Optimal. Leaf size=63 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e (m+1)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.102751, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((b*x)^m*(c + d*x)^n)/(e + f*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.4671, size = 46, normalized size = 0.73 \[ \frac{\left (b x\right )^{m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{f x}{e},- \frac{d x}{c} \right )}}{b e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x)**m*(d*x+c)**n/(f*x+e),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.402652, size = 153, normalized size = 2.43 \[ \frac{c e (m+2) x (b x)^m (c+d x)^n F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{(m+1) (e+f x) \left (c e (m+2) F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )+x \left (d e n F_1\left (m+2;1-n,1;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )-c f F_1\left (m+2;-n,2;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((b*x)^m*(c + d*x)^n)/(e + f*x),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n}}{fx+e}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x)^m*(d*x+c)^n/(f*x+e),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b x\right )^{m}{\left (d x + c\right )}^{n}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^m*(d*x + c)^n/(f*x + e),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (b x\right )^{m}{\left (d x + c\right )}^{n}}{f x + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^m*(d*x + c)^n/(f*x + e),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((b*x)**m*(d*x+c)**n/(f*x+e),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b x\right )^{m}{\left (d x + c\right )}^{n}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^m*(d*x + c)^n/(f*x + e),x, algorithm="giac")
[Out]