Optimal. Leaf size=304 \[ -\frac {(d+e x)^6 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{6 e^6}-\frac {(d+e x)^5 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{5 e^6}-\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{4 e^6}-\frac {(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {c (d+e x)^7 (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {B c^2 (d+e x)^8}{8 e^6} \]
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Rubi [A] time = 0.44, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \[ -\frac {(d+e x)^6 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{6 e^6}-\frac {(d+e x)^5 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{5 e^6}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{4 e^6}-\frac {(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {c (d+e x)^7 (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {B c^2 (d+e x)^8}{8 e^6} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}{e^5}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right ) (d+e x)^3}{e^5}+\frac {\left (-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^4}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^5}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^6}{e^5}+\frac {B c^2 (d+e x)^7}{e^5}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^6}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right ) (d+e x)^4}{4 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^5}{5 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^6}{6 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^7}{7 e^6}+\frac {B c^2 (d+e x)^8}{8 e^6}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 301, normalized size = 0.99 \[ a^2 A d^2 x+\frac {1}{6} x^6 \left (B \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+2 A c e (b e+c d)\right )+\frac {1}{5} x^5 \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )+c \left (2 a A e^2+4 a B d e+A c d^2\right )+b^2 e (A e+2 B d)\right )+\frac {1}{4} x^4 \left (2 b \left (a A e^2+2 a B d e+A c d^2\right )+a \left (a B e^2+4 A c d e+2 B c d^2\right )+b^2 d (2 A e+B d)\right )+\frac {1}{3} x^3 \left (A \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac {1}{2} a d x^2 (2 A (a e+b d)+a B d)+\frac {1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac {1}{8} B c^2 e^2 x^8 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 396, normalized size = 1.30 \[ \frac {1}{8} x^{8} e^{2} c^{2} B + \frac {2}{7} x^{7} e d c^{2} B + \frac {2}{7} x^{7} e^{2} c b B + \frac {1}{7} x^{7} e^{2} c^{2} A + \frac {1}{6} x^{6} d^{2} c^{2} B + \frac {2}{3} x^{6} e d c b B + \frac {1}{6} x^{6} e^{2} b^{2} B + \frac {1}{3} x^{6} e^{2} c a B + \frac {1}{3} x^{6} e d c^{2} A + \frac {1}{3} x^{6} e^{2} c b A + \frac {2}{5} x^{5} d^{2} c b B + \frac {2}{5} x^{5} e d b^{2} B + \frac {4}{5} x^{5} e d c a B + \frac {2}{5} x^{5} e^{2} b a B + \frac {1}{5} x^{5} d^{2} c^{2} A + \frac {4}{5} x^{5} e d c b A + \frac {1}{5} x^{5} e^{2} b^{2} A + \frac {2}{5} x^{5} e^{2} c a A + \frac {1}{4} x^{4} d^{2} b^{2} B + \frac {1}{2} x^{4} d^{2} c a B + x^{4} e d b a B + \frac {1}{4} x^{4} e^{2} a^{2} B + \frac {1}{2} x^{4} d^{2} c b A + \frac {1}{2} x^{4} e d b^{2} A + x^{4} e d c a A + \frac {1}{2} x^{4} e^{2} b a A + \frac {2}{3} x^{3} d^{2} b a B + \frac {2}{3} x^{3} e d a^{2} B + \frac {1}{3} x^{3} d^{2} b^{2} A + \frac {2}{3} x^{3} d^{2} c a A + \frac {4}{3} x^{3} e d b a A + \frac {1}{3} x^{3} e^{2} a^{2} A + \frac {1}{2} x^{2} d^{2} a^{2} B + x^{2} d^{2} b a A + x^{2} e d a^{2} A + x d^{2} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 396, normalized size = 1.30 \[ \frac {1}{8} \, B c^{2} x^{8} e^{2} + \frac {2}{7} \, B c^{2} d x^{7} e + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {2}{7} \, B b c x^{7} e^{2} + \frac {1}{7} \, A c^{2} x^{7} e^{2} + \frac {2}{3} \, B b c d x^{6} e + \frac {1}{3} \, A c^{2} d x^{6} e + \frac {2}{5} \, B b c d^{2} x^{5} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {1}{6} \, B b^{2} x^{6} e^{2} + \frac {1}{3} \, B a c x^{6} e^{2} + \frac {1}{3} \, A b c x^{6} e^{2} + \frac {2}{5} \, B b^{2} d x^{5} e + \frac {4}{5} \, B a c d x^{5} e + \frac {4}{5} \, A b c d x^{5} e + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {1}{2} \, B a c d^{2} x^{4} + \frac {1}{2} \, A b c d^{2} x^{4} + \frac {2}{5} \, B a b x^{5} e^{2} + \frac {1}{5} \, A b^{2} x^{5} e^{2} + \frac {2}{5} \, A a c x^{5} e^{2} + B a b d x^{4} e + \frac {1}{2} \, A b^{2} d x^{4} e + A a c d x^{4} e + \frac {2}{3} \, B a b d^{2} x^{3} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {2}{3} \, A a c d^{2} x^{3} + \frac {1}{4} \, B a^{2} x^{4} e^{2} + \frac {1}{2} \, A a b x^{4} e^{2} + \frac {2}{3} \, B a^{2} d x^{3} e + \frac {4}{3} \, A a b d x^{3} e + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + \frac {1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 293, normalized size = 0.96 \[ \frac {B \,c^{2} e^{2} x^{8}}{8}+\frac {\left (2 B b c \,e^{2}+\left (A \,e^{2}+2 B d e \right ) c^{2}\right ) x^{7}}{7}+A \,a^{2} d^{2} x +\frac {\left (\left (2 a c +b^{2}\right ) B \,e^{2}+2 \left (A \,e^{2}+2 B d e \right ) b c +\left (2 A d e +B \,d^{2}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{2}+2 B a b \,e^{2}+2 \left (2 A d e +B \,d^{2}\right ) b c +\left (A \,e^{2}+2 B d e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{2}+B \,a^{2} e^{2}+2 \left (A \,e^{2}+2 B d e \right ) a b +\left (2 A d e +B \,d^{2}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) A \,d^{2}+\left (A \,e^{2}+2 B d e \right ) a^{2}+2 \left (2 A d e +B \,d^{2}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{2}+\left (2 A d e +B \,d^{2}\right ) a^{2}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 300, normalized size = 0.99 \[ \frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} + 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.33, size = 310, normalized size = 1.02 \[ x^3\,\left (\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {4\,A\,a\,b\,d\,e}{3}+\frac {2\,A\,c\,a\,d^2}{3}+\frac {A\,b^2\,d^2}{3}\right )+x^6\,\left (\frac {B\,b^2\,e^2}{6}+\frac {2\,B\,b\,c\,d\,e}{3}+\frac {A\,b\,c\,e^2}{3}+\frac {B\,c^2\,d^2}{6}+\frac {A\,c^2\,d\,e}{3}+\frac {B\,a\,c\,e^2}{3}\right )+x^4\,\left (\frac {B\,a^2\,e^2}{4}+B\,a\,b\,d\,e+\frac {A\,a\,b\,e^2}{2}+\frac {B\,c\,a\,d^2}{2}+A\,c\,a\,d\,e+\frac {B\,b^2\,d^2}{4}+\frac {A\,b^2\,d\,e}{2}+\frac {A\,c\,b\,d^2}{2}\right )+x^5\,\left (\frac {2\,B\,b^2\,d\,e}{5}+\frac {A\,b^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^2}{5}+\frac {4\,A\,b\,c\,d\,e}{5}+\frac {2\,B\,a\,b\,e^2}{5}+\frac {A\,c^2\,d^2}{5}+\frac {4\,B\,a\,c\,d\,e}{5}+\frac {2\,A\,a\,c\,e^2}{5}\right )+\frac {a\,d\,x^2\,\left (2\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e\,x^7\,\left (A\,c\,e+2\,B\,b\,e+2\,B\,c\,d\right )}{7}+A\,a^2\,d^2\,x+\frac {B\,c^2\,e^2\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 405, normalized size = 1.33 \[ A a^{2} d^{2} x + \frac {B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B b c e^{2}}{7} + \frac {2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac {A b c e^{2}}{3} + \frac {A c^{2} d e}{3} + \frac {B a c e^{2}}{3} + \frac {B b^{2} e^{2}}{6} + \frac {2 B b c d e}{3} + \frac {B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac {2 A a c e^{2}}{5} + \frac {A b^{2} e^{2}}{5} + \frac {4 A b c d e}{5} + \frac {A c^{2} d^{2}}{5} + \frac {2 B a b e^{2}}{5} + \frac {4 B a c d e}{5} + \frac {2 B b^{2} d e}{5} + \frac {2 B b c d^{2}}{5}\right ) + x^{4} \left (\frac {A a b e^{2}}{2} + A a c d e + \frac {A b^{2} d e}{2} + \frac {A b c d^{2}}{2} + \frac {B a^{2} e^{2}}{4} + B a b d e + \frac {B a c d^{2}}{2} + \frac {B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {4 A a b d e}{3} + \frac {2 A a c d^{2}}{3} + \frac {A b^{2} d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac {B a^{2} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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