3.433 \(\int \frac{\left (a+\frac{b}{x}\right )^m}{(c+d x)^4} \, dx\)

Optimal. Leaf size=185 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left (m^2+3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{6 c^2 (m+1) (a c-b d)^4}+\frac{d^2 \left (a+\frac{b}{x}\right )^{m+1}}{3 c^2 \left (\frac{c}{x}+d\right )^3 (a c-b d)}-\frac{d \left (a+\frac{b}{x}\right )^{m+1} (6 a c-b d (m+4))}{6 c^2 \left (\frac{c}{x}+d\right )^2 (a c-b d)^2} \]

[Out]

(d^2*(a + b/x)^(1 + m))/(3*c^2*(a*c - b*d)*(d + c/x)^3) - (d*(6*a*c - b*d*(4 + m
))*(a + b/x)^(1 + m))/(6*c^2*(a*c - b*d)^2*(d + c/x)^2) - (b*(6*a^2*c^2 - 6*a*b*
c*d*(1 + m) + b^2*d^2*(2 + 3*m + m^2))*(a + b/x)^(1 + m)*Hypergeometric2F1[2, 1
+ m, 2 + m, (c*(a + b/x))/(a*c - b*d)])/(6*c^2*(a*c - b*d)^4*(1 + m))

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Rubi [A]  time = 0.40771, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left (m^2+3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{6 c^2 (m+1) (a c-b d)^4}+\frac{d^2 \left (a+\frac{b}{x}\right )^{m+1}}{3 c^2 \left (\frac{c}{x}+d\right )^3 (a c-b d)}-\frac{d \left (a+\frac{b}{x}\right )^{m+1} (6 a c-b d (m+4))}{6 c^2 \left (\frac{c}{x}+d\right )^2 (a c-b d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m/(c + d*x)^4,x]

[Out]

(d^2*(a + b/x)^(1 + m))/(3*c^2*(a*c - b*d)*(d + c/x)^3) - (d*(6*a*c - b*d*(4 + m
))*(a + b/x)^(1 + m))/(6*c^2*(a*c - b*d)^2*(d + c/x)^2) - (b*(6*a^2*c^2 - 6*a*b*
c*d*(1 + m) + b^2*d^2*(2 + 3*m + m^2))*(a + b/x)^(1 + m)*Hypergeometric2F1[2, 1
+ m, 2 + m, (c*(a + b/x))/(a*c - b*d)])/(6*c^2*(a*c - b*d)^4*(1 + m))

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Rubi in Sympy [A]  time = 21.976, size = 162, normalized size = 0.88 \[ - \frac{b \left (a + \frac{b}{x}\right )^{m + 1} \left (b d \left (- m + 1\right ) \left (3 a c - b d \left (m + 1\right )\right ) + \left (2 a c - b d \left (m + 1\right )\right ) \left (3 a c - 3 b d\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{6 c^{2} \left (m + 1\right ) \left (a c - b d\right )^{4}} + \frac{d^{2} \left (a + \frac{b}{x}\right )^{m + 1}}{3 c^{2} \left (a c - b d\right ) \left (\frac{c}{x} + d\right )^{3}} - \frac{d \left (a + \frac{b}{x}\right )^{m + 1} \left (6 a c - b d m - 4 b d\right )}{6 c^{2} \left (a c - b d\right )^{2} \left (\frac{c}{x} + d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**m/(d*x+c)**4,x)

[Out]

-b*(a + b/x)**(m + 1)*(b*d*(-m + 1)*(3*a*c - b*d*(m + 1)) + (2*a*c - b*d*(m + 1)
)*(3*a*c - 3*b*d))*hyper((2, m + 1), (m + 2,), c*(a + b/x)/(a*c - b*d))/(6*c**2*
(m + 1)*(a*c - b*d)**4) + d**2*(a + b/x)**(m + 1)/(3*c**2*(a*c - b*d)*(c/x + d)*
*3) - d*(a + b/x)**(m + 1)*(6*a*c - b*d*m - 4*b*d)/(6*c**2*(a*c - b*d)**2*(c/x +
d)**2)

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Mathematica [A]  time = 0.223834, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^m}{(c+d x)^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a + b/x)^m/(c + d*x)^4,x]

[Out]

Integrate[(a + b/x)^m/(c + d*x)^4, x]

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Maple [F]  time = 0.136, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( dx+c \right ) ^{4}} \left ( a+{\frac{b}{x}} \right ) ^{m}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m/(d*x+c)^4,x)

[Out]

int((a+b/x)^m/(d*x+c)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x}\right )}^{m}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^m/(d*x + c)^4,x, algorithm="maxima")

[Out]

integrate((a + b/x)^m/(d*x + c)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (\frac{a x + b}{x}\right )^{m}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^m/(d*x + c)^4,x, algorithm="fricas")

[Out]

integral(((a*x + b)/x)^m/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^
4), 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)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x}\right )}^{m}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^m/(d*x + c)^4,x, algorithm="giac")

[Out]

integrate((a + b/x)^m/(d*x + c)^4, x)