Optimal. Leaf size=180 \[ a^2 A d x+\frac{1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac{1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{7} B c^2 e x^7 \]
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Rubi [A] time = 0.297387, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ a^2 A d x+\frac{1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac{1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{7} B c^2 e x^7 \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A d+a (2 A b d+a B d+a A e) x+\left (a B (2 b d+a e)+A \left (b^2 d+2 a c d+2 a b e\right )\right ) x^2+\left (b^2 (B d+A e)+2 a c (B d+A e)+2 b (A c d+a B e)\right ) x^3+\left (b^2 B e+2 b c (B d+A e)+c (A c d+2 a B 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\\ &=a^2 A d x+\frac{1}{2} a (2 A b d+a B d+a A e) x^2+\frac{1}{3} \left (a B (2 b d+a e)+A \left (b^2 d+2 a c d+2 a b e\right )\right ) x^3+\frac{1}{4} \left (b^2 (B d+A e)+2 a c (B d+A e)+2 b (A c d+a B e)\right ) x^4+\frac{1}{5} \left (b^2 B e+2 b c (B d+A e)+c (A c d+2 a B 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*}
Mathematica [A] time = 0.0848326, size = 180, normalized size = 1. \[ a^2 A d x+\frac{1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac{1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{7} B c^2 e x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 167, normalized size = 0.9 \begin{align*}{\frac{B{c}^{2}e{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{2}+2\,Bebc \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}d+2\,bc \left ( Ae+Bd \right ) +Be \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abcd+ \left ( Ae+Bd \right ) \left ( 2\,ac+{b}^{2} \right ) +2\,Beab \right ){x}^{4}}{4}}+{\frac{ \left ( Ad \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( Ae+Bd \right ) ab+Be{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Adab+ \left ( Ae+Bd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{2}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989675, size = 248, normalized size = 1.38 \begin{align*} \frac{1}{7} \, B c^{2} e x^{7} + \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 \,{\left (B a + A b\right )} c\right )} e\right )} x^{5} + A a^{2} d x + \frac{1}{4} \,{\left ({\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d +{\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11372, size = 568, normalized size = 3.16 \begin{align*} \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{2}{5} x^{5} e c a 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 a B + \frac{1}{2} x^{4} e b a B + \frac{1}{2} x^{4} d c b A + \frac{1}{4} x^{4} e b^{2} A + \frac{1}{2} x^{4} e c a A + \frac{2}{3} x^{3} d b a B + \frac{1}{3} x^{3} e a^{2} B + \frac{1}{3} x^{3} d b^{2} A + \frac{2}{3} x^{3} d c a A + \frac{2}{3} x^{3} e b a A + \frac{1}{2} x^{2} d a^{2} B + x^{2} d b a A + \frac{1}{2} x^{2} e a^{2} A + x d a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.093143, size = 231, normalized size = 1.28 \begin{align*} A a^{2} d x + \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{2 B a c e}{5} + \frac{B b^{2} e}{5} + \frac{2 B b c d}{5}\right ) + x^{4} \left (\frac{A a c e}{2} + \frac{A b^{2} e}{4} + \frac{A b c d}{2} + \frac{B a b e}{2} + \frac{B a c d}{2} + \frac{B b^{2} d}{4}\right ) + x^{3} \left (\frac{2 A a b e}{3} + \frac{2 A a c d}{3} + \frac{A b^{2} d}{3} + \frac{B a^{2} e}{3} + \frac{2 B a b d}{3}\right ) + x^{2} \left (\frac{A a^{2} e}{2} + A a b d + \frac{B a^{2} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12674, size = 320, normalized size = 1.78 \begin{align*} \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} \, B a c x^{5} e + \frac{2}{5} \, A b c x^{5} e + \frac{1}{4} \, B b^{2} d x^{4} + \frac{1}{2} \, B a c d x^{4} + \frac{1}{2} \, A b c d x^{4} + \frac{1}{2} \, B a b x^{4} e + \frac{1}{4} \, A b^{2} x^{4} e + \frac{1}{2} \, A a c x^{4} e + \frac{2}{3} \, B a b d x^{3} + \frac{1}{3} \, A b^{2} d x^{3} + \frac{2}{3} \, A a c d x^{3} + \frac{1}{3} \, B a^{2} x^{3} e + \frac{2}{3} \, A a b x^{3} e + \frac{1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac{1}{2} \, A a^{2} x^{2} e + A a^{2} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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