Optimal. Leaf size=42 \[ \frac{2 x^{3/2} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{7}{4};\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a} \]
[Out]
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Rubi [A] time = 0.0358842, 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 \[ \frac{2}{3} x^{3/2} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(a + b*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 7.67244, size = 41, normalized size = 0.98 \[ \frac{2 x^{\frac{3}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(b*x**2+a)**p,x)
[Out]
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Mathematica [A] time = 0.0211915, 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.
[In] Integrate[Sqrt[x]*(a + b*x^2)^p,x]
[Out]
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Maple [F] time = 0.027, size = 0, normalized size = 0. \[ \int \sqrt{x} \left ( b{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(b*x^2+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} \sqrt{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} \sqrt{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(b*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} \sqrt{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p*sqrt(x),x, algorithm="giac")
[Out]