Optimal. Leaf size=42 \[ \frac {2 x^{3/2} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+\frac {7}{4};\frac {7}{4};-\frac {b x^2}{a}\right )}{3 a} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.21, 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} \[ \frac {2}{3} x^{3/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-\frac {b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \sqrt {x} \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {2}{3} x^{3/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-\frac {b x^2}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.21 \[ \frac {2}{3} x^{3/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-\frac {b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{p} \sqrt {x}, x\right ) \]
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 \[ \int {\left (b x^{2} + a\right )}^{p} \sqrt {x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \sqrt {x}\, \left (b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (b x^{2} + a\right )}^{p} \sqrt {x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {x}\,{\left (b\,x^2+a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 57.09, size = 37, normalized size = 0.88 \[ \frac {a^{p} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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