Optimal. Leaf size=167 \[ \frac{b^3 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a b^2 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.113862, antiderivative size = 167, 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} \[ \frac{b^3 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a b^2 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, 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 )^3 \, dx,x,x^2\right )}{2 b^2 \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^3 b^3 x^3+3 a^2 b^4 x^4+3 a b^5 x^5+b^6 x^6\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{a^3 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{a b^2 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0154582, size = 61, normalized size = 0.37 \[ \frac{x^8 \sqrt{\left (a+b x^2\right )^2} \left (84 a^2 b x^2+35 a^3+70 a b^2 x^4+20 b^3 x^6\right )}{280 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 58, normalized size = 0.4 \begin{align*}{\frac{{x}^{8} \left ( 20\,{b}^{3}{x}^{6}+70\,{b}^{2}a{x}^{4}+84\,{a}^{2}b{x}^{2}+35\,{a}^{3} \right ) }{280\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 1.48179, size = 86, normalized size = 0.51 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{1}{4} \, a b^{2} x^{12} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{8} \, a^{3} 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{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14016, size = 90, normalized size = 0.54 \begin{align*} \frac{1}{14} \, b^{3} x^{14} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{4} \, a b^{2} x^{12} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{3}{10} \, a^{2} b x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{8} \, a^{3} 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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