Optimal. Leaf size=255 \[ \frac{b^5 x^{29} \sqrt{a^2+2 a b x^3+b^2 x^6}}{29 \left (a+b x^3\right )}+\frac{5 a b^4 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{a^3 b^2 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{5 a^4 b x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{a^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0636303, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{b^5 x^{29} \sqrt{a^2+2 a b x^3+b^2 x^6}}{29 \left (a+b x^3\right )}+\frac{5 a b^4 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{a^3 b^2 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{5 a^4 b x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{a^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 270
Rubi steps
\begin{align*} \int x^{13} \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^{13} \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} \int \left (a^5 b^5 x^{13}+5 a^4 b^6 x^{16}+10 a^3 b^7 x^{19}+10 a^2 b^8 x^{22}+5 a b^9 x^{25}+b^{10} x^{28}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{5 a^4 b x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{a^3 b^2 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{5 a b^4 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac{b^5 x^{29} \sqrt{a^2+2 a b x^3+b^2 x^6}}{29 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0255999, size = 83, normalized size = 0.33 \[ \frac{x^{14} \sqrt{\left (a+b x^3\right )^2} \left (897260 a^2 b^3 x^9+1031849 a^3 b^2 x^6+606970 a^4 b x^3+147407 a^5+396865 a b^4 x^{12}+71162 b^5 x^{15}\right )}{2063698 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 80, normalized size = 0.3 \begin{align*}{\frac{{x}^{14} \left ( 71162\,{b}^{5}{x}^{15}+396865\,a{b}^{4}{x}^{12}+897260\,{a}^{2}{b}^{3}{x}^{9}+1031849\,{a}^{3}{b}^{2}{x}^{6}+606970\,{a}^{4}b{x}^{3}+147407\,{a}^{5} \right ) }{2063698\, \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 [A] time = 1.06016, size = 77, normalized size = 0.3 \begin{align*} \frac{1}{29} \, b^{5} x^{29} + \frac{5}{26} \, a b^{4} x^{26} + \frac{10}{23} \, a^{2} b^{3} x^{23} + \frac{1}{2} \, a^{3} b^{2} x^{20} + \frac{5}{17} \, a^{4} b x^{17} + \frac{1}{14} \, a^{5} x^{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65686, size = 144, normalized size = 0.56 \begin{align*} \frac{1}{29} \, b^{5} x^{29} + \frac{5}{26} \, a b^{4} x^{26} + \frac{10}{23} \, a^{2} b^{3} x^{23} + \frac{1}{2} \, a^{3} b^{2} x^{20} + \frac{5}{17} \, a^{4} b x^{17} + \frac{1}{14} \, a^{5} x^{14} \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^{13} \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 = 1.10066, size = 142, normalized size = 0.56 \begin{align*} \frac{1}{29} \, b^{5} x^{29} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{26} \, a b^{4} x^{26} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{23} \, a^{2} b^{3} x^{23} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{3} b^{2} x^{20} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{17} \, a^{4} b x^{17} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{14} \, a^{5} x^{14} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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