Optimal. Leaf size=246 \[ \frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{13/3}}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.158178, 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, 292, 31, 634, 617, 204, 628} \[ \frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{13/3}}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 290
Rule 325
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx &=\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{(10 A b-4 a B) \int \frac{1}{x^5 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac{(7 (5 A b-2 a B)) \int \frac{1}{x^5 \left (a+b x^3\right )} \, dx}{9 a^2 b}\\ &=-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{(7 (5 A b-2 a B)) \int \frac{1}{x^2 \left (a+b x^3\right )} \, dx}{9 a^3}\\ &=-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac{(7 b (5 A b-2 a B)) \int \frac{x}{a+b x^3} \, dx}{9 a^4}\\ &=-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{\left (7 b^{2/3} (5 A b-2 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{13/3}}+\frac{\left (7 b^{2/3} (5 A b-2 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 a^{13/3}}\\ &=-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac{\left (7 \sqrt [3]{b} (5 A b-2 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}{54 a^{13/3}}+\frac{\left (7 b^{2/3} (5 A b-2 a B)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^4}\\ &=-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}+\frac{\left (7 \sqrt [3]{b} (5 A b-2 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^{13/3}}\\ &=-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{7 (5 A b-2 a B)}{9 a^4 x}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}\\ \end{align*}
Mathematica [A] time = 0.167024, size = 214, normalized size = 0.87 \[ \frac{14 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{18 a^{4/3} b x^2 (a B-A b)}{\left (a+b x^3\right )^2}-\frac{27 a^{4/3} A}{x^4}-\frac{12 \sqrt [3]{a} b x^2 (5 a B-8 A b)}{a+b x^3}-\frac{108 \sqrt [3]{a} (a B-3 A b)}{x}+28 \sqrt [3]{b} (2 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt{3} \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{108 a^{13/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 299, normalized size = 1.2 \begin{align*}{\frac{8\,{b}^{3}A{x}^{5}}{9\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,{b}^{2}B{x}^{5}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{19\,A{x}^{2}{b}^{2}}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,bB{x}^{2}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{35\,Ab}{27\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{35\,Ab}{54\,{a}^{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{35\,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 ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,B}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,B}{27\,{a}^{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{14\,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 ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{A}{4\,{a}^{3}{x}^{4}}}+3\,{\frac{Ab}{{a}^{4}x}}-{\frac{B}{{a}^{3}x}} \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 = 1.5678, size = 817, normalized size = 3.32 \begin{align*} -\frac{84 \,{\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 147 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 27 \, A a^{3} + 54 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3} + 28 \, \sqrt{3}{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 14 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 28 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{108 \,{\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.87556, size = 189, normalized size = 0.77 \begin{align*} \operatorname{RootSum}{\left (19683 t^{3} a^{13} + 42875 A^{3} b^{4} - 51450 A^{2} B a b^{3} + 20580 A B^{2} a^{2} b^{2} - 2744 B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{9}}{1225 A^{2} b^{3} - 980 A B a b^{2} + 196 B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{9 A a^{3} + x^{9} \left (- 140 A b^{3} + 56 B a b^{2}\right ) + x^{6} \left (- 245 A a b^{2} + 98 B a^{2} b\right ) + x^{3} \left (- 90 A a^{2} b + 36 B a^{3}\right )}{36 a^{6} x^{4} + 72 a^{5} b x^{7} + 36 a^{4} b^{2} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15912, size = 343, normalized size = 1.39 \begin{align*} \frac{7 \,{\left (2 \, B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, A b^{2} \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^{5}} + \frac{7 \, \sqrt{3}{\left (2 \, \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 \, a^{5} b} - \frac{7 \,{\left (2 \, \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 )}{54 \, a^{5} b} - \frac{10 \, B a b^{2} x^{5} - 16 \, A b^{3} x^{5} + 13 \, B a^{2} b x^{2} - 19 \, A a b^{2} x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4}} - \frac{4 \, B a x^{3} - 12 \, A b x^{3} + A a}{4 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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