Optimal. Leaf size=87 \[ \frac {a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac {(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac {a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac {B (a+b x)^8}{8 b^4} \]
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Rubi [A] time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac {a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac {(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac {a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac {B (a+b x)^8}{8 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^2 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (-\frac {a^2 (-A b+a B) (a+b x)^4}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^5}{b^3}+\frac {(A b-3 a B) (a+b x)^6}{b^3}+\frac {B (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac {a^2 (A b-a B) (a+b x)^5}{5 b^4}-\frac {a (2 A b-3 a B) (a+b x)^6}{6 b^4}+\frac {(A b-3 a B) (a+b x)^7}{7 b^4}+\frac {B (a+b x)^8}{8 b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 88, normalized size = 1.01 \[ \frac {1}{840} x^3 \left (70 a^4 (4 A+3 B x)+168 a^3 b x (5 A+4 B x)+168 a^2 b^2 x^2 (6 A+5 B x)+80 a b^3 x^3 (7 A+6 B x)+15 b^4 x^4 (8 A+7 B x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 99, normalized size = 1.14 \[ \frac {1}{8} x^{8} b^{4} B + \frac {4}{7} x^{7} b^{3} a B + \frac {1}{7} x^{7} b^{4} A + x^{6} b^{2} a^{2} B + \frac {2}{3} x^{6} b^{3} a A + \frac {4}{5} x^{5} b a^{3} B + \frac {6}{5} x^{5} b^{2} a^{2} A + \frac {1}{4} x^{4} a^{4} B + x^{4} b a^{3} A + \frac {1}{3} x^{3} a^{4} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 99, normalized size = 1.14 \[ \frac {1}{8} \, B b^{4} x^{8} + \frac {4}{7} \, B a b^{3} x^{7} + \frac {1}{7} \, A b^{4} x^{7} + B a^{2} b^{2} x^{6} + \frac {2}{3} \, A a b^{3} x^{6} + \frac {4}{5} \, B a^{3} b x^{5} + \frac {6}{5} \, A a^{2} b^{2} x^{5} + \frac {1}{4} \, B a^{4} x^{4} + A a^{3} b x^{4} + \frac {1}{3} \, A a^{4} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 100, normalized size = 1.15 \[ \frac {B \,b^{4} x^{8}}{8}+\frac {A \,a^{4} x^{3}}{3}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{7}}{7}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{6}}{6}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{5}}{5}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 99, normalized size = 1.14 \[ \frac {1}{8} \, B b^{4} x^{8} + \frac {1}{3} \, A a^{4} x^{3} + \frac {1}{7} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 90, normalized size = 1.03 \[ x^4\,\left (\frac {B\,a^4}{4}+A\,b\,a^3\right )+x^7\,\left (\frac {A\,b^4}{7}+\frac {4\,B\,a\,b^3}{7}\right )+\frac {A\,a^4\,x^3}{3}+\frac {B\,b^4\,x^8}{8}+\frac {2\,a^2\,b\,x^5\,\left (3\,A\,b+2\,B\,a\right )}{5}+\frac {a\,b^2\,x^6\,\left (2\,A\,b+3\,B\,a\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 104, normalized size = 1.20 \[ \frac {A a^{4} x^{3}}{3} + \frac {B b^{4} x^{8}}{8} + x^{7} \left (\frac {A b^{4}}{7} + \frac {4 B a b^{3}}{7}\right ) + x^{6} \left (\frac {2 A a b^{3}}{3} + B a^{2} b^{2}\right ) + x^{5} \left (\frac {6 A a^{2} b^{2}}{5} + \frac {4 B a^{3} b}{5}\right ) + x^{4} \left (A a^{3} b + \frac {B a^{4}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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