3.706 \(\int \frac{x^m}{(a+b x)^3} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{a^3 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0197503, 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 \[ \frac{x^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.1199, size = 22, normalized size = 0.76 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0217755, size = 29, normalized size = 1. \[ \frac{x^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x)^3,x]

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( bx+a \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x+a)^3,x)

[Out]

int(x^m/(b*x+a)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x + a)^3,x, algorithm="maxima")

[Out]

integrate(x^m/(b*x + a)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x + a)^3,x, algorithm="fricas")

[Out]

integral(x^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

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Sympy [A]  time = 4.10526, size = 717, normalized size = 24.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(b*x+a)**3,x)

[Out]

a**2*m**3*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*
gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**2*m
**2*x*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*
b**2*x**2*gamma(m + 2)) - a**2*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1
)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2
*gamma(m + 2)) + a**2*m*x*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*ga
mma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*a**2*x*x**m*gamma(m + 1)/(2*a**5
*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*a*b
*m**3*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*g
amma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a*b*m**
2*x**2*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3
*b**2*x**2*gamma(m + 2)) - 2*a*b*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1,
m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2
*x**2*gamma(m + 2)) + a*b*x**2*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b
*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + b**2*m**3*x**3*x**m*lerchphi(
b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*
gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - b**2*m*x**3*x**m*lerchphi(b*x*ex
p_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(
m + 2) + 2*a**3*b**2*x**2*gamma(m + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x + a)^3,x, algorithm="giac")

[Out]

integrate(x^m/(b*x + a)^3, x)