Optimal. Leaf size=304 \[ -\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right ) (d+e x)^5}{5 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^6}{6 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^7}{7 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^8}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6} \]
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Rubi [A]
time = 0.33, 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 = {785}
\begin {gather*} -\frac {(d+e x)^7 \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 )}{7 e^6}-\frac {(d+e x)^6 \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 )}{6 e^6}+\frac {(d+e x)^5 \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 )}{5 e^6}-\frac {(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac {c (d+e x)^8 (-A c e-2 b B e+5 B c d)}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^3 \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)^3}{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)^4}{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)^5}{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)^6}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^7}{e^5}+\frac {B c^2 (d+e x)^8}{e^5}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 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)^5}{5 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)^6}{6 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)^7}{7 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^8}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 421, normalized size = 1.38 \begin {gather*} a^2 A d^3 x+\frac {1}{2} a d^2 (2 A b d+a B d+3 a A e) x^2+\frac {1}{3} d \left (a B d (2 b d+3 a e)+A \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right )\right ) x^3+\frac {1}{4} \left (b^2 d^2 (B d+3 A e)+2 b d \left (A c d^2+3 a B d e+3 a A e^2\right )+a \left (2 B c d^3+6 A c d^2 e+3 a B d e^2+a A e^3\right )\right ) x^4+\frac {1}{5} \left (3 b^2 d e (B d+A e)+a B e \left (6 c d^2+a e^2\right )+A c d \left (c d^2+6 a e^2\right )+2 b \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right )\right ) x^5+\frac {1}{6} \left (A e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right )+B \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right )\right ) x^6+\frac {1}{7} e \left (A c e (3 c d+2 b e)+B \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right )\right ) x^7+\frac {1}{8} c e^2 (3 B c d+2 b B e+A c e) x^8+\frac {1}{9} B c^2 e^3 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 419, normalized size = 1.38 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 438, normalized size = 1.44 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} e^{3} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + 2 \, B b c e^{3} + A c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + B b^{2} e^{3} + 2 \, {\left (B a e^{3} + A b e^{3}\right )} c + 3 \, {\left (2 \, B b c e^{2} + A c^{2} e^{2}\right )} d\right )} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (B c^{2} d^{3} + 2 \, B a b e^{3} + A b^{2} e^{3} + 2 \, A a c e^{3} + 3 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{2} + 3 \, {\left (B b^{2} e^{2} + 2 \, {\left (B a e^{2} + A b e^{2}\right )} c\right )} d\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d^{3} + B a^{2} e^{3} + 2 \, A a b e^{3} + 3 \, {\left (B b^{2} e + 2 \, {\left (B a e + A b e\right )} c\right )} d^{2} + 3 \, {\left (2 \, B a b e^{2} + A b^{2} e^{2} + 2 \, A a c e^{2}\right )} d\right )} x^{5} + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} + A a^{2} e^{3} + 3 \, {\left (2 \, B a b e + A b^{2} e + 2 \, A a c e\right )} d^{2} + 3 \, {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} d e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} + 3 \, {\left (B a^{2} e + 2 \, A a b e\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{2} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.59, size = 423, normalized size = 1.39 \begin {gather*} \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} x^{4} + A a^{2} d^{3} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} x^{2} + \frac {1}{2520} \, {\left (280 \, B c^{2} x^{9} + 315 \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + 360 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{7} + 630 \, A a^{2} x^{4} + 420 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{6} + 504 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} e^{3} + \frac {1}{280} \, {\left (105 \, B c^{2} d x^{8} + 120 \, {\left (2 \, B b c + A c^{2}\right )} d x^{7} + 140 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d x^{6} + 280 \, A a^{2} d x^{3} + 168 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d x^{5} + 210 \, {\left (B a^{2} + 2 \, A a b\right )} d x^{4}\right )} e^{2} + \frac {1}{140} \, {\left (60 \, B c^{2} d^{2} x^{7} + 70 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{6} + 84 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} x^{5} + 210 \, A a^{2} d^{2} x^{2} + 105 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} x^{4} + 140 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} x^{3}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 583, normalized size = 1.92 \begin {gather*} A a^{2} d^{3} x + \frac {B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{2} e^{3}}{8} + \frac {B b c e^{3}}{4} + \frac {3 B c^{2} d e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {2 A b c e^{3}}{7} + \frac {3 A c^{2} d e^{2}}{7} + \frac {2 B a c e^{3}}{7} + \frac {B b^{2} e^{3}}{7} + \frac {6 B b c d e^{2}}{7} + \frac {3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A a c e^{3}}{3} + \frac {A b^{2} e^{3}}{6} + A b c d e^{2} + \frac {A c^{2} d^{2} e}{2} + \frac {B a b e^{3}}{3} + B a c d e^{2} + \frac {B b^{2} d e^{2}}{2} + B b c d^{2} e + \frac {B c^{2} d^{3}}{6}\right ) + x^{5} \cdot \left (\frac {2 A a b e^{3}}{5} + \frac {6 A a c d e^{2}}{5} + \frac {3 A b^{2} d e^{2}}{5} + \frac {6 A b c d^{2} e}{5} + \frac {A c^{2} d^{3}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a b d e^{2}}{5} + \frac {6 B a c d^{2} e}{5} + \frac {3 B b^{2} d^{2} e}{5} + \frac {2 B b c d^{3}}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a b d e^{2}}{2} + \frac {3 A a c d^{2} e}{2} + \frac {3 A b^{2} d^{2} e}{4} + \frac {A b c d^{3}}{2} + \frac {3 B a^{2} d e^{2}}{4} + \frac {3 B a b d^{2} e}{2} + \frac {B a c d^{3}}{2} + \frac {B b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + 2 A a b d^{2} e + \frac {2 A a c d^{3}}{3} + \frac {A b^{2} d^{3}}{3} + B a^{2} d^{2} e + \frac {2 B a b d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} d^{2} e}{2} + A a b d^{3} + \frac {B a^{2} d^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 556, normalized size = 1.83 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} e^{3} + \frac {3}{8} \, B c^{2} d x^{8} e^{2} + \frac {3}{7} \, B c^{2} d^{2} x^{7} e + \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{4} \, B b c x^{8} e^{3} + \frac {1}{8} \, A c^{2} x^{8} e^{3} + \frac {6}{7} \, B b c d x^{7} e^{2} + \frac {3}{7} \, A c^{2} d x^{7} e^{2} + B b c d^{2} x^{6} e + \frac {1}{2} \, A c^{2} d^{2} x^{6} e + \frac {2}{5} \, B b c d^{3} x^{5} + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {1}{7} \, B b^{2} x^{7} e^{3} + \frac {2}{7} \, B a c x^{7} e^{3} + \frac {2}{7} \, A b c x^{7} e^{3} + \frac {1}{2} \, B b^{2} d x^{6} e^{2} + B a c d x^{6} e^{2} + A b c d x^{6} e^{2} + \frac {3}{5} \, B b^{2} d^{2} x^{5} e + \frac {6}{5} \, B a c d^{2} x^{5} e + \frac {6}{5} \, A b c d^{2} x^{5} e + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {1}{2} \, B a c d^{3} x^{4} + \frac {1}{2} \, A b c d^{3} x^{4} + \frac {1}{3} \, B a b x^{6} e^{3} + \frac {1}{6} \, A b^{2} x^{6} e^{3} + \frac {1}{3} \, A a c x^{6} e^{3} + \frac {6}{5} \, B a b d x^{5} e^{2} + \frac {3}{5} \, A b^{2} d x^{5} e^{2} + \frac {6}{5} \, A a c d x^{5} e^{2} + \frac {3}{2} \, B a b d^{2} x^{4} e + \frac {3}{4} \, A b^{2} d^{2} x^{4} e + \frac {3}{2} \, A a c d^{2} x^{4} e + \frac {2}{3} \, B a b d^{3} x^{3} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {2}{3} \, A a c d^{3} x^{3} + \frac {1}{5} \, B a^{2} x^{5} e^{3} + \frac {2}{5} \, A a b x^{5} e^{3} + \frac {3}{4} \, B a^{2} d x^{4} e^{2} + \frac {3}{2} \, A a b d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + 2 \, A a b d^{2} x^{3} e + \frac {1}{2} \, B a^{2} d^{3} x^{2} + A a b d^{3} x^{2} + \frac {1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac {3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 450, normalized size = 1.48 \begin {gather*} x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,b\,d\,e^2}{5}+\frac {2\,A\,a\,b\,e^3}{5}+\frac {6\,B\,a\,c\,d^2\,e}{5}+\frac {6\,A\,a\,c\,d\,e^2}{5}+\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}+\frac {2\,B\,b\,c\,d^3}{5}+\frac {6\,A\,b\,c\,d^2\,e}{5}+\frac {A\,c^2\,d^3}{5}\right )+x^3\,\left (B\,a^2\,d^2\,e+A\,a^2\,d\,e^2+\frac {2\,B\,a\,b\,d^3}{3}+2\,A\,a\,b\,d^2\,e+\frac {2\,A\,c\,a\,d^3}{3}+\frac {A\,b^2\,d^3}{3}\right )+x^7\,\left (\frac {B\,b^2\,e^3}{7}+\frac {6\,B\,b\,c\,d\,e^2}{7}+\frac {2\,A\,b\,c\,e^3}{7}+\frac {3\,B\,c^2\,d^2\,e}{7}+\frac {3\,A\,c^2\,d\,e^2}{7}+\frac {2\,B\,a\,c\,e^3}{7}\right )+x^4\,\left (\frac {3\,B\,a^2\,d\,e^2}{4}+\frac {A\,a^2\,e^3}{4}+\frac {3\,B\,a\,b\,d^2\,e}{2}+\frac {3\,A\,a\,b\,d\,e^2}{2}+\frac {B\,c\,a\,d^3}{2}+\frac {3\,A\,c\,a\,d^2\,e}{2}+\frac {B\,b^2\,d^3}{4}+\frac {3\,A\,b^2\,d^2\,e}{4}+\frac {A\,c\,b\,d^3}{2}\right )+x^6\,\left (\frac {B\,b^2\,d\,e^2}{2}+\frac {A\,b^2\,e^3}{6}+B\,b\,c\,d^2\,e+A\,b\,c\,d\,e^2+\frac {B\,a\,b\,e^3}{3}+\frac {B\,c^2\,d^3}{6}+\frac {A\,c^2\,d^2\,e}{2}+B\,a\,c\,d\,e^2+\frac {A\,a\,c\,e^3}{3}\right )+A\,a^2\,d^3\,x+\frac {a\,d^2\,x^2\,\left (3\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e^2\,x^8\,\left (A\,c\,e+2\,B\,b\,e+3\,B\,c\,d\right )}{8}+\frac {B\,c^2\,e^3\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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