Optimal. Leaf size=36 \[ \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b} \]
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Rubi [A]
time = 0.02, 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 = {1121, 623}
\begin {gather*} \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 1121
Rubi steps
\begin {align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, 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 )^{3/2}}{8 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.75 \begin {gather*} \frac {\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^{3/2}}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 24, normalized size = 0.67
method | result | size |
default | \(\frac {\left (b \,x^{2}+a \right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{8 b}\) | \(24\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b \,x^{2}+a \right )^{3}}{8 b}\) | \(26\) |
gosper | \(\frac {x^{2} \left (b^{3} x^{6}+4 a \,b^{2} x^{4}+6 a^{2} b \,x^{2}+4 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{8 \left (b \,x^{2}+a \right )^{3}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 35, normalized size = 0.97 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, a^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 35, normalized size = 0.97 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, a^{3} x^{2} \end {gather*}
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 \begin {gather*} \int x \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.65, size = 44, normalized size = 1.22 \begin {gather*} \frac {1}{8} \, {\left (2 \, {\left (b x^{4} + 2 \, a x^{2}\right )} a^{2} + {\left (b x^{4} + 2 \, a x^{2}\right )}^{2} b\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.25, size = 36, normalized size = 1.00 \begin {gather*} \frac {\left (b^2\,x^2+a\,b\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{8\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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