Optimal. Leaf size=79 \[ \frac{b x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{a x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0643209, antiderivative size = 79, 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} \[ \frac{b x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{a x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right ) \, dx,x,x^2\right )}{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 b x^2+b^2 x^3\right ) \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac{a x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac{b x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0142518, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (4 a x^6+3 b x^8\right )}{24 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.172, size = 36, normalized size = 0.5 \begin{align*}{\frac{{x}^{6} \left ( 3\,b{x}^{2}+4\,a \right ) }{24\,b{x}^{2}+24\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{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 = 1.68822, size = 31, normalized size = 0.39 \begin{align*} \frac{1}{8} \, b x^{8} + \frac{1}{6} \, a x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.093207, size = 12, normalized size = 0.15 \begin{align*} \frac{a x^{6}}{6} + \frac{b x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14404, size = 39, normalized size = 0.49 \begin{align*} \frac{1}{8} \, b x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{6} \, a x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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