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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]