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

Optimal. Leaf size=64 \[ \frac{x^m \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (-m;-n,1;1-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d m} \]

[Out]

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

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Rubi [A]  time = 0.196939, antiderivative size = 64, 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 \[ \frac{x^m \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (-m;-n,1;1-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.0663, size = 53, normalized size = 0.83 \[ \frac{x^{m - 1} \left (1 + \frac{b}{a x}\right )^{- n} \left (a + \frac{b}{x}\right )^{n} \left (\frac{1}{x}\right )^{- m} \left (\frac{1}{x}\right )^{m - 1} \operatorname{appellf_{1}}{\left (- m,1,- n,- m + 1,- \frac{c}{d x},- \frac{b}{a x} \right )}}{d m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(m - 1)*(1 + b/(a*x))**(-n)*(a + b/x)**n*(1/x)**(-m)*(1/x)**(m - 1)*appellf1(
-m, 1, -n, -m + 1, -c/(d*x), -b/(a*x))/(d*m)

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Mathematica [A]  time = 0.0959344, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n x^m}{c+d x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{dx+c} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n*x^m/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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