Optimal. Leaf size=304 \[ -\frac{(d+e x)^7 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{7 e^6}-\frac{(d+e x)^6 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{6 e^6}-\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{5 e^6}-\frac{(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac{c (d+e x)^8 (-A c e-2 b B e+5 B c d)}{8 e^6}+\frac{B c^2 (d+e x)^9}{9 e^6} \]
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Rubi [A] time = 0.494012, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{(d+e x)^7 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{7 e^6}-\frac{(d+e x)^6 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{6 e^6}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{5 e^6}-\frac{(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac{c (d+e x)^8 (-A c e-2 b B e+5 B c d)}{8 e^6}+\frac{B c^2 (d+e x)^9}{9 e^6} \]
Antiderivative was successfully verified.
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Rule 771
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*}
Mathematica [A] time = 0.20943, size = 421, normalized size = 1.38 \[ a^2 A d^3 x+\frac{1}{7} e x^7 \left (B \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+A c e (2 b e+3 c d)\right )+\frac{1}{6} x^6 \left (A e \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+B \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )\right )+\frac{1}{5} x^5 \left (2 b \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+A c d \left (6 a e^2+c d^2\right )+a B e \left (a e^2+6 c d^2\right )+3 b^2 d e (A e+B d)\right )+\frac{1}{4} x^4 \left (2 b d \left (3 a A e^2+3 a B d e+A c d^2\right )+a \left (a A e^3+3 a B d e^2+6 A c d^2 e+2 B c d^3\right )+b^2 d^2 (3 A e+B d)\right )+\frac{1}{3} d x^3 \left (A \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+a B d (3 a e+2 b d)\right )+\frac{1}{2} a d^2 x^2 (3 a A e+a B d+2 A b d)+\frac{1}{8} c e^2 x^8 (A c e+2 b B e+3 B c d)+\frac{1}{9} B c^2 e^3 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 419, normalized size = 1.4 \begin{align*}{\frac{B{c}^{2}{e}^{3}{x}^{9}}{9}}+{\frac{ \left ( \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){c}^{2}+2\,B{e}^{3}bc \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){c}^{2}+2\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) bc+B{e}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){c}^{2}+2\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) bc+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\,B{e}^{3}ab \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{3}{c}^{2}+2\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) bc+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) ab+{a}^{2}B{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,A{d}^{3}bc+ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) ab+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{3} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) ab+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{3}ab+ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{3}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00016, size = 562, normalized size = 1.85 \begin{align*} \frac{1}{9} \, B c^{2} e^{3} x^{9} + \frac{1}{8} \,{\left (3 \, B c^{2} d e^{2} +{\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, B c^{2} d^{2} e + 3 \,{\left (2 \, B b c + A c^{2}\right )} d e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac{1}{6} \,{\left (B c^{2} d^{3} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left ({\left (2 \, B b c + A c^{2}\right )} d^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e + 3 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{2} e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} + 3 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\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} + 2 \, A a b\right )} d^{2} e\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18228, size = 1316, normalized size = 4.33 \begin{align*} \frac{1}{9} x^{9} e^{3} c^{2} B + \frac{3}{8} x^{8} e^{2} d c^{2} B + \frac{1}{4} x^{8} e^{3} c b B + \frac{1}{8} x^{8} e^{3} c^{2} A + \frac{3}{7} x^{7} e d^{2} c^{2} B + \frac{6}{7} x^{7} e^{2} d c b B + \frac{1}{7} x^{7} e^{3} b^{2} B + \frac{2}{7} x^{7} e^{3} c a B + \frac{3}{7} x^{7} e^{2} d c^{2} A + \frac{2}{7} x^{7} e^{3} c b A + \frac{1}{6} x^{6} d^{3} c^{2} B + x^{6} e d^{2} c b B + \frac{1}{2} x^{6} e^{2} d b^{2} B + x^{6} e^{2} d c a B + \frac{1}{3} x^{6} e^{3} b a B + \frac{1}{2} x^{6} e d^{2} c^{2} A + x^{6} e^{2} d c b A + \frac{1}{6} x^{6} e^{3} b^{2} A + \frac{1}{3} x^{6} e^{3} c a A + \frac{2}{5} x^{5} d^{3} c b B + \frac{3}{5} x^{5} e d^{2} b^{2} B + \frac{6}{5} x^{5} e d^{2} c a B + \frac{6}{5} x^{5} e^{2} d b a B + \frac{1}{5} x^{5} e^{3} a^{2} B + \frac{1}{5} x^{5} d^{3} c^{2} A + \frac{6}{5} x^{5} e d^{2} c b A + \frac{3}{5} x^{5} e^{2} d b^{2} A + \frac{6}{5} x^{5} e^{2} d c a A + \frac{2}{5} x^{5} e^{3} b a A + \frac{1}{4} x^{4} d^{3} b^{2} B + \frac{1}{2} x^{4} d^{3} c a B + \frac{3}{2} x^{4} e d^{2} b a B + \frac{3}{4} x^{4} e^{2} d a^{2} B + \frac{1}{2} x^{4} d^{3} c b A + \frac{3}{4} x^{4} e d^{2} b^{2} A + \frac{3}{2} x^{4} e d^{2} c a A + \frac{3}{2} x^{4} e^{2} d b a A + \frac{1}{4} x^{4} e^{3} a^{2} A + \frac{2}{3} x^{3} d^{3} b a B + x^{3} e d^{2} a^{2} B + \frac{1}{3} x^{3} d^{3} b^{2} A + \frac{2}{3} x^{3} d^{3} c a A + 2 x^{3} e d^{2} b a A + x^{3} e^{2} d a^{2} A + \frac{1}{2} x^{2} d^{3} a^{2} B + x^{2} d^{3} b a A + \frac{3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.130991, size = 583, normalized size = 1.92 \begin{align*} 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} \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} \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} \left (\frac{3 A a^{2} d^{2} e}{2} + A a b d^{3} + \frac{B a^{2} d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15467, size = 751, normalized size = 2.47 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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