Optimal. Leaf size=159 \[ \frac{b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}+\frac{3 a b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac{a^2 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac{a^3 x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.0332733, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1088, 194} \[ \frac{b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}+\frac{3 a b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac{a^2 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac{a^3 x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 1088
Rule 194
Rubi steps
\begin{align*} \int \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \int \left (2 a b+2 b^2 x^2\right )^3 \, dx}{\left (2 a b+2 b^2 x^2\right )^3}\\ &=\frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \int \left (8 a^3 b^3+24 a^2 b^4 x^2+24 a b^5 x^4+8 b^6 x^6\right ) \, dx}{\left (2 a b+2 b^2 x^2\right )^3}\\ &=\frac{a^3 x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac{a^2 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac{3 a b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac{b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0122895, size = 59, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (35 a^2 b x^3+35 a^3 x+21 a b^2 x^5+5 b^3 x^7\right )}{35 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 56, normalized size = 0.4 \begin{align*}{\frac{x \left ( 5\,{b}^{3}{x}^{6}+21\,a{x}^{4}{b}^{2}+35\,{a}^{2}b{x}^{2}+35\,{a}^{3} \right ) }{35\, \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 [A] time = 1.00336, size = 42, normalized size = 0.26 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{3}{5} \, a b^{2} x^{5} + a^{2} b x^{3} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4736, size = 66, normalized size = 0.42 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{3}{5} \, a b^{2} x^{5} + a^{2} b x^{3} + a^{3} x \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 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09579, size = 85, normalized size = 0.53 \begin{align*} \frac{1}{7} \, b^{3} x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{3}{5} \, a b^{2} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + a^{2} b x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + a^{3} x \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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