3.292 \(\int \frac{\left (a+\frac{b}{x}\right )^n x^m}{(c+d x)^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{x^{m-1} \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (1-m;-n,2;2-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d^2 (1-m)} \]

[Out]

-(((a + b/x)^n*x^(-1 + m)*AppellF1[1 - m, -n, 2, 2 - m, -(b/(a*x)), -(c/(d*x))])
/(d^2*(1 - m)*(1 + b/(a*x))^n))

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Rubi [A]  time = 0.191042, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x^{m-1} \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (1-m;-n,2;2-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d^2 (1-m)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b/x)^n*x^m)/(c + d*x)^2,x]

[Out]

-(((a + b/x)^n*x^(-1 + m)*AppellF1[1 - m, -n, 2, 2 - m, -(b/(a*x)), -(c/(d*x))])
/(d^2*(1 - m)*(1 + b/(a*x))^n))

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Rubi in Sympy [A]  time = 23.5547, size = 60, normalized size = 0.82 \[ - \frac{x^{m - 2} \left (1 + \frac{b}{a x}\right )^{- n} \left (a + \frac{b}{x}\right )^{n} \left (\frac{1}{x}\right )^{- m + 1} \left (\frac{1}{x}\right )^{m - 2} \operatorname{appellf_{1}}{\left (- m + 1,2,- n,- m + 2,- \frac{c}{d x},- \frac{b}{a x} \right )}}{d^{2} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(m - 2)*(1 + b/(a*x))**(-n)*(a + b/x)**n*(1/x)**(-m + 1)*(1/x)**(m - 2)*appe
llf1(-m + 1, 2, -n, -m + 2, -c/(d*x), -b/(a*x))/(d**2*(-m + 1))

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Mathematica [A]  time = 0.102033, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n x^m}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[((a + b/x)^n*x^m)/(c + d*x)^2, x]

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n*x^m/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^n*x^m/(d*x + c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**n*x**m/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^n*x^m/(d*x + c)^2, x)