Optimal. Leaf size=36 \[ \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1107, 609} \[ \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
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Rule 609
Rule 1107
Rubi steps
\begin {align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.75 \[ \frac {\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 57, normalized size = 1.58 \[ \frac {1}{12} \, b^{5} x^{12} + \frac {1}{2} \, a b^{4} x^{10} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {5}{4} \, a^{4} b x^{4} + \frac {1}{2} \, a^{5} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 66, normalized size = 1.83 \[ \frac {1}{12} \, {\left (3 \, {\left (b x^{4} + 2 \, a x^{2}\right )} a^{4} + 3 \, {\left (b x^{4} + 2 \, a x^{2}\right )}^{2} a^{2} b + {\left (b x^{4} + 2 \, a x^{2}\right )}^{3} b^{2}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 79, normalized size = 2.19 \[ \frac {\left (b^{5} x^{10}+6 a \,b^{4} x^{8}+15 a^{2} b^{3} x^{6}+20 a^{3} b^{2} x^{4}+15 a^{4} b \,x^{2}+6 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{2}}{12 \left (b \,x^{2}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 57, normalized size = 1.58 \[ \frac {1}{12} \, b^{5} x^{12} + \frac {1}{2} \, a b^{4} x^{10} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {5}{4} \, a^{4} b x^{4} + \frac {1}{2} \, a^{5} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 36, normalized size = 1.00 \[ \frac {\left (b^2\,x^2+a\,b\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}}{12\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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