Optimal. Leaf size=66 \[ \frac{2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac{(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{a c^2 e (m+1)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0843184, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac{(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{a c^2 e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.3447, size = 49, normalized size = 0.74 \[ \frac{2 \left (e x\right )^{m + 1}}{c^{2} e \left (a - b x\right )} - \frac{\left (e x\right )^{m + 1} \left (2 m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x}{a}} \right )}}{a c^{2} e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.119589, size = 103, normalized size = 1.56 \[ \frac{(e x)^m \left (\frac{b x}{b x-a}\right )^{1-m} \left (2 a m \, _2F_1\left (1-m,-m;2-m;\frac{a}{a-b x}\right )-(m-1) (a-b x) \, _2F_1\left (-m,-m;1-m;\frac{a}{a-b x}\right )\right )}{b^2 c^2 (m-1) m x} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( bx+a \right ) }{ \left ( -bcx+ac \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )} \left (e x\right )^{m}}{b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + a^{2} c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 13.1543, size = 799, normalized size = 12.11 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2,x, algorithm="giac")
[Out]