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

Optimal. Leaf size=202 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^3 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d (2-n)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^3 (n+1) (a c-b d)^2}+\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{n+1}}{a d^2 \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d \left (\frac{c}{x}+d\right )} \]

[Out]

(c*(2*a*c - b*d)*(a + b/x)^(1 + n))/(a*d^2*(a*c - b*d)*(d + c/x)) + ((a + b/x)^(
1 + n)*x)/(a*d*(d + c/x)) + (c^2*(2*a*c - b*d*(2 - n))*(a + b/x)^(1 + n)*Hyperge
ometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(d^3*(a*c - b*d)^2*(1 +
n)) - ((2*a*c - b*d*n)*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 +
b/(a*x)])/(a^2*d^3*(1 + n))

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Rubi [A]  time = 0.622069, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^3 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d (2-n)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^3 (n+1) (a c-b d)^2}+\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{n+1}}{a d^2 \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d \left (\frac{c}{x}+d\right )} \]

Antiderivative was successfully verified.

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

[Out]

(c*(2*a*c - b*d)*(a + b/x)^(1 + n))/(a*d^2*(a*c - b*d)*(d + c/x)) + ((a + b/x)^(
1 + n)*x)/(a*d*(d + c/x)) + (c^2*(2*a*c - b*d*(2 - n))*(a + b/x)^(1 + n)*Hyperge
ometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(d^3*(a*c - b*d)^2*(1 +
n)) - ((2*a*c - b*d*n)*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 +
b/(a*x)])/(a^2*d^3*(1 + n))

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Rubi in Sympy [A]  time = 85.6146, size = 155, normalized size = 0.77 \[ \frac{c^{2} \left (a + \frac{b}{x}\right )^{n + 1} \left (2 a c + b d n - 2 b d\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{d^{3} \left (n + 1\right ) \left (a c - b d\right )^{2}} + \frac{c \left (a + \frac{b}{x}\right )^{n + 1} \left (2 a c - b d\right )}{a d^{2} \left (a c - b d\right ) \left (\frac{c}{x} + d\right )} + \frac{x \left (a + \frac{b}{x}\right )^{n + 1}}{a d \left (\frac{c}{x} + d\right )} - \frac{\left (a + \frac{b}{x}\right )^{n + 1} \left (2 a c - b d n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{a^{2} d^{3} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*(a + b/x)**(n + 1)*(2*a*c + b*d*n - 2*b*d)*hyper((1, n + 1), (n + 2,), c*(a
+ b/x)/(a*c - b*d))/(d**3*(n + 1)*(a*c - b*d)**2) + c*(a + b/x)**(n + 1)*(2*a*c
- b*d)/(a*d**2*(a*c - b*d)*(c/x + d)) + x*(a + b/x)**(n + 1)/(a*d*(c/x + d)) -
(a + b/x)**(n + 1)*(2*a*c - b*d*n)*hyper((1, n + 1), (n + 2,), 1 + b/(a*x))/(a**
2*d**3*(n + 1))

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Mathematica [A]  time = 0.13735, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n x^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{ \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n*x^2/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2*((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**2/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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