\left (729 t^{3} a^{11} - 512 A^{3} b^{5} + 960 A^{2} B a b^{4} - 600 A B^{2} a^{2} b^{3} + 125 B^{3} a^{3} b^{2}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{4}}{- 8 A b^{2} + 5 B a b} + x \right )} \right )\right )} - \frac{6 A a^{2} + x^{6} \left (- 40 A b^{2} + 25 B a b\right ) + x^{3} \left (- 24 A a b + 15 B a^{2}\right )}{30 a^{4} x^{5} + 30 a^{3} b x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14912, size = 278, normalized size = 1.29 \begin{align*} -\frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 8 \, \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 )}{9 \, a^{4}} + \frac{{\left (5 \, B a b - 8 \, 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 )}{9 \, a^{4}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 8 \, \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 )}{18 \, a^{4}} - \frac{B a b x - A b^{2} x}{3 \,{\left (b x^{3} + a\right )} a^{3}} - \frac{5 \, B a x^{3} - 10 \, A b x^{3} + 2 \, A a}{10 \, a^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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