3.2267 \(\int \frac{\left (a+b \sqrt{x}\right )^p}{x^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 b^2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a^3 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.053777, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 b^2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a^3 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.83301, size = 39, normalized size = 0.85 \[ - \frac{2 b^{2} \left (a + b \sqrt{x}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b \sqrt{x}}{a}} \right )}}{a^{3} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0319637, size = 62, normalized size = 1.35 \[ \frac{2 \left (\frac{a}{b \sqrt{x}}+1\right )^{-p} \left (a+b \sqrt{x}\right )^p \, _2F_1\left (2-p,-p;3-p;-\frac{a}{b \sqrt{x}}\right )}{(p-2) x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+b\sqrt{x} \right ) ^{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^p/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*sqrt(x) + a)^p/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*sqrt(x) + a)^p/x^2, x)

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Sympy [A]  time = 18.198, size = 42, normalized size = 0.91 \[ - \frac{2 b^{p} x^{\frac{p}{2}} \Gamma \left (- p + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p + 2 \\ - p + 3 \end{matrix}\middle |{\frac{a e^{i \pi }}{b \sqrt{x}}} \right )}}{x \Gamma \left (- p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b**p*x**(p/2)*gamma(-p + 2)*hyper((-p, -p + 2), (-p + 3,), a*exp_polar(I*pi)/
(b*sqrt(x)))/(x*gamma(-p + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*sqrt(x) + a)^p/x^2, x)