Optimal. Leaf size=78 \[ \frac{\left (a+b x^3\right )^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^2}-\frac{a \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^2} \]
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Rubi [A] time = 0.056291, 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} \[ \frac{\left (a+b x^3\right )^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^2}-\frac{a \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^2} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int x^5 \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int x \left (a b+b^2 x\right )^5 \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (-\frac{a \left (a b+b^2 x\right )^5}{b}+\frac{\left (a b+b^2 x\right )^6}{b^2}\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{a \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^2}+\frac{\left (a+b x^3\right )^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^2}\\ \end{align*}
Mathematica [A] time = 0.0206595, size = 83, normalized size = 1.06 \[ \frac{x^6 \sqrt{\left (a+b x^3\right )^2} \left (84 a^2 b^3 x^9+105 a^3 b^2 x^6+70 a^4 b x^3+21 a^5+35 a b^4 x^{12}+6 b^5 x^{15}\right )}{126 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 80, normalized size = 1. \begin{align*}{\frac{{x}^{6} \left ( 6\,{b}^{5}{x}^{15}+35\,a{b}^{4}{x}^{12}+84\,{a}^{2}{b}^{3}{x}^{9}+105\,{a}^{3}{b}^{2}{x}^{6}+70\,{a}^{4}b{x}^{3}+21\,{a}^{5} \right ) }{126\, \left ( b{x}^{3}+a \right ) ^{5}} \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 [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 = 1.7183, size = 136, normalized size = 1.74 \begin{align*} \frac{1}{21} \, b^{5} x^{21} + \frac{5}{18} \, a b^{4} x^{18} + \frac{2}{3} \, a^{2} b^{3} x^{15} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{6} \, a^{5} x^{6} \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 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 [A] time = 1.11771, size = 90, normalized size = 1.15 \begin{align*} \frac{1}{126} \,{\left (6 \, b^{5} x^{21} + 35 \, a b^{4} x^{18} + 84 \, a^{2} b^{3} x^{15} + 105 \, a^{3} b^{2} x^{12} + 70 \, a^{4} b x^{9} + 21 \, a^{5} x^{6}\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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