Optimal. Leaf size=255 \[ \frac{b^5 x^{24} \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 \left (a+b x^2\right )}+\frac{5 a b^4 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{5 a^4 b x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{a^5 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.161685, antiderivative size = 255, 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 = {1111, 646, 43} \[ \frac{b^5 x^{24} \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 \left (a+b x^2\right )}+\frac{5 a b^4 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{5 a^4 b x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{a^5 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^{13} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^6 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int x^6 \left (a b+b^2 x\right )^5 \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \left (a^5 b^5 x^6+5 a^4 b^6 x^7+10 a^3 b^7 x^8+10 a^2 b^8 x^9+5 a b^9 x^{10}+b^{10} x^{11}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac{a^5 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{5 a^4 b x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{5 a b^4 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac{b^5 x^{24} \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0246346, size = 83, normalized size = 0.33 \[ \frac{x^{14} \sqrt{\left (a+b x^2\right )^2} \left (5544 a^2 b^3 x^6+6160 a^3 b^2 x^4+3465 a^4 b x^2+792 a^5+2520 a b^4 x^8+462 b^5 x^{10}\right )}{11088 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 80, normalized size = 0.3 \begin{align*}{\frac{{x}^{14} \left ( 462\,{b}^{5}{x}^{10}+2520\,a{b}^{4}{x}^{8}+5544\,{a}^{2}{b}^{3}{x}^{6}+6160\,{b}^{2}{a}^{3}{x}^{4}+3465\,{a}^{4}b{x}^{2}+792\,{a}^{5} \right ) }{11088\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+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 = 1.42539, size = 142, normalized size = 0.56 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{22} \, a b^{4} x^{22} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{5}{16} \, a^{4} b x^{16} + \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^{2}\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 = 1.11661, size = 142, normalized size = 0.56 \begin{align*} \frac{1}{24} \, b^{5} x^{24} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{22} \, a b^{4} x^{22} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{2} \, a^{2} b^{3} x^{20} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{9} \, a^{3} b^{2} x^{18} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{16} \, a^{4} b x^{16} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{14} \, a^{5} x^{14} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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