3.1071 \(\int x^{3-2 p} \left (a+b x^2\right )^p \, dx\)

Optimal. Leaf size=49 \[ \frac{x^{4-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,3;3-p;-\frac{b x^2}{a}\right )}{2 a (2-p)} \]

[Out]

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

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Rubi [A]  time = 0.0534336, antiderivative size = 64, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^{4-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (2-p,-p;3-p;-\frac{b x^2}{a}\right )}{2 (2-p)} \]

Antiderivative was successfully verified.

[In]  Int[x^(3 - 2*p)*(a + b*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.65634, size = 44, normalized size = 0.9 \[ \frac{x^{- 2 p + 4} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p + 2 \\ - p + 3 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 \left (- p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3-2*p)*(b*x**2+a)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0504031, size = 62, normalized size = 1.27 \[ -\frac{x^{4-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (2-p,-p;3-p;-\frac{b x^2}{a}\right )}{2 (p-2)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3 - 2*p)*(a + b*x^2)^p,x]

[Out]

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

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{x}^{3-2\,p} \left ( b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3-2*p)*(b*x^2+a)^p,x)

[Out]

int(x^(3-2*p)*(b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^(-2*p + 3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*x^(-2*p + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^(-2*p + 3),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*x^(-2*p + 3), 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(x**(3-2*p)*(b*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*x^(-2*p + 3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*x^(-2*p + 3), x)