Optimal. Leaf size=51 \[ \frac {2 a^2 (d x)^{5/2}}{5 d}+\frac {4 a b (d x)^{9/2}}{9 d^3}+\frac {2 b^2 (d x)^{13/2}}{13 d^5} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14} \[ \frac {2 a^2 (d x)^{5/2}}{5 d}+\frac {4 a b (d x)^{9/2}}{9 d^3}+\frac {2 b^2 (d x)^{13/2}}{13 d^5} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 (d x)^{3/2}+\frac {2 a b (d x)^{7/2}}{d^2}+\frac {b^2 (d x)^{11/2}}{d^4}\right ) \, dx\\ &=\frac {2 a^2 (d x)^{5/2}}{5 d}+\frac {4 a b (d x)^{9/2}}{9 d^3}+\frac {2 b^2 (d x)^{13/2}}{13 d^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.65 \[ \frac {2}{585} x (d x)^{3/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 34, normalized size = 0.67 \[ \frac {2}{585} \, {\left (45 \, b^{2} d x^{6} + 130 \, a b d x^{4} + 117 \, a^{2} d x^{2}\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 42, normalized size = 0.82 \[ \frac {2}{585} \, {\left (45 \, \sqrt {d x} b^{2} x^{6} + 130 \, \sqrt {d x} a b x^{4} + 117 \, \sqrt {d x} a^{2} x^{2}\right )} d \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.59 \[ \frac {2 \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right ) \left (d x \right )^{\frac {3}{2}} x}{585} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 41, normalized size = 0.80 \[ \frac {2 \, {\left (45 \, \left (d x\right )^{\frac {13}{2}} b^{2} + 130 \, \left (d x\right )^{\frac {9}{2}} a b d^{2} + 117 \, \left (d x\right )^{\frac {5}{2}} a^{2} d^{4}\right )}}{585 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 41, normalized size = 0.80 \[ \frac {90\,b^2\,{\left (d\,x\right )}^{13/2}+234\,a^2\,d^4\,{\left (d\,x\right )}^{5/2}+260\,a\,b\,d^2\,{\left (d\,x\right )}^{9/2}}{585\,d^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 49, normalized size = 0.96 \[ \frac {2 a^{2} d^{\frac {3}{2}} x^{\frac {5}{2}}}{5} + \frac {4 a b d^{\frac {3}{2}} x^{\frac {9}{2}}}{9} + \frac {2 b^{2} d^{\frac {3}{2}} x^{\frac {13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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