Optimal. Leaf size=55 \[ \frac {1}{3} A b^2 x^3+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{4} b x^4 (2 A c+b B)+\frac {1}{6} B c^2 x^6 \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \[ \frac {1}{3} A b^2 x^3+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{4} b x^4 (2 A c+b B)+\frac {1}{6} B c^2 x^6 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^2+b (b B+2 A c) x^3+c (2 b B+A c) x^4+B c^2 x^5\right ) \, dx\\ &=\frac {1}{3} A b^2 x^3+\frac {1}{4} b (b B+2 A c) x^4+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{6} B c^2 x^6\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.89 \[ \frac {1}{60} x^3 \left (20 A b^2+12 c x^2 (A c+2 b B)+15 b x (2 A c+b B)+10 B c^2 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 53, normalized size = 0.96 \[ \frac {1}{6} x^{6} c^{2} B + \frac {2}{5} x^{5} c b B + \frac {1}{5} x^{5} c^{2} A + \frac {1}{4} x^{4} b^{2} B + \frac {1}{2} x^{4} c b A + \frac {1}{3} x^{3} b^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.96 \[ \frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {1}{3} \, A b^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 0.95 \[ \frac {B \,c^{2} x^{6}}{6}+\frac {A \,b^{2} x^{3}}{3}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{5}}{5}+\frac {\left (2 A b c +b^{2} B \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 51, normalized size = 0.93 \[ \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{3} \, A b^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, A b c\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.93 \[ x^4\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+\frac {A\,b^2\,x^3}{3}+\frac {B\,c^2\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 54, normalized size = 0.98 \[ \frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {B b^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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