Optimal. Leaf size=61 \[ \frac {(a+b x)^6 (A b-2 a B)}{6 b^3}-\frac {a (a+b x)^5 (A b-a B)}{5 b^3}+\frac {B (a+b x)^7}{7 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 76} \begin {gather*} \frac {(a+b x)^6 (A b-2 a B)}{6 b^3}-\frac {a (a+b x)^5 (A b-a B)}{5 b^3}+\frac {B (a+b x)^7}{7 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x (a+b x)^4 (A+B x) \, dx\\ &=\int \left (\frac {a (-A b+a B) (a+b x)^4}{b^2}+\frac {(A b-2 a B) (a+b x)^5}{b^2}+\frac {B (a+b x)^6}{b^2}\right ) \, dx\\ &=-\frac {a (A b-a B) (a+b x)^5}{5 b^3}+\frac {(A b-2 a B) (a+b x)^6}{6 b^3}+\frac {B (a+b x)^7}{7 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 88, normalized size = 1.44 \begin {gather*} \frac {1}{210} x^2 \left (35 a^4 (3 A+2 B x)+70 a^3 b x (4 A+3 B x)+63 a^2 b^2 x^2 (5 A+4 B x)+28 a b^3 x^3 (6 A+5 B x)+5 b^4 x^4 (7 A+6 B x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 100, normalized size = 1.64 \begin {gather*} \frac {1}{7} x^{7} b^{4} B + \frac {2}{3} x^{6} b^{3} a B + \frac {1}{6} x^{6} b^{4} A + \frac {6}{5} x^{5} b^{2} a^{2} B + \frac {4}{5} x^{5} b^{3} a A + x^{4} b a^{3} B + \frac {3}{2} x^{4} b^{2} a^{2} A + \frac {1}{3} x^{3} a^{4} B + \frac {4}{3} x^{3} b a^{3} A + \frac {1}{2} x^{2} a^{4} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 100, normalized size = 1.64 \begin {gather*} \frac {1}{7} \, B b^{4} x^{7} + \frac {2}{3} \, B a b^{3} x^{6} + \frac {1}{6} \, A b^{4} x^{6} + \frac {6}{5} \, B a^{2} b^{2} x^{5} + \frac {4}{5} \, A a b^{3} x^{5} + B a^{3} b x^{4} + \frac {3}{2} \, A a^{2} b^{2} x^{4} + \frac {1}{3} \, B a^{4} x^{3} + \frac {4}{3} \, A a^{3} b x^{3} + \frac {1}{2} \, A a^{4} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 100, normalized size = 1.64 \begin {gather*} \frac {B \,b^{4} x^{7}}{7}+\frac {A \,a^{4} x^{2}}{2}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{6}}{6}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{5}}{5}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{4}}{4}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 99, normalized size = 1.62 \begin {gather*} \frac {1}{7} \, B b^{4} x^{7} + \frac {1}{2} \, A a^{4} x^{2} + \frac {1}{6} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{6} + \frac {2}{5} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 91, normalized size = 1.49 \begin {gather*} x^3\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,b\,a^3}{3}\right )+x^6\,\left (\frac {A\,b^4}{6}+\frac {2\,B\,a\,b^3}{3}\right )+\frac {A\,a^4\,x^2}{2}+\frac {B\,b^4\,x^7}{7}+\frac {a^2\,b\,x^4\,\left (3\,A\,b+2\,B\,a\right )}{2}+\frac {2\,a\,b^2\,x^5\,\left (2\,A\,b+3\,B\,a\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 107, normalized size = 1.75 \begin {gather*} \frac {A a^{4} x^{2}}{2} + \frac {B b^{4} x^{7}}{7} + x^{6} \left (\frac {A b^{4}}{6} + \frac {2 B a b^{3}}{3}\right ) + x^{5} \left (\frac {4 A a b^{3}}{5} + \frac {6 B a^{2} b^{2}}{5}\right ) + x^{4} \left (\frac {3 A a^{2} b^{2}}{2} + B a^{3} b\right ) + x^{3} \left (\frac {4 A a^{3} b}{3} + \frac {B a^{4}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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