Optimal. Leaf size=45 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0582823, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]
Antiderivative was successfully verified.
[In] Int[((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^n,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.9038, size = 34, normalized size = 0.76 \[ \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} - m, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*(b*x+a)/(a*d-b*c))**m*(d*x+c)**n,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.103873, size = 46, normalized size = 1.02 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b c+b d x}{b c-a d}\right )}{d n+d} \]
Antiderivative was successfully verified.
[In] Integrate[((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^n,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.174, size = 0, normalized size = 0. \[ \int \left ({\frac{d \left ( bx+a \right ) }{ad-bc}} \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{n} \left (-\frac{{\left (b x + a\right )} d}{b c - a d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d x + c\right )}^{n} \left (-\frac{b d x + a d}{b c - a d}\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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((d*(b*x+a)/(a*d-b*c))**m*(d*x+c)**n,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{n} \left (-\frac{{\left (b x + a\right )} d}{b c - a d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m,x, algorithm="giac")
[Out]