Optimal. Leaf size=127 \[ \frac{x^4 \left (a+b x^2\right )}{4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a x^2 \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^2 \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.101596, antiderivative size = 127, 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} \[ \frac{x^4 \left (a+b x^2\right )}{4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a x^2 \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^2 \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx,x,x^2\right )\\ &=\frac{\left (a b+b^2 x^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+b^2 x} \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (a b+b^2 x^2\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b^3}+\frac{x}{b^2}+\frac{a^2}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{a x^2 \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^4 \left (a+b x^2\right )}{4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^2 \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0240429, size = 55, normalized size = 0.43 \[ \frac{\left (a+b x^2\right ) \left (2 a^2 \log \left (a+b x^2\right )+b x^2 \left (b x^2-2 a\right )\right )}{4 b^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.211, size = 52, normalized size = 0.4 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) \left ({b}^{2}{x}^{4}-2\,ab{x}^{2}+2\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) \right ) }{4\,{b}^{3}}{\frac{1}{\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 [A] time = 0.999292, size = 62, normalized size = 0.49 \begin{align*} \frac{x^{4}}{4 \, \sqrt{b^{2}}} - \frac{a b x^{2}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{a^{2} b^{2} \log \left (x^{2} + \frac{a}{b}\right )}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22679, size = 73, normalized size = 0.57 \begin{align*} \frac{b^{2} x^{4} - 2 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.327823, size = 32, normalized size = 0.25 \begin{align*} \frac{a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} - \frac{a x^{2}}{2 b^{2}} + \frac{x^{4}}{4 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13911, size = 80, normalized size = 0.63 \begin{align*} \frac{a^{2} \log \left ({\left | b x^{2} + a \right |}\right ) \mathrm{sgn}\left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{b x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) - 2 \, a x^{2} \mathrm{sgn}\left (b x^{2} + a\right )}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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