Optimal. Leaf size=75 \[ \frac {1}{5} A b^3 x^5+\frac {1}{6} b^2 x^6 (3 A c+b B)+\frac {1}{8} c^2 x^8 (A c+3 b B)+\frac {3}{7} b c x^7 (A c+b B)+\frac {1}{9} B c^3 x^9 \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} \frac {1}{6} b^2 x^6 (3 A c+b B)+\frac {1}{5} A b^3 x^5+\frac {1}{8} c^2 x^8 (A c+3 b B)+\frac {3}{7} b c x^7 (A c+b B)+\frac {1}{9} B c^3 x^9 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int x (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^4+b^2 (b B+3 A c) x^5+3 b c (b B+A c) x^6+c^2 (3 b B+A c) x^7+B c^3 x^8\right ) \, dx\\ &=\frac {1}{5} A b^3 x^5+\frac {1}{6} b^2 (b B+3 A c) x^6+\frac {3}{7} b c (b B+A c) x^7+\frac {1}{8} c^2 (3 b B+A c) x^8+\frac {1}{9} B c^3 x^9\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{5} A b^3 x^5+\frac {1}{6} b^2 x^6 (3 A c+b B)+\frac {1}{8} c^2 x^8 (A c+3 b B)+\frac {3}{7} b c x^7 (A c+b B)+\frac {1}{9} B c^3 x^9 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (A+B x) \left (b x+c x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.34, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{9} x^{9} c^{3} B + \frac {3}{8} x^{8} c^{2} b B + \frac {1}{8} x^{8} c^{3} A + \frac {3}{7} x^{7} c b^{2} B + \frac {3}{7} x^{7} c^{2} b A + \frac {1}{6} x^{6} b^{3} B + \frac {1}{2} x^{6} c b^{2} A + \frac {1}{5} x^{5} b^{3} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{9} \, B c^{3} x^{9} + \frac {3}{8} \, B b c^{2} x^{8} + \frac {1}{8} \, A c^{3} x^{8} + \frac {3}{7} \, B b^{2} c x^{7} + \frac {3}{7} \, A b c^{2} x^{7} + \frac {1}{6} \, B b^{3} x^{6} + \frac {1}{2} \, A b^{2} c x^{6} + \frac {1}{5} \, A b^{3} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 76, normalized size = 1.01 \begin {gather*} \frac {B \,c^{3} x^{9}}{9}+\frac {A \,b^{3} x^{5}}{5}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{8}}{8}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{7}}{7}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.88, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{5} \, A b^{3} x^{5} + \frac {1}{8} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + \frac {3}{7} \, {\left (B b^{2} c + A b c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 69, normalized size = 0.92 \begin {gather*} x^6\,\left (\frac {B\,b^3}{6}+\frac {A\,c\,b^2}{2}\right )+x^8\,\left (\frac {A\,c^3}{8}+\frac {3\,B\,b\,c^2}{8}\right )+\frac {A\,b^3\,x^5}{5}+\frac {B\,c^3\,x^9}{9}+\frac {3\,b\,c\,x^7\,\left (A\,c+B\,b\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 82, normalized size = 1.09 \begin {gather*} \frac {A b^{3} x^{5}}{5} + \frac {B c^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{3}}{8} + \frac {3 B b c^{2}}{8}\right ) + x^{7} \left (\frac {3 A b c^{2}}{7} + \frac {3 B b^{2} c}{7}\right ) + x^{6} \left (\frac {A b^{2} c}{2} + \frac {B b^{3}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________