Optimal. Leaf size=276 \[ \frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.133187, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {1355, 288, 199, 200, 31, 634, 617, 204, 628} \[ \frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 288
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac{x^3}{\left (a b+b^2 x^3\right )^3} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{\left (a b+b^2 x^3\right )^2} \, dx}{6 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{a b+b^2 x^3} \, dx}{9 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{27 a^{5/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{2 \sqrt [3]{a} \sqrt [3]{b}-b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{27 a^{5/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a b+b^2 x^3\right ) \int \frac{-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{54 a^{5/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{18 a^{4/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\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^{5/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}
Mathematica [A] time = 0.0639138, size = 235, normalized size = 0.85 \[ \frac{3 a^{2/3} b^{4/3} x^4-b^2 x^6 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 a b x^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-6 a^{5/3} \sqrt [3]{b} x+2 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{5/3} b^{4/3} \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 297, normalized size = 1.1 \begin{align*} -{\frac{b{x}^{3}+a}{54\,a{b}^{2}} \left ( 2\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){x}^{6}{b}^{2}-2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{6}{b}^{2}+\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){x}^{6}{b}^{2}-3\, \left ({\frac{a}{b}} \right ) ^{2/3}{x}^{4}{b}^{2}+4\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){x}^{3}ab-4\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{3}ab+2\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{3}ab+6\, \left ({\frac{a}{b}} \right ) ^{2/3}xab+2\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){a}^{2}-2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){a}^{2}+\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){a}^{2} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{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 = 1.88762, size = 1152, normalized size = 4.17 \begin{align*} \left [\frac{3 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x + 3 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) -{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{54 \,{\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}, \frac{3 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) -{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{54 \,{\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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