Optimal. Leaf size=160 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]
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Rubi [A] time = 0.118328, antiderivative size = 160, 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} \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 \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^3 \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 (-\frac{a^3 \left (a b+b^2 x\right )^5}{b^3}+\frac{3 a^2 \left (a b+b^2 x\right )^6}{b^4}-\frac{3 a \left (a b+b^2 x\right )^7}{b^5}+\frac{\left (a b+b^2 x\right )^8}{b^6}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac{a^3 \left (a+b x^2\right )^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 b^4}+\frac{3 a^2 \left (a+b x^2\right )^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 b^4}-\frac{3 a \left (a+b x^2\right )^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 b^4}+\frac{\left (a+b x^2\right )^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{18 b^4}\\ \end{align*}
Mathematica [A] time = 0.0215497, size = 83, normalized size = 0.52 \[ \frac{x^8 \sqrt{\left (a+b x^2\right )^2} \left (720 a^2 b^3 x^6+840 a^3 b^2 x^4+504 a^4 b x^2+126 a^5+315 a b^4 x^8+56 b^5 x^{10}\right )}{1008 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.162, size = 80, normalized size = 0.5 \begin{align*}{\frac{{x}^{8} \left ( 56\,{b}^{5}{x}^{10}+315\,a{b}^{4}{x}^{8}+720\,{a}^{2}{b}^{3}{x}^{6}+840\,{b}^{2}{a}^{3}{x}^{4}+504\,{a}^{4}b{x}^{2}+126\,{a}^{5} \right ) }{1008\, \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 = 1.47461, size = 138, normalized size = 0.86 \begin{align*} \frac{1}{18} \, b^{5} x^{18} + \frac{5}{16} \, a b^{4} x^{16} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{8} \, a^{5} x^{8} \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^{7} \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 = 1.13432, size = 142, normalized size = 0.89 \begin{align*} \frac{1}{18} \, b^{5} x^{18} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{16} \, a b^{4} x^{16} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{7} \, a^{2} b^{3} x^{14} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{6} \, a^{3} b^{2} x^{12} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{2} \, a^{4} b x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{8} \, a^{5} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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