Optimal. Leaf size=162 \[ \frac {1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac {1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^2 x^3+\frac {1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac {1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac {1}{8} B c^2 e^2 x^8 \]
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Rubi [A] time = 0.22, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac {1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^2 x^3+\frac {1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac {1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac {1}{8} B c^2 e^2 x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 d^2 x^2+b d (b B d+2 A c d+2 A b e) x^3+\left (A c^2 d^2+b^2 e (2 B d+A e)+2 b c d (B d+2 A e)\right ) x^4+\left (2 A c e (c d+b e)+B \left (c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^5+c e (A c e+2 B (c d+b e)) x^6+B c^2 e^2 x^7\right ) \, dx\\ &=\frac {1}{3} A b^2 d^2 x^3+\frac {1}{4} b d (b B d+2 A c d+2 A b e) x^4+\frac {1}{5} \left (A c^2 d^2+b^2 e (2 B d+A e)+2 b c d (B d+2 A e)\right ) x^5+\frac {1}{6} \left (2 A c e (c d+b e)+B \left (c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^6+\frac {1}{7} c e (A c e+2 B (c d+b e)) x^7+\frac {1}{8} B c^2 e^2 x^8\\ \end {align*}
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Mathematica [A] time = 0.05, size = 162, normalized size = 1.00 \begin {gather*} \frac {1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac {1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^2 x^3+\frac {1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac {1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac {1}{8} B c^2 e^2 x^8 \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)^2 \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 = 0.35, size = 205, normalized size = 1.27 \begin {gather*} \frac {1}{8} x^{8} e^{2} c^{2} B + \frac {2}{7} x^{7} e d c^{2} B + \frac {2}{7} x^{7} e^{2} c b B + \frac {1}{7} x^{7} e^{2} c^{2} A + \frac {1}{6} x^{6} d^{2} c^{2} B + \frac {2}{3} x^{6} e d c b B + \frac {1}{6} x^{6} e^{2} b^{2} B + \frac {1}{3} x^{6} e d c^{2} A + \frac {1}{3} x^{6} e^{2} c b A + \frac {2}{5} x^{5} d^{2} c b B + \frac {2}{5} x^{5} e d b^{2} B + \frac {1}{5} x^{5} d^{2} c^{2} A + \frac {4}{5} x^{5} e d c b A + \frac {1}{5} x^{5} e^{2} b^{2} A + \frac {1}{4} x^{4} d^{2} b^{2} B + \frac {1}{2} x^{4} d^{2} c b A + \frac {1}{2} x^{4} e d b^{2} A + \frac {1}{3} x^{3} d^{2} 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 = 205, normalized size = 1.27 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} e^{2} + \frac {2}{7} \, B c^{2} d x^{7} e + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {2}{7} \, B b c x^{7} e^{2} + \frac {1}{7} \, A c^{2} x^{7} e^{2} + \frac {2}{3} \, B b c d x^{6} e + \frac {1}{3} \, A c^{2} d x^{6} e + \frac {2}{5} \, B b c d^{2} x^{5} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {1}{6} \, B b^{2} x^{6} e^{2} + \frac {1}{3} \, A b c x^{6} e^{2} + \frac {2}{5} \, B b^{2} d x^{5} e + \frac {4}{5} \, A b c d x^{5} e + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {1}{2} \, A b c d^{2} x^{4} + \frac {1}{5} \, A b^{2} x^{5} e^{2} + \frac {1}{2} \, A b^{2} d x^{4} e + \frac {1}{3} \, A b^{2} d^{2} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 172, normalized size = 1.06 \begin {gather*} \frac {B \,c^{2} e^{2} x^{8}}{8}+\frac {A \,b^{2} d^{2} x^{3}}{3}+\frac {\left (2 B b c \,e^{2}+\left (A \,e^{2}+2 B d e \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (B \,b^{2} e^{2}+2 \left (A \,e^{2}+2 B d e \right ) b c +\left (2 A d e +B \,d^{2}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{2}+\left (A \,e^{2}+2 B d e \right ) b^{2}+2 \left (2 A d e +B \,d^{2}\right ) b c \right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{2}+\left (2 A d e +B \,d^{2}\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 = 0.64, size = 171, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (A b^{2} e^{2} + {\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} d e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, A b^{2} d e + {\left (B b^{2} + 2 \, A b c\right )} d^{2}\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 161, normalized size = 0.99 \begin {gather*} x^5\,\left (\frac {2\,B\,b^2\,d\,e}{5}+\frac {A\,b^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^2}{5}+\frac {4\,A\,b\,c\,d\,e}{5}+\frac {A\,c^2\,d^2}{5}\right )+x^6\,\left (\frac {B\,b^2\,e^2}{6}+\frac {2\,B\,b\,c\,d\,e}{3}+\frac {A\,b\,c\,e^2}{3}+\frac {B\,c^2\,d^2}{6}+\frac {A\,c^2\,d\,e}{3}\right )+\frac {b\,d\,x^4\,\left (2\,A\,b\,e+2\,A\,c\,d+B\,b\,d\right )}{4}+\frac {c\,e\,x^7\,\left (A\,c\,e+2\,B\,b\,e+2\,B\,c\,d\right )}{7}+\frac {A\,b^2\,d^2\,x^3}{3}+\frac {B\,c^2\,e^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 212, normalized size = 1.31 \begin {gather*} \frac {A b^{2} d^{2} x^{3}}{3} + \frac {B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B b c e^{2}}{7} + \frac {2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac {A b c e^{2}}{3} + \frac {A c^{2} d e}{3} + \frac {B b^{2} e^{2}}{6} + \frac {2 B b c d e}{3} + \frac {B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac {A b^{2} e^{2}}{5} + \frac {4 A b c d e}{5} + \frac {A c^{2} d^{2}}{5} + \frac {2 B b^{2} d e}{5} + \frac {2 B b c d^{2}}{5}\right ) + x^{4} \left (\frac {A b^{2} d e}{2} + \frac {A b c d^{2}}{2} + \frac {B b^{2} d^{2}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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