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

Optimal. Leaf size=32 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]

[Out]

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

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Rubi [A]  time = 0.0461447, antiderivative size = 32, 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 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.3979, size = 22, normalized size = 0.69 \[ - \frac{\left (d + e x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, m \\ m + 1 \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{c d m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0715492, size = 57, normalized size = 1.78 \[ -\frac{(d+e x)^m \left (1-\left (\frac{d}{e x}+1\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{d}{e x}\right )\right )}{c d m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ce{x}^{2}+cdx}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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