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

Optimal. Leaf size=105 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} (a c-b d (n+1)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)^2}-\frac{d \left (a+\frac{b}{x}\right )^{n+1}}{c \left (\frac{c}{x}+d\right ) (a c-b d)} \]

[Out]

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

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Rubi [A]  time = 0.182437, antiderivative size = 105, 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 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} (a c-b d (n+1)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)^2}-\frac{d \left (a+\frac{b}{x}\right )^{n+1}}{c \left (\frac{c}{x}+d\right ) (a c-b d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.9283, size = 73, normalized size = 0.7 \[ - \frac{d \left (a + \frac{b}{x}\right )^{n + 1}}{c \left (a c - b d\right ) \left (\frac{c}{x} + d\right )} + \frac{\left (a + \frac{b}{x}\right )^{n + 1} \left (a c - b d \left (n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{c \left (n + 1\right ) \left (a c - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0875272, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n}{x (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n/x/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((a*x + b)/x)^n/(d^2*x^3 + 2*c*d*x^2 + c^2*x), 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/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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