3.2266 \(\int \frac{\left (a+b \sqrt{x}\right )^p}{x} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.047178, antiderivative size = 43, 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 \[ -\frac{2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.65795, size = 34, normalized size = 0.79 \[ - \frac{2 \left (a + b \sqrt{x}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b \sqrt{x}}{a}} \right )}}{a \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.034323, size = 55, normalized size = 1.28 \[ \frac{2 \left (\frac{a}{b \sqrt{x}}+1\right )^{-p} \left (a+b \sqrt{x}\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a}{b \sqrt{x}}\right )}{p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( a+b\sqrt{x} \right ) ^{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^p/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*sqrt(x) + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*sqrt(x) + a)^p/x, x)

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Sympy [A]  time = 7.17471, size = 41, normalized size = 0.95 \[ - \frac{2 b^{p} x^{\frac{p}{2}} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{a e^{i \pi }}{b \sqrt{x}}} \right )}}{\Gamma \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b**p*x**(p/2)*gamma(-p)*hyper((-p, -p), (-p + 1,), a*exp_polar(I*pi)/(b*sqrt(
x)))/gamma(-p + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*sqrt(x) + a)^p/x, x)