3.2260 \(\int \frac{x^m}{\left (a+b \sqrt{x}\right )^2} \, dx\)

Optimal. Leaf size=37 \[ \frac{x^{m+1} \, _2F_1\left (2,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0467434, antiderivative size = 37, 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 \[ \frac{x^{m+1} \, _2F_1\left (2,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*Sqrt[x])^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.06195, size = 29, normalized size = 0.78 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, 2 m + 2 \\ 2 m + 3 \end{matrix}\middle |{- \frac{b \sqrt{x}}{a}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0645262, size = 71, normalized size = 1.92 \[ \frac{2 x^{m+\frac{1}{2}} \left (\, _2F_1\left (1,2 m+1;2 m+2;-\frac{b \sqrt{x}}{a}\right )-\, _2F_1\left (2,2 m+1;2 m+2;-\frac{b \sqrt{x}}{a}\right )\right )}{a b (2 m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(1/2 + m)*(Hypergeometric2F1[1, 1 + 2*m, 2 + 2*m, -((b*Sqrt[x])/a)] - Hyper
geometric2F1[2, 1 + 2*m, 2 + 2*m, -((b*Sqrt[x])/a)]))/(a*b*(1 + 2*m))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( a+b\sqrt{x} \right ) ^{-2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -{\left (2 \, m + 1\right )} \int \frac{x^{m}}{a b \sqrt{x} + a^{2}}\,{d x} + \frac{2 \, x x^{m}}{a b \sqrt{x} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*m + 1)*integrate(x^m/(a*b*sqrt(x) + a^2), x) + 2*x*x^m/(a*b*sqrt(x) + a^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b^2*x + 2*a*b*sqrt(x) + a^2), x)

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Sympy [A]  time = 11.0901, size = 473, normalized size = 12.78 \[ - \frac{8 a m^{2} x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{12 a m x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} + \frac{4 a m x x^{m} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{4 a x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} + \frac{4 a x x^{m} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{8 b m^{2} x^{\frac{3}{2}} x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{12 b m x^{\frac{3}{2}} x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{4 b x^{\frac{3}{2}} x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a*m**2*x*x**m*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2
)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 12*a*m*x*x**m*lerchphi
(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3) +
a**2*b*sqrt(x)*gamma(2*m + 3)) + 4*a*m*x*x**m*gamma(2*m + 2)/(a**3*gamma(2*m + 3
) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 4*a*x*x**m*lerchphi(b*sqrt(x)*exp_polar(I*p
i)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m
+ 3)) + 4*a*x*x**m*gamma(2*m + 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2
*m + 3)) - 8*b*m**2*x**(3/2)*x**m*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m +
2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 12*b*
m*x**(3/2)*x**m*lerchphi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)
/(a**3*gamma(2*m + 3) + a**2*b*sqrt(x)*gamma(2*m + 3)) - 4*b*x**(3/2)*x**m*lerch
phi(b*sqrt(x)*exp_polar(I*pi)/a, 1, 2*m + 2)*gamma(2*m + 2)/(a**3*gamma(2*m + 3)
+ a**2*b*sqrt(x)*gamma(2*m + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*sqrt(x) + a)^2, x)