Optimal. Leaf size=114 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b (c f (m+1)+d e))}{d (m+1) (m+2) (b c-a d)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.171169, antiderivative size = 112, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+2)+b c f (m+1)+b d e)}{d (m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-3 - m)*(e + f*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 23.3263, size = 92, normalized size = 0.81 \[ \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (c f - d e\right )}{d \left (m + 2\right ) \left (a d - b c\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- b d e + f \left (a d \left (m + 2\right ) - b c \left (m + 1\right )\right )\right )}{d \left (m + 1\right ) \left (m + 2\right ) \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-3-m)*(f*x+e),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.254548, size = 82, normalized size = 0.72 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} (b (c e (m+2)+c f (m+1) x+d e x)-a (c f+d e (m+1)+d f (m+2) x))}{(m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-3 - m)*(e + f*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 158, normalized size = 1.4 \[ -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-2-m} \left ( adfmx-bcfmx+adem+2\,adfx-bcem-bcfx-bdex+acf+ade-2\,bce \right ) }{{a}^{2}{d}^{2}{m}^{2}-2\,abcd{m}^{2}+{b}^{2}{c}^{2}{m}^{2}+3\,{a}^{2}{d}^{2}m-6\,abcdm+3\,{b}^{2}{c}^{2}m+2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.241899, size = 454, normalized size = 3.98 \[ -\frac{{\left (a^{2} c^{2} f -{\left (b^{2} d^{2} e +{\left (b^{2} c d - a b d^{2}\right )} f m +{\left (b^{2} c d - 2 \, a b d^{2}\right )} f\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} e m -{\left (3 \, b^{2} c d e +{\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2}\right )} f +{\left ({\left (b^{2} c d - a b d^{2}\right )} e +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} f\right )} m\right )} x^{2} -{\left (2 \, a b c^{2} - a^{2} c d\right )} e +{\left (3 \, a^{2} c d f -{\left (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2}\right )} e -{\left ({\left (b^{2} c^{2} - a^{2} d^{2}\right )} e +{\left (a b c^{2} - a^{2} c d\right )} f\right )} m\right )} x\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m^{2} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 3),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+a)**m*(d*x+c)**(-3-m)*(f*x+e),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 3),x, algorithm="giac")
[Out]