Optimal. Leaf size=40 \[ \frac {(d+e x)^{m+1}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {644, 32} \[ \frac {(d+e x)^{m+1}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 644
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac {(d+e x) \int (d+e x)^{-1+m} \, dx}{\sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac {(d+e x)^{1+m}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 29, normalized size = 0.72 \[ \frac {(d+e x)^{m+1}}{e m \sqrt {c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.21, size = 45, normalized size = 1.12 \[ \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )}^{m}}{c e^{2} m x + c d e m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{m}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 39, normalized size = 0.98 \[ \frac {\left (e x +d \right )^{m +1}}{\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, e m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.43, size = 17, normalized size = 0.42 \[ \frac {{\left (e x + d\right )}^{m}}{\sqrt {c} e m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.49, size = 48, normalized size = 1.20 \[ \frac {{\left (d+e\,x\right )}^m\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c\,e^2\,m\,\left (x+\frac {d}{e}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{m}}{\sqrt {c \left (d + e x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________