Optimal. Leaf size=94 \[ \frac {a^4 c^3 (e x)^{m+1}}{e (m+1)}-\frac {2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac {2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {75} \begin {gather*} -\frac {2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac {a^4 c^3 (e x)^{m+1}}{e (m+1)}+\frac {2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rubi steps
\begin {align*} \int (e x)^m (a+b x) (a c-b c x)^3 \, dx &=\int \left (a^4 c^3 (e x)^m-\frac {2 a^3 b c^3 (e x)^{1+m}}{e}+\frac {2 a b^3 c^3 (e x)^{3+m}}{e^3}-\frac {b^4 c^3 (e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {a^4 c^3 (e x)^{1+m}}{e (1+m)}-\frac {2 a^3 b c^3 (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a b^3 c^3 (e x)^{4+m}}{e^4 (4+m)}-\frac {b^4 c^3 (e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 112, normalized size = 1.19 \begin {gather*} -\frac {c^3 x (e x)^m \left (-\left (a^4 \left (m^3+11 m^2+38 m+40\right )\right )+2 a^3 b \left (m^3+10 m^2+29 m+20\right ) x-2 a b^3 \left (m^3+8 m^2+17 m+10\right ) x^3+b^4 \left (m^3+7 m^2+14 m+8\right ) x^4\right )}{(m+1) (m+2) (m+4) (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m (a+b x) (a c-b c x)^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.09, size = 209, normalized size = 2.22 \begin {gather*} -\frac {{\left ({\left (b^{4} c^{3} m^{3} + 7 \, b^{4} c^{3} m^{2} + 14 \, b^{4} c^{3} m + 8 \, b^{4} c^{3}\right )} x^{5} - 2 \, {\left (a b^{3} c^{3} m^{3} + 8 \, a b^{3} c^{3} m^{2} + 17 \, a b^{3} c^{3} m + 10 \, a b^{3} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} b c^{3} m^{3} + 10 \, a^{3} b c^{3} m^{2} + 29 \, a^{3} b c^{3} m + 20 \, a^{3} b c^{3}\right )} x^{2} - {\left (a^{4} c^{3} m^{3} + 11 \, a^{4} c^{3} m^{2} + 38 \, a^{4} c^{3} m + 40 \, a^{4} c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.26, size = 306, normalized size = 3.26 \begin {gather*} -\frac {b^{4} c^{3} m^{3} x^{5} x^{m} e^{m} - 2 \, a b^{3} c^{3} m^{3} x^{4} x^{m} e^{m} + 7 \, b^{4} c^{3} m^{2} x^{5} x^{m} e^{m} - 16 \, a b^{3} c^{3} m^{2} x^{4} x^{m} e^{m} + 14 \, b^{4} c^{3} m x^{5} x^{m} e^{m} + 2 \, a^{3} b c^{3} m^{3} x^{2} x^{m} e^{m} - 34 \, a b^{3} c^{3} m x^{4} x^{m} e^{m} + 8 \, b^{4} c^{3} x^{5} x^{m} e^{m} - a^{4} c^{3} m^{3} x x^{m} e^{m} + 20 \, a^{3} b c^{3} m^{2} x^{2} x^{m} e^{m} - 20 \, a b^{3} c^{3} x^{4} x^{m} e^{m} - 11 \, a^{4} c^{3} m^{2} x x^{m} e^{m} + 58 \, a^{3} b c^{3} m x^{2} x^{m} e^{m} - 38 \, a^{4} c^{3} m x x^{m} e^{m} + 40 \, a^{3} b c^{3} x^{2} x^{m} e^{m} - 40 \, a^{4} c^{3} x x^{m} e^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 175, normalized size = 1.86 \begin {gather*} \frac {\left (-b^{4} m^{3} x^{4}+2 a \,b^{3} m^{3} x^{3}-7 b^{4} m^{2} x^{4}+16 a \,b^{3} m^{2} x^{3}-14 b^{4} m \,x^{4}-2 a^{3} b \,m^{3} x +34 a \,b^{3} m \,x^{3}-8 b^{4} x^{4}+a^{4} m^{3}-20 a^{3} b \,m^{2} x +20 a \,b^{3} x^{3}+11 a^{4} m^{2}-58 a^{3} b m x +38 a^{4} m -40 a^{3} b x +40 a^{4}\right ) c^{3} x \left (e x \right )^{m}}{\left (m +5\right ) \left (m +4\right ) \left (m +2\right ) \left (m +1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.11, size = 91, normalized size = 0.97 \begin {gather*} -\frac {b^{4} c^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b^{3} c^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {2 \, a^{3} b c^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{4} c^{3}}{e {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.46, size = 182, normalized size = 1.94 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {a^4\,c^3\,x\,\left (m^3+11\,m^2+38\,m+40\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {b^4\,c^3\,x^5\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}+\frac {2\,a\,b^3\,c^3\,x^4\,\left (m^3+8\,m^2+17\,m+10\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {2\,a^3\,b\,c^3\,x^2\,\left (m^3+10\,m^2+29\,m+20\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.73, size = 838, normalized size = 8.91 \begin {gather*} \begin {cases} \frac {- \frac {a^{4} c^{3}}{4 x^{4}} + \frac {2 a^{3} b c^{3}}{3 x^{3}} - \frac {2 a b^{3} c^{3}}{x} - b^{4} c^{3} \log {\relax (x )}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a^{4} c^{3}}{3 x^{3}} + \frac {a^{3} b c^{3}}{x^{2}} + 2 a b^{3} c^{3} \log {\relax (x )} - b^{4} c^{3} x}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {a^{4} c^{3}}{x} - 2 a^{3} b c^{3} \log {\relax (x )} + a b^{3} c^{3} x^{2} - \frac {b^{4} c^{3} x^{3}}{3}}{e^{2}} & \text {for}\: m = -2 \\\frac {a^{4} c^{3} \log {\relax (x )} - 2 a^{3} b c^{3} x + \frac {2 a b^{3} c^{3} x^{3}}{3} - \frac {b^{4} c^{3} x^{4}}{4}}{e} & \text {for}\: m = -1 \\\frac {a^{4} c^{3} e^{m} m^{3} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {11 a^{4} c^{3} e^{m} m^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {38 a^{4} c^{3} e^{m} m x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 a^{4} c^{3} e^{m} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {2 a^{3} b c^{3} e^{m} m^{3} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {20 a^{3} b c^{3} e^{m} m^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {58 a^{3} b c^{3} e^{m} m x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {40 a^{3} b c^{3} e^{m} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {2 a b^{3} c^{3} e^{m} m^{3} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {16 a b^{3} c^{3} e^{m} m^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {34 a b^{3} c^{3} e^{m} m x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {20 a b^{3} c^{3} e^{m} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {b^{4} c^{3} e^{m} m^{3} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {7 b^{4} c^{3} e^{m} m^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {14 b^{4} c^{3} e^{m} m x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {8 b^{4} c^{3} e^{m} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________