3.432 \(\int \frac{\left (a+\frac{b}{x}\right )^m}{(c+d x)^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (m+1)) \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{2 c (m+1) (a c-b d)^3}-\frac{d \left (a+\frac{b}{x}\right )^{m+1}}{2 c \left (\frac{c}{x}+d\right )^2 (a c-b d)} \]

[Out]

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

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Rubi [A]  time = 0.165579, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (m+1)) \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{2 c (m+1) (a c-b d)^3}-\frac{d \left (a+\frac{b}{x}\right )^{m+1}}{2 c \left (\frac{c}{x}+d\right )^2 (a c-b d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m/(c + d*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.2758, size = 83, normalized size = 0.74 \[ - \frac{b \left (a + \frac{b}{x}\right )^{m + 1} \left (2 a c - b d \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{2 c \left (m + 1\right ) \left (a c - b d\right )^{3}} - \frac{d \left (a + \frac{b}{x}\right )^{m + 1}}{2 c \left (a c - b d\right ) \left (\frac{c}{x} + d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0937313, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^m}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b/x)^m/(c + d*x)^3, x]

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( dx+c \right ) ^{3}} \left ( a+{\frac{b}{x}} \right ) ^{m}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m/(d*x+c)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^m/(d*x + c)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^m/(d*x + c)^3, x)