3.1054 \(\int \frac{\left (a+b x^2\right )^p}{\sqrt{x}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \sqrt{x} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{4};\frac{5}{4};-\frac{b x^2}{a}\right )}{a} \]

[Out]

(2*Sqrt[x]*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 5/4 + p, 5/4, -((b*x^2)/a)])
/a

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Rubi [A]  time = 0.0359386, 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 \[ 2 \sqrt{x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^p/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(a + b*x^2)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^2)/a)])/(1 + (b*
x^2)/a)^p

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Rubi in Sympy [A]  time = 7.93054, size = 39, normalized size = 0.98 \[ 2 \sqrt{x} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**p/x**(1/2),x)

[Out]

2*sqrt(x)*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, 1/4), (5/4,), -b*x**2/
a)

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Mathematica [A]  time = 0.0211848, size = 49, normalized size = 1.22 \[ 2 \sqrt{x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^p/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(a + b*x^2)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^2)/a)])/(1 + (b*
x^2)/a)^p

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{ \left ( b{x}^{2}+a \right ) ^{p}{\frac{1}{\sqrt{x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p/x^(1/2),x)

[Out]

int((b*x^2+a)^p/x^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p/sqrt(x),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p/sqrt(x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{p}}{\sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p/sqrt(x),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p/sqrt(x), x)

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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((b*x**2+a)**p/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p/sqrt(x),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p/sqrt(x), x)