3.298 \(\int \frac{\left (a+\frac{b}{x}\right )^n}{x^3 (c+d x)^2} \, dx\)

Optimal. Leaf size=217 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} \left (d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac{c (a c-b d) (a c+b d (n+3))}{x}\right )}{b^2 c^3 (n+1) (n+2) \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{d^2 \left (a+\frac{b}{x}\right )^{n+1} (3 a c-b d (n+3)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^3 (n+1) (a c-b d)^2}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+2) x^2 \left (\frac{c}{x}+d\right )} \]

[Out]

-(((a + b/x)^(1 + n)*(d*(b*d*(2 + n)*(a*c + b*d*(3 + n)) - a*c*(a*c + b*d*(5 + 3
*n))) - (c*(a*c - b*d)*(a*c + b*d*(3 + n)))/x))/(b^2*c^3*(a*c - b*d)*(1 + n)*(2
+ n)*(d + c/x))) - (a + b/x)^(1 + n)/(b*c*(2 + n)*(d + c/x)*x^2) + (d^2*(3*a*c -
b*d*(3 + n))*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))
/(a*c - b*d)])/(c^3*(a*c - b*d)^2*(1 + n))

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Rubi [A]  time = 0.618771, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} \left (d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac{c (a c-b d) (a c+b d (n+3))}{x}\right )}{b^2 c^3 (n+1) (n+2) \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{d^2 \left (a+\frac{b}{x}\right )^{n+1} (3 a c-b d (n+3)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^3 (n+1) (a c-b d)^2}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+2) x^2 \left (\frac{c}{x}+d\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^n/(x^3*(c + d*x)^2),x]

[Out]

-(((a + b/x)^(1 + n)*(d*(b*d*(2 + n)*(a*c + b*d*(3 + n)) - a*c*(a*c + b*d*(5 + 3
*n))) - (c*(a*c - b*d)*(a*c + b*d*(3 + n)))/x))/(b^2*c^3*(a*c - b*d)*(1 + n)*(2
+ n)*(d + c/x))) - (a + b/x)^(1 + n)/(b*c*(2 + n)*(d + c/x)*x^2) + (d^2*(3*a*c -
b*d*(3 + n))*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))
/(a*c - b*d)])/(c^3*(a*c - b*d)^2*(1 + n))

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Rubi in Sympy [A]  time = 53.0992, size = 178, normalized size = 0.82 \[ \frac{d^{2} \left (a + \frac{b}{x}\right )^{n + 1} \left (3 a c - b d n - 3 b d\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{c^{3} \left (n + 1\right ) \left (a c - b d\right )^{2}} - \frac{\left (a + \frac{b}{x}\right )^{n + 1}}{b c x^{2} \left (n + 2\right ) \left (\frac{c}{x} + d\right )} + \frac{\left (a + \frac{b}{x}\right )^{n + 1} \left (\frac{c \left (a c - b d\right ) \left (a c + b d \left (n + 3\right )\right )}{x} + d \left (a c \left (a c + 2 b d \left (n + 1\right ) + b d \left (n + 3\right )\right ) - b d \left (n + 2\right ) \left (a c + b d \left (n + 3\right )\right )\right )\right )}{b^{2} c^{3} \left (n + 1\right ) \left (n + 2\right ) \left (a c - b d\right ) \left (\frac{c}{x} + d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**n/x**3/(d*x+c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.438179, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n}{x^3 (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a + b/x)^n/(x^3*(c + d*x)^2),x]

[Out]

Integrate[(a + b/x)^n/(x^3*(c + d*x)^2), x]

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n/x^3/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^n/((d*x + c)^2*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (\frac{a x + b}{x}\right )^{n}}{d^{2} x^{5} + 2 \, c d x^{4} + c^{2} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^n/((d*x + c)^2*x^3),x, algorithm="fricas")

[Out]

integral(((a*x + b)/x)^n/(d^2*x^5 + 2*c*d*x^4 + c^2*x^3), 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**3/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^n/((d*x + c)^2*x^3),x, algorithm="giac")

[Out]

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