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

Optimal. Leaf size=80 \[ \frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac{a x}{b},-\frac{d x}{c}\right )}{1-m} \]

[Out]

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

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Rubi [A]  time = 0.148823, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac{a x}{b},-\frac{d x}{c}\right )}{1-m} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m*(c + d*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.3662, size = 61, normalized size = 0.76 \[ \frac{x^{m} x^{- m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (a + \frac{b}{x}\right )^{m} \left (c + d x\right )^{n} \left (\frac{a x}{b} + 1\right )^{- m} \operatorname{appellf_{1}}{\left (- m + 1,- m,- n,- m + 2,- \frac{a x}{b},- \frac{d x}{c} \right )}}{- m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**m*x**(-m + 1)*(1 + d*x/c)**(-n)*(a + b/x)**m*(c + d*x)**n*(a*x/b + 1)**(-m)*a
ppellf1(-m + 1, -m, -n, -m + 2, -a*x/b, -d*x/c)/(-m + 1)

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Mathematica [A]  time = 0.0849088, size = 0, normalized size = 0. \[ \int \left (a+\frac{b}{x}\right )^m (c+d x)^n \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b/x)^m*(c + d*x)^n, x]

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m*(d*x+c)^n,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n*(a + b/x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n*(a + b/x)^m, x)