Optimal. Leaf size=272 \[ \frac{3 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7}+\frac{3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{3 (d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7} \]
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Rubi [A] time = 0.271228, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{3 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7}+\frac{3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{3 (d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}{e^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{e^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^4}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^5}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^7}{e^6}+\frac{c^3 (d+e x)^8}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}{3 e^7}-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^7}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^5}{5 e^7}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^6}{6 e^7}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^7}-\frac{3 c^2 (2 c d-b e) (d+e x)^8}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7}\\ \end{align*}
Mathematica [A] time = 0.0948533, size = 282, normalized size = 1.04 \[ \frac{1}{4} x^4 \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+\frac{1}{2} a^2 d x^2 (2 a e+3 b d)+a^3 d^2 x+\frac{1}{7} c x^7 \left (3 c e (a e+2 b d)+3 b^2 e^2+c^2 d^2\right )+\frac{1}{6} x^6 \left (3 b c \left (2 a e^2+c d^2\right )+6 a c^2 d e+6 b^2 c d e+b^3 e^2\right )+\frac{1}{5} x^5 \left (3 b^2 \left (a e^2+c d^2\right )+12 a b c d e+3 a c \left (a e^2+c d^2\right )+2 b^3 d e\right )+\frac{1}{3} a x^3 \left (6 a b d e+a \left (a e^2+3 c d^2\right )+3 b^2 d^2\right )+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 359, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{ \left ( 3\,{e}^{2}b{c}^{2}+2\,de{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ({d}^{2}{c}^{3}+6\,deb{c}^{2}+{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{2}b{c}^{2}+2\,de \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,de \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,de \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +3\,{a}^{2}b{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +6\,deb{a}^{2}+{e}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{3}+3\,{d}^{2}b{a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978291, size = 371, normalized size = 1.36 \begin{align*} \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{8} \,{\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, b c^{2} d^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e +{\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{6} + a^{3} d^{2} x + \frac{1}{5} \,{\left (3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d e + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, a^{2} b e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \,{\left (a b^{2} + a^{2} c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80794, size = 743, normalized size = 2.73 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{3}{8} x^{8} e^{2} c^{2} b + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{6}{7} x^{7} e d c^{2} b + \frac{3}{7} x^{7} e^{2} c b^{2} + \frac{3}{7} x^{7} e^{2} c^{2} a + \frac{1}{2} x^{6} d^{2} c^{2} b + x^{6} e d c b^{2} + \frac{1}{6} x^{6} e^{2} b^{3} + x^{6} e d c^{2} a + x^{6} e^{2} c b a + \frac{3}{5} x^{5} d^{2} c b^{2} + \frac{2}{5} x^{5} e d b^{3} + \frac{3}{5} x^{5} d^{2} c^{2} a + \frac{12}{5} x^{5} e d c b a + \frac{3}{5} x^{5} e^{2} b^{2} a + \frac{3}{5} x^{5} e^{2} c a^{2} + \frac{1}{4} x^{4} d^{2} b^{3} + \frac{3}{2} x^{4} d^{2} c b a + \frac{3}{2} x^{4} e d b^{2} a + \frac{3}{2} x^{4} e d c a^{2} + \frac{3}{4} x^{4} e^{2} b a^{2} + x^{3} d^{2} b^{2} a + x^{3} d^{2} c a^{2} + 2 x^{3} e d b a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + \frac{3}{2} x^{2} d^{2} b a^{2} + x^{2} e d a^{3} + x d^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.116787, size = 332, normalized size = 1.22 \begin{align*} a^{3} d^{2} x + \frac{c^{3} e^{2} x^{9}}{9} + x^{8} \left (\frac{3 b c^{2} e^{2}}{8} + \frac{c^{3} d e}{4}\right ) + x^{7} \left (\frac{3 a c^{2} e^{2}}{7} + \frac{3 b^{2} c e^{2}}{7} + \frac{6 b c^{2} d e}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{6} \left (a b c e^{2} + a c^{2} d e + \frac{b^{3} e^{2}}{6} + b^{2} c d e + \frac{b c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} c e^{2}}{5} + \frac{3 a b^{2} e^{2}}{5} + \frac{12 a b c d e}{5} + \frac{3 a c^{2} d^{2}}{5} + \frac{2 b^{3} d e}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac{3 a^{2} b e^{2}}{4} + \frac{3 a^{2} c d e}{2} + \frac{3 a b^{2} d e}{2} + \frac{3 a b c d^{2}}{2} + \frac{b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + 2 a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac{3 a^{2} b d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1392, size = 446, normalized size = 1.64 \begin{align*} \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{8} \, b c^{2} x^{8} e^{2} + \frac{6}{7} \, b c^{2} d x^{7} e + \frac{1}{2} \, b c^{2} d^{2} x^{6} + \frac{3}{7} \, b^{2} c x^{7} e^{2} + \frac{3}{7} \, a c^{2} x^{7} e^{2} + b^{2} c d x^{6} e + a c^{2} d x^{6} e + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{3}{5} \, a c^{2} d^{2} x^{5} + \frac{1}{6} \, b^{3} x^{6} e^{2} + a b c x^{6} e^{2} + \frac{2}{5} \, b^{3} d x^{5} e + \frac{12}{5} \, a b c d x^{5} e + \frac{1}{4} \, b^{3} d^{2} x^{4} + \frac{3}{2} \, a b c d^{2} x^{4} + \frac{3}{5} \, a b^{2} x^{5} e^{2} + \frac{3}{5} \, a^{2} c x^{5} e^{2} + \frac{3}{2} \, a b^{2} d x^{4} e + \frac{3}{2} \, a^{2} c d x^{4} e + a b^{2} d^{2} x^{3} + a^{2} c d^{2} x^{3} + \frac{3}{4} \, a^{2} b x^{4} e^{2} + 2 \, a^{2} b d x^{3} e + \frac{3}{2} \, a^{2} b d^{2} x^{2} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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