3.1053 \(\int \sqrt{x} \left (a+b x^2\right )^p \, dx\)

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]

(2*x^(3/2)*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 7/4 + p, 7/4, -((b*x^2)/a)])
/(3*a)

_______________________________________________________________________________________

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]

(2*x^(3/2)*(a + b*x^2)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^2)/a)])/(3*(1 +
(b*x^2)/a)^p)

_______________________________________________________________________________________

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]

2*x**(3/2)*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, 3/4), (7/4,), -b*x**2
/a)/3

_______________________________________________________________________________________

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]

(2*x^(3/2)*(a + b*x^2)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^2)/a)])/(3*(1 +
(b*x^2)/a)^p)

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

integrate((b*x^2 + a)^p*sqrt(x), x)

_______________________________________________________________________________________

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]

integral((b*x^2 + a)^p*sqrt(x), x)

_______________________________________________________________________________________

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]

Timed out

_______________________________________________________________________________________

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]

integrate((b*x^2 + a)^p*sqrt(x), x)