Optimal. Leaf size=53 \[ 4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac {4}{3} a b^3 e^{3 x}+\frac {1}{4} b^4 e^{4 x}+a^4 x \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2320, 45}
\begin {gather*} a^4 x+4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac {4}{3} a b^3 e^{3 x}+\frac {1}{4} b^4 e^{4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2320
Rubi steps
\begin {align*} \int \left (a+b e^x\right )^4 \, dx &=\text {Subst}\left (\int \frac {(a+b x)^4}{x} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (4 a^3 b+\frac {a^4}{x}+6 a^2 b^2 x+4 a b^3 x^2+b^4 x^3\right ) \, dx,x,e^x\right )\\ &=4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac {4}{3} a b^3 e^{3 x}+\frac {1}{4} b^4 e^{4 x}+a^4 x\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 53, normalized size = 1.00 \begin {gather*} \frac {1}{12} b e^x \left (48 a^3+36 a^2 b e^x+16 a b^2 e^{2 x}+3 b^3 e^{3 x}\right )+a^4 \log \left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 48, normalized size = 0.91
method | result | size |
norman | \(4 a^{3} b \,{\mathrm e}^{x}+3 a^{2} b^{2} {\mathrm e}^{2 x}+\frac {4 a \,b^{3} {\mathrm e}^{3 x}}{3}+\frac {b^{4} {\mathrm e}^{4 x}}{4}+a^{4} x\) | \(46\) |
risch | \(4 a^{3} b \,{\mathrm e}^{x}+3 a^{2} b^{2} {\mathrm e}^{2 x}+\frac {4 a \,b^{3} {\mathrm e}^{3 x}}{3}+\frac {b^{4} {\mathrm e}^{4 x}}{4}+a^{4} x\) | \(46\) |
derivativedivides | \(\frac {b^{4} {\mathrm e}^{4 x}}{4}+\frac {4 a \,b^{3} {\mathrm e}^{3 x}}{3}+3 a^{2} b^{2} {\mathrm e}^{2 x}+4 a^{3} b \,{\mathrm e}^{x}+a^{4} \ln \left ({\mathrm e}^{x}\right )\) | \(48\) |
default | \(\frac {b^{4} {\mathrm e}^{4 x}}{4}+\frac {4 a \,b^{3} {\mathrm e}^{3 x}}{3}+3 a^{2} b^{2} {\mathrm e}^{2 x}+4 a^{3} b \,{\mathrm e}^{x}+a^{4} \ln \left ({\mathrm e}^{x}\right )\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 45, normalized size = 0.85 \begin {gather*} a^{4} x + \frac {1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac {4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 45, normalized size = 0.85 \begin {gather*} a^{4} x + \frac {1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac {4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.06, size = 51, normalized size = 0.96 \begin {gather*} a^{4} x + 4 a^{3} b e^{x} + 3 a^{2} b^{2} e^{2 x} + \frac {4 a b^{3} e^{3 x}}{3} + \frac {b^{4} e^{4 x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.31, size = 45, normalized size = 0.85 \begin {gather*} a^{4} x + \frac {1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac {4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.43, size = 45, normalized size = 0.85 \begin {gather*} x\,a^4+4\,{\mathrm {e}}^x\,a^3\,b+3\,{\mathrm {e}}^{2\,x}\,a^2\,b^2+\frac {4\,{\mathrm {e}}^{3\,x}\,a\,b^3}{3}+\frac {{\mathrm {e}}^{4\,x}\,b^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________