Optimal. Leaf size=87 \[ \frac{a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac{(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac{a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac{B (a+b x)^{10}}{10 b^4} \]
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Rubi [A] time = 0.0636993, antiderivative size = 87, normalized size of antiderivative = 1., 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} \[ \frac{a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac{(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac{a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac{B (a+b x)^{10}}{10 b^4} \]
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 )^3 \, dx &=\int x^2 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (-\frac{a^2 (-A b+a B) (a+b x)^6}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^7}{b^3}+\frac{(A b-3 a B) (a+b x)^8}{b^3}+\frac{B (a+b x)^9}{b^3}\right ) \, dx\\ &=\frac{a^2 (A b-a B) (a+b x)^7}{7 b^4}-\frac{a (2 A b-3 a B) (a+b x)^8}{8 b^4}+\frac{(A b-3 a B) (a+b x)^9}{9 b^4}+\frac{B (a+b x)^{10}}{10 b^4}\\ \end{align*}
Mathematica [A] time = 0.0178557, size = 143, normalized size = 1.64 \[ \frac{5}{7} a^2 b^3 x^7 (4 a B+3 A b)+\frac{5}{6} a^3 b^2 x^6 (3 a B+4 A b)+\frac{3}{5} a^4 b x^5 (2 a B+5 A b)+\frac{1}{4} a^5 x^4 (a B+6 A b)+\frac{1}{3} a^6 A x^3+\frac{1}{9} b^5 x^9 (6 a B+A b)+\frac{3}{8} a b^4 x^8 (5 a B+2 A b)+\frac{1}{10} b^6 B x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 148, normalized size = 1.7 \begin{align*}{\frac{B{b}^{6}{x}^{10}}{10}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{a}^{4}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,A{a}^{4}{b}^{2}+6\,B{a}^{5}b \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{4}}{4}}+{\frac{A{a}^{6}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983454, size = 198, normalized size = 2.28 \begin{align*} \frac{1}{10} \, B b^{6} x^{10} + \frac{1}{3} \, A a^{6} x^{3} + \frac{1}{9} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + \frac{3}{8} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + \frac{5}{7} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + \frac{5}{6} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + \frac{3}{5} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33808, size = 350, normalized size = 4.02 \begin{align*} \frac{1}{10} x^{10} b^{6} B + \frac{2}{3} x^{9} b^{5} a B + \frac{1}{9} x^{9} b^{6} A + \frac{15}{8} x^{8} b^{4} a^{2} B + \frac{3}{4} x^{8} b^{5} a A + \frac{20}{7} x^{7} b^{3} a^{3} B + \frac{15}{7} x^{7} b^{4} a^{2} A + \frac{5}{2} x^{6} b^{2} a^{4} B + \frac{10}{3} x^{6} b^{3} a^{3} A + \frac{6}{5} x^{5} b a^{5} B + 3 x^{5} b^{2} a^{4} A + \frac{1}{4} x^{4} a^{6} B + \frac{3}{2} x^{4} b a^{5} A + \frac{1}{3} x^{3} a^{6} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.101986, size = 163, normalized size = 1.87 \begin{align*} \frac{A a^{6} x^{3}}{3} + \frac{B b^{6} x^{10}}{10} + x^{9} \left (\frac{A b^{6}}{9} + \frac{2 B a b^{5}}{3}\right ) + x^{8} \left (\frac{3 A a b^{5}}{4} + \frac{15 B a^{2} b^{4}}{8}\right ) + x^{7} \left (\frac{15 A a^{2} b^{4}}{7} + \frac{20 B a^{3} b^{3}}{7}\right ) + x^{6} \left (\frac{10 A a^{3} b^{3}}{3} + \frac{5 B a^{4} b^{2}}{2}\right ) + x^{5} \left (3 A a^{4} b^{2} + \frac{6 B a^{5} b}{5}\right ) + x^{4} \left (\frac{3 A a^{5} b}{2} + \frac{B a^{6}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12979, size = 201, normalized size = 2.31 \begin{align*} \frac{1}{10} \, B b^{6} x^{10} + \frac{2}{3} \, B a b^{5} x^{9} + \frac{1}{9} \, A b^{6} x^{9} + \frac{15}{8} \, B a^{2} b^{4} x^{8} + \frac{3}{4} \, A a b^{5} x^{8} + \frac{20}{7} \, B a^{3} b^{3} x^{7} + \frac{15}{7} \, A a^{2} b^{4} x^{7} + \frac{5}{2} \, B a^{4} b^{2} x^{6} + \frac{10}{3} \, A a^{3} b^{3} x^{6} + \frac{6}{5} \, B a^{5} b x^{5} + 3 \, A a^{4} b^{2} x^{5} + \frac{1}{4} \, B a^{6} x^{4} + \frac{3}{2} \, A a^{5} b x^{4} + \frac{1}{3} \, A a^{6} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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