3.444 \(\int \frac{(d+e x)^m}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=190 \[ -\frac{c^2 (d+e x)^{m+1} (2 c d-b e (2-m)) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{b^3 (m+1) (c d-b e)^2}+\frac{(d+e x)^{m+1} (2 c d-b e m) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{b^3 d^2 (m+1)}-\frac{c (2 c d-b e) (d+e x)^{m+1}}{b^2 d (b+c x) (c d-b e)}-\frac{(d+e x)^{m+1}}{b d x (b+c x)} \]

[Out]

-((c*(2*c*d - b*e)*(d + e*x)^(1 + m))/(b^2*d*(c*d - b*e)*(b + c*x))) - (d + e*x)
^(1 + m)/(b*d*x*(b + c*x)) - (c^2*(2*c*d - b*e*(2 - m))*(d + e*x)^(1 + m)*Hyperg
eometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e)])/(b^3*(c*d - b*e)^2*(1 +
m)) + ((2*c*d - b*e*m)*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 +
(e*x)/d])/(b^3*d^2*(1 + m))

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Rubi [A]  time = 0.590525, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{c^2 (d+e x)^{m+1} (2 c d-b e (2-m)) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{b^3 (m+1) (c d-b e)^2}+\frac{(d+e x)^{m+1} (2 c d-b e m) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{b^3 d^2 (m+1)}-\frac{c (2 c d-b e) (d+e x)^{m+1}}{b^2 d (b+c x) (c d-b e)}-\frac{(d+e x)^{m+1}}{b d x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(2*c*d - b*e)*(d + e*x)^(1 + m))/(b^2*d*(c*d - b*e)*(b + c*x))) - (d + e*x)
^(1 + m)/(b*d*x*(b + c*x)) - (c^2*(2*c*d - b*e*(2 - m))*(d + e*x)^(1 + m)*Hyperg
eometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e)])/(b^3*(c*d - b*e)^2*(1 +
m)) + ((2*c*d - b*e*m)*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 +
(e*x)/d])/(b^3*d^2*(1 + m))

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Rubi in Sympy [A]  time = 77.9409, size = 158, normalized size = 0.83 \[ - \frac{c \left (d + e x\right )^{m + 1}}{b x \left (b + c x\right ) \left (b e - c d\right )} - \frac{\left (d + e x\right )^{m + 1} \left (b e - 2 c d\right )}{b^{2} d x \left (b e - c d\right )} + \frac{c^{2} \left (d + e x\right )^{m + 1} \left (- b e m + 2 b e - 2 c d\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (- d - e x\right )}{b e - c d}} \right )}}{b^{3} \left (m + 1\right ) \left (b e - c d\right )^{2}} + \frac{\left (d + e x\right )^{m + 1} \left (- b e m + 2 c d\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{b^{3} d^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(c*x**2+b*x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0941896, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (b x+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(d + e*x)^m/(b*x + c*x^2)^2, x]

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Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+b*x)^2,x)

[Out]

int((e*x+d)^m/(c*x^2+b*x)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(c*x^2 + b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(c*x^2 + b*x)^2, x)