Optimal. Leaf size=87 \[ \frac {a^2 (a+b x)^6 (A b-a B)}{6 b^4}+\frac {(a+b x)^8 (A b-3 a B)}{8 b^4}-\frac {a (a+b x)^7 (2 A b-3 a B)}{7 b^4}+\frac {B (a+b x)^9}{9 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \[ \frac {a^2 (a+b x)^6 (A b-a B)}{6 b^4}+\frac {(a+b x)^8 (A b-3 a B)}{8 b^4}-\frac {a (a+b x)^7 (2 A b-3 a B)}{7 b^4}+\frac {B (a+b x)^9}{9 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rubi steps
\begin {align*} \int x^2 (a+b x)^5 (A+B x) \, dx &=\int \left (-\frac {a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac {(A b-3 a B) (a+b x)^7}{b^3}+\frac {B (a+b x)^8}{b^3}\right ) \, dx\\ &=\frac {a^2 (A b-a B) (a+b x)^6}{6 b^4}-\frac {a (2 A b-3 a B) (a+b x)^7}{7 b^4}+\frac {(A b-3 a B) (a+b x)^8}{8 b^4}+\frac {B (a+b x)^9}{9 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 114, normalized size = 1.31 \[ \frac {1}{3} a^5 A x^3+\frac {1}{4} a^4 x^4 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac {5}{3} a^2 b^2 x^6 (a B+A b)+\frac {1}{8} b^4 x^8 (5 a B+A b)+\frac {5}{7} a b^3 x^7 (2 a B+A b)+\frac {1}{9} b^5 B x^9 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 124, normalized size = 1.43 \[ \frac {1}{9} x^{9} b^{5} B + \frac {5}{8} x^{8} b^{4} a B + \frac {1}{8} x^{8} b^{5} A + \frac {10}{7} x^{7} b^{3} a^{2} B + \frac {5}{7} x^{7} b^{4} a A + \frac {5}{3} x^{6} b^{2} a^{3} B + \frac {5}{3} x^{6} b^{3} a^{2} A + x^{5} b a^{4} B + 2 x^{5} b^{2} a^{3} A + \frac {1}{4} x^{4} a^{5} B + \frac {5}{4} x^{4} b a^{4} A + \frac {1}{3} x^{3} a^{5} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.23, size = 124, normalized size = 1.43 \[ \frac {1}{9} \, B b^{5} x^{9} + \frac {5}{8} \, B a b^{4} x^{8} + \frac {1}{8} \, A b^{5} x^{8} + \frac {10}{7} \, B a^{2} b^{3} x^{7} + \frac {5}{7} \, A a b^{4} x^{7} + \frac {5}{3} \, B a^{3} b^{2} x^{6} + \frac {5}{3} \, A a^{2} b^{3} x^{6} + B a^{4} b x^{5} + 2 \, A a^{3} b^{2} x^{5} + \frac {1}{4} \, B a^{5} x^{4} + \frac {5}{4} \, A a^{4} b x^{4} + \frac {1}{3} \, A a^{5} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 124, normalized size = 1.43 \[ \frac {B \,b^{5} x^{9}}{9}+\frac {A \,a^{5} x^{3}}{3}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{8}}{8}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{7}}{7}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{6}}{6}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{5}}{5}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.05, size = 118, normalized size = 1.36 \[ \frac {1}{9} \, B b^{5} x^{9} + \frac {1}{3} \, A a^{5} x^{3} + \frac {1}{8} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 106, normalized size = 1.22 \[ x^4\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )+x^8\,\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )+\frac {A\,a^5\,x^3}{3}+\frac {B\,b^5\,x^9}{9}+\frac {5\,a^2\,b^2\,x^6\,\left (A\,b+B\,a\right )}{3}+a^3\,b\,x^5\,\left (2\,A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^7\,\left (A\,b+2\,B\,a\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 133, normalized size = 1.53 \[ \frac {A a^{5} x^{3}}{3} + \frac {B b^{5} x^{9}}{9} + x^{8} \left (\frac {A b^{5}}{8} + \frac {5 B a b^{4}}{8}\right ) + x^{7} \left (\frac {5 A a b^{4}}{7} + \frac {10 B a^{2} b^{3}}{7}\right ) + x^{6} \left (\frac {5 A a^{2} b^{3}}{3} + \frac {5 B a^{3} b^{2}}{3}\right ) + x^{5} \left (2 A a^{3} b^{2} + B a^{4} b\right ) + x^{4} \left (\frac {5 A a^{4} b}{4} + \frac {B a^{5}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________