Optimal. Leaf size=251 \[ \frac{b^5 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
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Rubi [A] time = 0.070981, antiderivative size = 251, 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 = {1112, 266, 43} \[ \frac{b^5 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{\left (a b+b^2 x^2\right )^5}{x} \, dx}{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 \frac{\left (a b+b^2 x\right )^5}{x} \, 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 (5 a^4 b^6+\frac{a^5 b^5}{x}+10 a^3 b^7 x+10 a^2 b^8 x^2+5 a b^9 x^3+b^{10} x^4\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac{5 a^4 b x^2 \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^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{b^5 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2}\\ \end{align*}
Mathematica [A] time = 0.0249113, size = 82, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (b x^2 \left (200 a^2 b^2 x^4+300 a^3 b x^2+300 a^4+75 a b^3 x^6+12 b^4 x^8\right )+120 a^5 \log (x)\right )}{120 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.211, size = 79, normalized size = 0.3 \begin{align*}{\frac{12\,{b}^{5}{x}^{10}+75\,a{b}^{4}{x}^{8}+200\,{a}^{2}{b}^{3}{x}^{6}+300\,{b}^{2}{a}^{3}{x}^{4}+300\,{a}^{4}b{x}^{2}+120\,{a}^{5}\ln \left ( x \right ) }{120\, \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 = 1.55215, size = 130, normalized size = 0.52 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} \log \left (x\right ) \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 \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12498, size = 143, normalized size = 0.57 \begin{align*} \frac{1}{10} \, b^{5} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{8} \, a b^{4} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{3} \, a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{2} \, a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{2} \, a^{4} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{2} \, a^{5} \log \left (x^{2}\right ) \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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