Optimal. Leaf size=34 \[ \frac {a}{3 b^2 \left (a+b e^x\right )^3}-\frac {1}{2 b^2 \left (a+b e^x\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac {a}{3 b^2 \left (a+b e^x\right )^3}-\frac {1}{2 b^2 \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx &=\operatorname {Subst}\left (\int \frac {x}{(a+b x)^4} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^4}+\frac {1}{b (a+b x)^3}\right ) \, dx,x,e^x\right )\\ &=\frac {a}{3 b^2 \left (a+b e^x\right )^3}-\frac {1}{2 b^2 \left (a+b e^x\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 24, normalized size = 0.71 \[ -\frac {a+3 b e^x}{6 b^2 \left (a+b e^x\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 47, normalized size = 1.38 \[ -\frac {3 \, b e^{x} + a}{6 \, {\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 20, normalized size = 0.59 \[ -\frac {3 \, b e^{x} + a}{6 \, {\left (b e^{x} + a\right )}^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 29, normalized size = 0.85 \[ \frac {a}{3 \left (b \,{\mathrm e}^{x}+a \right )^{3} b^{2}}-\frac {1}{2 \left (b \,{\mathrm e}^{x}+a \right )^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.46, size = 85, normalized size = 2.50 \[ -\frac {b e^{x}}{2 \, {\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} - \frac {a}{6 \, {\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.60, size = 53, normalized size = 1.56 \[ \frac {\frac {{\mathrm {e}}^{2\,x}}{2\,a}+\frac {b\,{\mathrm {e}}^{3\,x}}{6\,a^2}}{a^3+3\,{\mathrm {e}}^x\,a^2\,b+3\,{\mathrm {e}}^{2\,x}\,a\,b^2+{\mathrm {e}}^{3\,x}\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.16, size = 51, normalized size = 1.50 \[ \frac {- a - 3 b e^{x}}{6 a^{3} b^{2} + 18 a^{2} b^{3} e^{x} + 18 a b^{4} e^{2 x} + 6 b^{5} e^{3 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________