Optimal. Leaf size=100 \[ \frac {1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac {1}{3} A b^2 d x^3+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{4} b x^4 (A b e+2 A c d+b B d)+\frac {1}{7} B c^2 e x^7 \]
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Rubi [A] time = 0.11, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \begin {gather*} \frac {1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac {1}{3} A b^2 d x^3+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{4} b x^4 (A b e+2 A c d+b B d)+\frac {1}{7} B c^2 e x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 d x^2+b (b B d+2 A c d+A b e) x^3+\left (A c^2 d+b^2 B e+2 b c (B d+A e)\right ) x^4+c (B c d+2 b B e+A c e) x^5+B c^2 e x^6\right ) \, dx\\ &=\frac {1}{3} A b^2 d x^3+\frac {1}{4} b (b B d+2 A c d+A b e) x^4+\frac {1}{5} \left (A c^2 d+b^2 B e+2 b c (B d+A e)\right ) x^5+\frac {1}{6} c (B c d+2 b B e+A c e) x^6+\frac {1}{7} B c^2 e x^7\\ \end {align*}
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Mathematica [A] time = 0.03, size = 101, normalized size = 1.01 \begin {gather*} \frac {1}{5} x^5 \left (2 A b c e+A c^2 d+b^2 B e+2 b B c d\right )+\frac {1}{3} A b^2 d x^3+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{4} b x^4 (A b e+2 A c d+b B d)+\frac {1}{7} B c^2 e x^7 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 117, normalized size = 1.17 \begin {gather*} \frac {1}{7} x^{7} e c^{2} B + \frac {1}{6} x^{6} d c^{2} B + \frac {1}{3} x^{6} e c b B + \frac {1}{6} x^{6} e c^{2} A + \frac {2}{5} x^{5} d c b B + \frac {1}{5} x^{5} e b^{2} B + \frac {1}{5} x^{5} d c^{2} A + \frac {2}{5} x^{5} e c b A + \frac {1}{4} x^{4} d b^{2} B + \frac {1}{2} x^{4} d c b A + \frac {1}{4} x^{4} e b^{2} A + \frac {1}{3} x^{3} d b^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 123, normalized size = 1.23 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} e + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{3} \, B b c x^{6} e + \frac {1}{6} \, A c^{2} x^{6} e + \frac {2}{5} \, B b c d x^{5} + \frac {1}{5} \, A c^{2} d x^{5} + \frac {1}{5} \, B b^{2} x^{5} e + \frac {2}{5} \, A b c x^{5} e + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, A b c d x^{4} + \frac {1}{4} \, A b^{2} x^{4} e + \frac {1}{3} \, A b^{2} d x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 97, normalized size = 0.97 \begin {gather*} \frac {B \,c^{2} e \,x^{7}}{7}+\frac {A \,b^{2} d \,x^{3}}{3}+\frac {\left (2 B b c e +\left (A e +B d \right ) c^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d +B \,b^{2} e +2 \left (A e +B d \right ) b c \right ) x^{5}}{5}+\frac {\left (2 A b c d +\left (A e +B d \right ) b^{2}\right ) x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 103, normalized size = 1.03 \begin {gather*} \frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {1}{6} \, {\left (B c^{2} d + {\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d + {\left (B b^{2} + 2 \, A b c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left (A b^{2} e + {\left (B b^{2} + 2 \, A b c\right )} d\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 102, normalized size = 1.02 \begin {gather*} x^5\,\left (\frac {A\,c^2\,d}{5}+\frac {B\,b^2\,e}{5}+\frac {2\,A\,b\,c\,e}{5}+\frac {2\,B\,b\,c\,d}{5}\right )+x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {A\,b\,c\,d}{2}\right )+x^6\,\left (\frac {A\,c^2\,e}{6}+\frac {B\,c^2\,d}{6}+\frac {B\,b\,c\,e}{3}\right )+\frac {A\,b^2\,d\,x^3}{3}+\frac {B\,c^2\,e\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 121, normalized size = 1.21 \begin {gather*} \frac {A b^{2} d x^{3}}{3} + \frac {B c^{2} e x^{7}}{7} + x^{6} \left (\frac {A c^{2} e}{6} + \frac {B b c e}{3} + \frac {B c^{2} d}{6}\right ) + x^{5} \left (\frac {2 A b c e}{5} + \frac {A c^{2} d}{5} + \frac {B b^{2} e}{5} + \frac {2 B b c d}{5}\right ) + x^{4} \left (\frac {A b^{2} e}{4} + \frac {A b c d}{2} + \frac {B b^{2} d}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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