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

Optimal. Leaf size=39 \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{e x}{d}+1\right )}{c^2 d^2 (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.0583751, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{e x}{d}+1\right )}{c^2 d^2 (1-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.5945, size = 31, normalized size = 0.79 \[ - \frac{e \left (d + e x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m - 1 \\ m \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{c^{2} d^{2} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*(d + e*x)**(m - 1)*hyper((2, m - 1), (m,), 1 + e*x/d)/(c**2*d**2*(-m + 1))

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Mathematica [A]  time = 0.040084, size = 61, normalized size = 1.56 \[ \frac{\left (\frac{d}{e x}+1\right )^{-m} (d+e x)^m \, _2F_1\left (2-m,3-m;4-m;-\frac{d}{e x}\right )}{c^2 e^2 (m-3) x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ce{x}^{2}+cdx \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**m/(d**2*x**2 + 2*d*e*x**3 + e**2*x**4), x)/c**2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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