Optimal. Leaf size=78 \[ -\frac {(d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \log (a e+c d x) \left (-a e^3 g-c d e^2 g x\right )^m}{c d e^2 g} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {891, 23, 31} \[ -\frac {(d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \log (a e+c d x) \left (-a e^3 g-c d e^2 g x\right )^m}{c d e^2 g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 23
Rule 31
Rule 891
Rubi steps
\begin {align*} \int (d+e x)^m \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (a e+c d x)^{-m} \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^{-1+m} \, dx\\ &=\left ((d+e x)^m \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac {1}{c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x} \, dx\\ &=-\frac {(d+e x)^m \left (-a e^3 g-c d e^2 g x\right )^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \log (a e+c d x)}{c d e^2 g}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 64, normalized size = 0.82 \[ -\frac {(d+e x)^m ((d+e x) (a e+c d x))^{-m} \log (a e+c d x) \left (-e^2 g (a e+c d x)\right )^m}{c d e^2 g} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 35, normalized size = 0.45 \[ -\frac {\log \left (c d x + a e\right )}{c d e^{2} g \left (-\frac {1}{e^{2} g}\right )^{m}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c d e^{2} g x + c d^{2} e g - {\left (c d^{2} + a e^{2}\right )} e g\right )}^{m - 1} {\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{-m} \left (-c d \,e^{2} g x +c \,d^{2} e g -\left (a \,e^{2}+c \,d^{2}\right ) e g \right )^{m -1} \left (e x +d \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 32, normalized size = 0.41 \[ -\frac {e^{2 \, m - 2} \left (-g\right )^{m} \log \left (c d x + a e\right )}{c d g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^m\,{\left (c\,d^2\,e\,g-e\,g\,\left (c\,d^2+a\,e^2\right )-c\,d\,e^2\,g\,x\right )}^{m-1}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________