Optimal. Leaf size=118 \[ -\frac {(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac {(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac {d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac {B c (d+e x)^8}{8 e^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 118, 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 = {771} \begin {gather*} -\frac {(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac {(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac {d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac {B c (d+e x)^8}{8 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^4}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^5}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^6}{e^3}+\frac {B c (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac {d (B d-A e) (c d-b e) (d+e x)^5}{5 e^4}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^6}{6 e^4}-\frac {(3 B c d-b B e-A c e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 177, normalized size = 1.50 \begin {gather*} \frac {1}{3} d^3 x^3 (4 A b e+A c d+b B d)+\frac {1}{4} d^2 x^4 (2 A e (3 b e+2 c d)+B d (4 b e+c d))+\frac {1}{7} e^3 x^7 (A c e+b B e+4 B c d)+\frac {1}{6} e^2 x^6 (A e (b e+4 c d)+2 B d (2 b e+3 c d))+\frac {2}{5} d e x^5 (A e (2 b e+3 c d)+B d (3 b e+2 c d))+\frac {1}{2} A b d^4 x^2+\frac {1}{8} B c e^4 x^8 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.35, size = 218, normalized size = 1.85 \begin {gather*} \frac {1}{8} x^{8} e^{4} c B + \frac {4}{7} x^{7} e^{3} d c B + \frac {1}{7} x^{7} e^{4} b B + \frac {1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac {2}{3} x^{6} e^{3} d b B + \frac {2}{3} x^{6} e^{3} d c A + \frac {1}{6} x^{6} e^{4} b A + \frac {4}{5} x^{5} e d^{3} c B + \frac {6}{5} x^{5} e^{2} d^{2} b B + \frac {6}{5} x^{5} e^{2} d^{2} c A + \frac {4}{5} x^{5} e^{3} d b A + \frac {1}{4} x^{4} d^{4} c B + x^{4} e d^{3} b B + x^{4} e d^{3} c A + \frac {3}{2} x^{4} e^{2} d^{2} b A + \frac {1}{3} x^{3} d^{4} b B + \frac {1}{3} x^{3} d^{4} c A + \frac {4}{3} x^{3} e d^{3} b A + \frac {1}{2} x^{2} d^{4} b A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 210, normalized size = 1.78 \begin {gather*} \frac {1}{8} \, B c x^{8} e^{4} + \frac {4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac {4}{5} \, B c d^{3} x^{5} e + \frac {1}{4} \, B c d^{4} x^{4} + \frac {1}{7} \, B b x^{7} e^{4} + \frac {1}{7} \, A c x^{7} e^{4} + \frac {2}{3} \, B b d x^{6} e^{3} + \frac {2}{3} \, A c d x^{6} e^{3} + \frac {6}{5} \, B b d^{2} x^{5} e^{2} + \frac {6}{5} \, A c d^{2} x^{5} e^{2} + B b d^{3} x^{4} e + A c d^{3} x^{4} e + \frac {1}{3} \, B b d^{4} x^{3} + \frac {1}{3} \, A c d^{4} x^{3} + \frac {1}{6} \, A b x^{6} e^{4} + \frac {4}{5} \, A b d x^{5} e^{3} + \frac {3}{2} \, A b d^{2} x^{4} e^{2} + \frac {4}{3} \, A b d^{3} x^{3} e + \frac {1}{2} \, A b d^{4} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 200, normalized size = 1.69 \begin {gather*} \frac {B c \,e^{4} x^{8}}{8}+\frac {A b \,d^{4} x^{2}}{2}+\frac {\left (B b \,e^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) c \right ) x^{7}}{7}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) b +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c \right ) x^{6}}{6}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b +\left (4 A \,d^{3} e +B \,d^{4}\right ) c \right ) x^{4}}{4}+\frac {\left (A c \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) b \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 178, normalized size = 1.51 \begin {gather*} \frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{2} \, A b d^{4} x^{2} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + {\left (B b + A c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + A b e^{4} + 4 \, {\left (B b + A c\right )} d e^{3}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, B c d^{3} e + 2 \, A b d e^{3} + 3 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 6 \, A b d^{2} e^{2} + 4 \, {\left (B b + A c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (4 \, A b d^{3} e + {\left (B b + A c\right )} d^{4}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 182, normalized size = 1.54 \begin {gather*} x^4\,\left (\frac {B\,c\,d^4}{4}+A\,c\,d^3\,e+B\,b\,d^3\,e+\frac {3\,A\,b\,d^2\,e^2}{2}\right )+x^6\,\left (\frac {A\,b\,e^4}{6}+\frac {2\,A\,c\,d\,e^3}{3}+\frac {2\,B\,b\,d\,e^3}{3}+B\,c\,d^2\,e^2\right )+x^3\,\left (\frac {A\,c\,d^4}{3}+\frac {B\,b\,d^4}{3}+\frac {4\,A\,b\,d^3\,e}{3}\right )+x^7\,\left (\frac {A\,c\,e^4}{7}+\frac {B\,b\,e^4}{7}+\frac {4\,B\,c\,d\,e^3}{7}\right )+\frac {2\,d\,e\,x^5\,\left (2\,A\,b\,e^2+2\,B\,c\,d^2+3\,A\,c\,d\,e+3\,B\,b\,d\,e\right )}{5}+\frac {A\,b\,d^4\,x^2}{2}+\frac {B\,c\,e^4\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.10, size = 230, normalized size = 1.95 \begin {gather*} \frac {A b d^{4} x^{2}}{2} + \frac {B c e^{4} x^{8}}{8} + x^{7} \left (\frac {A c e^{4}}{7} + \frac {B b e^{4}}{7} + \frac {4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac {A b e^{4}}{6} + \frac {2 A c d e^{3}}{3} + \frac {2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac {4 A b d e^{3}}{5} + \frac {6 A c d^{2} e^{2}}{5} + \frac {6 B b d^{2} e^{2}}{5} + \frac {4 B c d^{3} e}{5}\right ) + x^{4} \left (\frac {3 A b d^{2} e^{2}}{2} + A c d^{3} e + B b d^{3} e + \frac {B c d^{4}}{4}\right ) + x^{3} \left (\frac {4 A b d^{3} e}{3} + \frac {A c d^{4}}{3} + \frac {B b d^{4}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________