Optimal. Leaf size=78 \[ \frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.0526029, antiderivative size = 78, 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, 43} \[ \frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac{x^5}{\left (a b+b^2 x^3\right )^5} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a b+b^2 x\right )^5} \, dx,x,x^3\right )}{3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b^6 (a+b x)^5}+\frac{1}{b^6 (a+b x)^4}\right ) \, dx,x,x^3\right )}{3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}
Mathematica [A] time = 0.016172, size = 39, normalized size = 0.5 \[ \frac{-a-4 b x^3}{36 b^2 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 32, normalized size = 0.4 \begin{align*} -{\frac{ \left ( b{x}^{3}+a \right ) \left ( 4\,b{x}^{3}+a \right ) }{36\,{b}^{2}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02845, size = 65, normalized size = 0.83 \begin{align*} -\frac{1}{9 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{a}{12 \,{\left (x^{3} + \frac{a}{b}\right )}^{4}{\left (b^{2}\right )}^{\frac{5}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68398, size = 119, normalized size = 1.53 \begin{align*} -\frac{4 \, b x^{3} + a}{36 \,{\left (b^{6} x^{12} + 4 \, a b^{5} x^{9} + 6 \, a^{2} b^{4} x^{6} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\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^{5}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{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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