Optimal. Leaf size=227 \[ \frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac{5 (4 A b-a B)}{18 a^3 b x^2}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{5 (4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt [3]{b}}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.131091, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac{5 (4 A b-a B)}{18 a^3 b x^2}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{5 (4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt [3]{b}}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2} \]
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^3 \left (a+b x^3\right )^3} \, dx &=\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{(8 A b-2 a B) \int \frac{1}{x^3 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac{(5 (4 A b-a B)) \int \frac{1}{x^3 \left (a+b x^3\right )} \, dx}{9 a^2 b}\\ &=-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac{(5 (4 A b-a B)) \int \frac{1}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac{(5 (4 A b-a B)) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}}-\frac{(5 (4 A b-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^{11/3}}\\ &=-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}-\frac{(5 (4 A b-a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3}}+\frac{(5 (4 A b-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}{54 a^{11/3} \sqrt [3]{b}}\\ &=-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{(5 (4 A b-a B)) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{11/3} \sqrt [3]{b}}\\ &=-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac{5 (4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt [3]{b}}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.156448, size = 189, normalized size = 0.83 \[ \frac{\frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{9 a^{5/3} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac{3 a^{2/3} x (5 a B-11 A b)}{a+b x^3}-\frac{27 a^{2/3} A}{x^2}+\frac{10 (a B-4 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{10 \sqrt{3} (4 A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{54 a^{11/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 277, normalized size = 1.2 \begin{align*} -{\frac{11\,A{x}^{4}{b}^{2}}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,bB{x}^{4}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,Abx}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{4\,Bx}{9\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{20\,A}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,A}{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\,A\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{5\,B}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,B}{54\,{a}^{2}b}\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}}{27\,{a}^{2}b}\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}{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 [B] time = 1.64808, size = 1785, normalized size = 7.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.01203, size = 143, normalized size = 0.63 \begin{align*} \frac{- 9 A a^{2} + x^{6} \left (- 20 A b^{2} + 5 B a b\right ) + x^{3} \left (- 32 A a b + 8 B a^{2}\right )}{18 a^{5} x^{2} + 36 a^{4} b x^{5} + 18 a^{3} b^{2} x^{8}} + \operatorname{RootSum}{\left (19683 t^{3} a^{11} b + 8000 A^{3} b^{3} - 6000 A^{2} B a b^{2} + 1500 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{27 t a^{4}}{- 20 A b + 5 B a} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14006, size = 282, normalized size = 1.24 \begin{align*} -\frac{5 \,{\left (B a - 4 \, A b\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^{4}} + \frac{5 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \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^{4} b} + \frac{5 \,{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \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 )}{54 \, a^{4} b} + \frac{5 \, B a b x^{6} - 20 \, A b^{2} x^{6} + 8 \, B a^{2} x^{3} - 32 \, A a b x^{3} - 9 \, A a^{2}}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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