3.705 \(\int \frac{x^m}{(a+b x)^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{b x}{a}\right )}{a^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0202472, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{b x}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.07959, size = 22, normalized size = 0.76 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0190924, size = 29, normalized size = 1. \[ \frac{x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{b x}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x)^2,x]

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( bx+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x+a)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.91052, size = 262, normalized size = 9.03 \[ - \frac{a m^{2} x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac{a m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac{a m x x^{m} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac{a x x^{m} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac{b m^{2} x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac{b m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(b*x+a)**2,x)

[Out]

-a*m**2*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamm
a(m + 2) + a**2*b*x*gamma(m + 2)) - a*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1
, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) + a*m*x*x**m*g
amma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) + a*x*x**m*gamma(m + 1)/
(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) - b*m**2*x**2*x**m*lerchphi(b*x*exp_
polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)
) - b*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*g
amma(m + 2) + a**2*b*x*gamma(m + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x + a)^2, x)