Optimal. Leaf size=122 \[ -\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left (a+b x^3\right )}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.0505751, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 44} \[ -\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left (a+b x^3\right )}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{x^4 \left (a b+b^2 x^3\right )} \, dx}{\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}{x^2 \left (a b+b^2 x\right )} \, dx,x,x^3\right )}{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 \left (\frac{1}{a b x^2}-\frac{1}{a^2 x}+\frac{b}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )}{3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \left (a+b x^3\right ) \log (x)}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}
Mathematica [A] time = 0.0157451, size = 54, normalized size = 0.44 \[ -\frac{\left (a+b x^3\right ) \left (-b x^3 \log \left (a+b x^3\right )+a+3 b x^3 \log (x)\right )}{3 a^2 x^3 \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 51, normalized size = 0.4 \begin{align*} -{\frac{ \left ( b{x}^{3}+a \right ) \left ( 3\,b\ln \left ( x \right ){x}^{3}-b\ln \left ( b{x}^{3}+a \right ){x}^{3}+a \right ) }{3\,{a}^{2}{x}^{3}}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{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 = 1.73329, size = 80, normalized size = 0.66 \begin{align*} \frac{b x^{3} \log \left (b x^{3} + a\right ) - 3 \, b x^{3} \log \left (x\right ) - a}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.531288, size = 31, normalized size = 0.25 \begin{align*} - \frac{1}{3 a x^{3}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1288, size = 68, normalized size = 0.56 \begin{align*} \frac{1}{3} \,{\left (\frac{b \log \left ({\left | b x^{3} + a \right |}\right )}{a^{2}} - \frac{3 \, b \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{3} - a}{a^{2} x^{3}}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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