Optimal. Leaf size=67 \[ \frac {2 (d x)^{5/2} \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {5}{4},-2 p;\frac {9}{4};-\frac {b x^2}{a}\right )}{5 d} \]
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Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1113, 364} \[ \frac {2 (d x)^{5/2} \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {5}{4},-2 p;\frac {9}{4};-\frac {b x^2}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1113
Rubi steps
\begin {align*} \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx &=\left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \int (d x)^{3/2} \left (1+\frac {b x^2}{a}\right )^{2 p} \, dx\\ &=\frac {2 (d x)^{5/2} \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {5}{4},-2 p;\frac {9}{4};-\frac {b x^2}{a}\right )}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 0.84 \[ \frac {2}{5} x (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac {5}{4},-2 p;\frac {9}{4};-\frac {b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d x} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} d x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{\frac {3}{2}} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{\frac {3}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{\frac {3}{2}} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \,d x \]
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 \[ \int \left (d x\right )^{\frac {3}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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