Optimal. Leaf size=196 \[ \frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}-\frac{x^2 (2 A b-5 a B)}{6 a b^2}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{8/3}}+\frac{x^5 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.110349, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {457, 321, 292, 31, 634, 617, 204, 628} \[ \frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}-\frac{x^2 (2 A b-5 a B)}{6 a b^2}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{8/3}}+\frac{x^5 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 321
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) x^5}{3 a b \left (a+b x^3\right )}+\frac{(-2 A b+5 a B) \int \frac{x^4}{a+b x^3} \, dx}{3 a b}\\ &=-\frac{(2 A b-5 a B) x^2}{6 a b^2}+\frac{(A b-a B) x^5}{3 a b \left (a+b x^3\right )}+\frac{(2 A b-5 a B) \int \frac{x}{a+b x^3} \, dx}{3 b^2}\\ &=-\frac{(2 A b-5 a B) x^2}{6 a b^2}+\frac{(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac{(2 A b-5 a B) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{7/3}}+\frac{(2 A b-5 a B) \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 \sqrt [3]{a} b^{7/3}}\\ &=-\frac{(2 A b-5 a B) x^2}{6 a b^2}+\frac{(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}+\frac{(2 A b-5 a B) \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 \sqrt [3]{a} b^{8/3}}+\frac{(2 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 b^{7/3}}\\ &=-\frac{(2 A b-5 a B) x^2}{6 a b^2}+\frac{(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}+\frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}+\frac{(2 A b-5 a B) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a} b^{8/3}}\\ &=-\frac{(2 A b-5 a B) x^2}{6 a b^2}+\frac{(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac{(2 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} \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}+\frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}\\ \end{align*}
Mathematica [A] time = 0.11884, size = 165, normalized size = 0.84 \[ \frac{\frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{6 b^{2/3} x^2 (A b-a B)}{a+b x^3}+\frac{2 (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} (5 a B-2 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+9 b^{2/3} B x^2}{18 b^{8/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 235, normalized size = 1.2 \begin{align*}{\frac{B{x}^{2}}{2\,{b}^{2}}}-{\frac{{x}^{2}A}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{B{x}^{2}a}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,Ba}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,Ba}{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\,Ba\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{2\,A}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{9\,{b}^{2}}\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{2\,A\sqrt{3}}{9\,{b}^{2}}\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 = 1.79822, size = 1314, normalized size = 6.7 \begin{align*} \left [\frac{9 \, B a b^{3} x^{5} + 3 \,{\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2} +{\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3}\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) -{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) + 2 \,{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{18 \,{\left (a b^{5} x^{3} + a^{2} b^{4}\right )}}, \frac{9 \, B a b^{3} x^{5} + 3 \,{\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{2} - 6 \, \sqrt{\frac{1}{3}}{\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2} +{\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) -{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) + 2 \,{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{18 \,{\left (a b^{5} x^{3} + a^{2} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.24371, size = 126, normalized size = 0.64 \begin{align*} \frac{B x^{2}}{2 b^{2}} + \frac{x^{2} \left (- A b + B a\right )}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a b^{8} + 8 A^{3} b^{3} - 60 A^{2} B a b^{2} + 150 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a b^{5}}{4 A^{2} b^{2} - 20 A B a b + 25 B^{2} a^{2}} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13116, size = 285, normalized size = 1.45 \begin{align*} \frac{B x^{2}}{2 \, b^{2}} + \frac{{\left (5 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, 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^{2}} + \frac{B a x^{2} - A b x^{2}}{3 \,{\left (b x^{3} + a\right )} b^{2}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 2 \, \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 \, a b^{4}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 2 \, \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 \, a b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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