Optimal. Leaf size=156 \[ \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^5}-\frac {2 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}{4 e^5}-\frac {2 c (d+e x)^7 (2 c d-b e)}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5} \]
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Rubi [A] time = 0.17, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \[ \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^5}-\frac {2 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}{4 e^5}-\frac {2 c (d+e x)^7 (2 c d-b e)}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^6}{e^4}+\frac {c^2 (d+e x)^7}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^5}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^6}{6 e^5}-\frac {2 c (2 c d-b e) (d+e x)^7}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 223, normalized size = 1.43 \[ \frac {1}{4} x^4 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac {1}{6} e x^6 \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac {1}{3} d x^3 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac {1}{5} x^5 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac {1}{2} a d^2 x^2 (3 a e+2 b d)+\frac {1}{7} c e^2 x^7 (2 b e+3 c d)+\frac {1}{8} c^2 e^3 x^8 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 259, normalized size = 1.66 \[ \frac {1}{8} x^{8} e^{3} c^{2} + \frac {3}{7} x^{7} e^{2} d c^{2} + \frac {2}{7} x^{7} e^{3} c b + \frac {1}{2} x^{6} e d^{2} c^{2} + x^{6} e^{2} d c b + \frac {1}{6} x^{6} e^{3} b^{2} + \frac {1}{3} x^{6} e^{3} c a + \frac {1}{5} x^{5} d^{3} c^{2} + \frac {6}{5} x^{5} e d^{2} c b + \frac {3}{5} x^{5} e^{2} d b^{2} + \frac {6}{5} x^{5} e^{2} d c a + \frac {2}{5} x^{5} e^{3} b a + \frac {1}{2} x^{4} d^{3} c b + \frac {3}{4} x^{4} e d^{2} b^{2} + \frac {3}{2} x^{4} e d^{2} c a + \frac {3}{2} x^{4} e^{2} d b a + \frac {1}{4} x^{4} e^{3} a^{2} + \frac {1}{3} x^{3} d^{3} b^{2} + \frac {2}{3} x^{3} d^{3} c a + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac {3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 253, normalized size = 1.62 \[ \frac {1}{8} \, c^{2} x^{8} e^{3} + \frac {3}{7} \, c^{2} d x^{7} e^{2} + \frac {1}{2} \, c^{2} d^{2} x^{6} e + \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {2}{7} \, b c x^{7} e^{3} + b c d x^{6} e^{2} + \frac {6}{5} \, b c d^{2} x^{5} e + \frac {1}{2} \, b c d^{3} x^{4} + \frac {1}{6} \, b^{2} x^{6} e^{3} + \frac {1}{3} \, a c x^{6} e^{3} + \frac {3}{5} \, b^{2} d x^{5} e^{2} + \frac {6}{5} \, a c d x^{5} e^{2} + \frac {3}{4} \, b^{2} d^{2} x^{4} e + \frac {3}{2} \, a c d^{2} x^{4} e + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {2}{3} \, a c d^{3} x^{3} + \frac {2}{5} \, a b x^{5} e^{3} + \frac {3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 219, normalized size = 1.40 \[ \frac {c^{2} e^{3} x^{8}}{8}+\frac {\left (2 e^{3} b c +3 d \,e^{2} c^{2}\right ) x^{7}}{7}+a^{2} d^{3} x +\frac {\left (6 b c d \,e^{2}+3 c^{2} d^{2} e +\left (2 a c +b^{2}\right ) e^{3}\right ) x^{6}}{6}+\frac {\left (2 a b \,e^{3}+6 b c \,d^{2} e +c^{2} d^{3}+3 \left (2 a c +b^{2}\right ) d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 a b d \,e^{2}+2 b c \,d^{3}+3 \left (2 a c +b^{2}\right ) d^{2} e \right ) x^{4}}{4}+\frac {\left (3 a^{2} d \,e^{2}+6 a b \,d^{2} e +\left (2 a c +b^{2}\right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{2}+2 d^{3} a b \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 218, normalized size = 1.40 \[ \frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 218, normalized size = 1.40 \[ x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,a\,b\,d\,e^2}{2}+\frac {3\,c\,a\,d^2\,e}{2}+\frac {3\,b^2\,d^2\,e}{4}+\frac {c\,b\,d^3}{2}\right )+x^5\,\left (\frac {3\,b^2\,d\,e^2}{5}+\frac {6\,b\,c\,d^2\,e}{5}+\frac {2\,a\,b\,e^3}{5}+\frac {c^2\,d^3}{5}+\frac {6\,a\,c\,d\,e^2}{5}\right )+x^3\,\left (a^2\,d\,e^2+2\,a\,b\,d^2\,e+\frac {2\,c\,a\,d^3}{3}+\frac {b^2\,d^3}{3}\right )+x^6\,\left (\frac {b^2\,e^3}{6}+b\,c\,d\,e^2+\frac {c^2\,d^2\,e}{2}+\frac {a\,c\,e^3}{3}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^8}{8}+\frac {a\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )}{2}+\frac {c\,e^2\,x^7\,\left (2\,b\,e+3\,c\,d\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 260, normalized size = 1.67 \[ a^{2} d^{3} x + \frac {c^{2} e^{3} x^{8}}{8} + x^{7} \left (\frac {2 b c e^{3}}{7} + \frac {3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac {a c e^{3}}{3} + \frac {b^{2} e^{3}}{6} + b c d e^{2} + \frac {c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac {2 a b e^{3}}{5} + \frac {6 a c d e^{2}}{5} + \frac {3 b^{2} d e^{2}}{5} + \frac {6 b c d^{2} e}{5} + \frac {c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a b d e^{2}}{2} + \frac {3 a c d^{2} e}{2} + \frac {3 b^{2} d^{2} e}{4} + \frac {b c d^{3}}{2}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac {2 a c d^{3}}{3} + \frac {b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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