Optimal. Leaf size=40 \[ -\frac{2 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{3}{4};\frac{3}{4};-\frac{b x^2}{a}\right )}{a \sqrt{x}} \]
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Rubi [A] time = 0.0131761, antiderivative size = 49, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac{2 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{4},-p;\frac{3}{4};-\frac{b x^2}{a}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^p}{x^{3/2}} \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b x^2}{a}\right )^p}{x^{3/2}} \, dx\\ &=-\frac{2 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (-\frac{1}{4},-p;\frac{3}{4};-\frac{b x^2}{a}\right )}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0071059, size = 49, normalized size = 1.22 \[ -\frac{2 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{4},-p;\frac{3}{4};-\frac{b x^2}{a}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{p}{x}^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{p}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{p}}{x^{\frac{3}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{p}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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