Optimal. Leaf size=111 \[ -\frac {3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac {3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \]
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Rubi [A] time = 0.23, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \[ -\frac {3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac {3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^5 \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^5}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{e^3}+\frac {c^3 d^3 (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 255, normalized size = 2.30 \[ \frac {1}{504} x \left (84 a^3 e^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 c d e^2 x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 329, normalized size = 2.96 \[ \frac {1}{9} x^{9} e^{5} d^{3} c^{3} + \frac {5}{8} x^{8} e^{4} d^{4} c^{3} + \frac {3}{8} x^{8} e^{6} d^{2} c^{2} a + \frac {10}{7} x^{7} e^{3} d^{5} c^{3} + \frac {15}{7} x^{7} e^{5} d^{3} c^{2} a + \frac {3}{7} x^{7} e^{7} d c a^{2} + \frac {5}{3} x^{6} e^{2} d^{6} c^{3} + 5 x^{6} e^{4} d^{4} c^{2} a + \frac {5}{2} x^{6} e^{6} d^{2} c a^{2} + \frac {1}{6} x^{6} e^{8} a^{3} + x^{5} e d^{7} c^{3} + 6 x^{5} e^{3} d^{5} c^{2} a + 6 x^{5} e^{5} d^{3} c a^{2} + x^{5} e^{7} d a^{3} + \frac {1}{4} x^{4} d^{8} c^{3} + \frac {15}{4} x^{4} e^{2} d^{6} c^{2} a + \frac {15}{2} x^{4} e^{4} d^{4} c a^{2} + \frac {5}{2} x^{4} e^{6} d^{2} a^{3} + x^{3} e d^{7} c^{2} a + 5 x^{3} e^{3} d^{5} c a^{2} + \frac {10}{3} x^{3} e^{5} d^{3} a^{3} + \frac {3}{2} x^{2} e^{2} d^{6} c a^{2} + \frac {5}{2} x^{2} e^{4} d^{4} a^{3} + x e^{3} d^{5} a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 310, normalized size = 2.79 \[ \frac {1}{9} \, c^{3} d^{3} x^{9} e^{5} + \frac {5}{8} \, c^{3} d^{4} x^{8} e^{4} + \frac {10}{7} \, c^{3} d^{5} x^{7} e^{3} + \frac {5}{3} \, c^{3} d^{6} x^{6} e^{2} + c^{3} d^{7} x^{5} e + \frac {1}{4} \, c^{3} d^{8} x^{4} + \frac {3}{8} \, a c^{2} d^{2} x^{8} e^{6} + \frac {15}{7} \, a c^{2} d^{3} x^{7} e^{5} + 5 \, a c^{2} d^{4} x^{6} e^{4} + 6 \, a c^{2} d^{5} x^{5} e^{3} + \frac {15}{4} \, a c^{2} d^{6} x^{4} e^{2} + a c^{2} d^{7} x^{3} e + \frac {3}{7} \, a^{2} c d x^{7} e^{7} + \frac {5}{2} \, a^{2} c d^{2} x^{6} e^{6} + 6 \, a^{2} c d^{3} x^{5} e^{5} + \frac {15}{2} \, a^{2} c d^{4} x^{4} e^{4} + 5 \, a^{2} c d^{5} x^{3} e^{3} + \frac {3}{2} \, a^{2} c d^{6} x^{2} e^{2} + \frac {1}{6} \, a^{3} x^{6} e^{8} + a^{3} d x^{5} e^{7} + \frac {5}{2} \, a^{3} d^{2} x^{4} e^{6} + \frac {10}{3} \, a^{3} d^{3} x^{3} e^{5} + \frac {5}{2} \, a^{3} d^{4} x^{2} e^{4} + a^{3} d^{5} x e^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 801, normalized size = 7.22 \[ \frac {c^{3} d^{3} e^{5} x^{9}}{9}+a^{3} d^{5} e^{3} x +\frac {\left (2 c^{3} d^{4} e^{4}+3 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{2} e^{4}\right ) x^{8}}{8}+\frac {\left (c^{3} d^{5} e^{3}+6 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{3} e^{3}+\left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} c d e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) c d e \right ) e^{2}\right ) x^{7}}{7}+\frac {\left (3 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{4} e^{2}+2 \left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} c d e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) c d e \right ) d e +\left (4 \left (a \,e^{2}+c \,d^{2}\right ) a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) e^{2}\right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} c d e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) c d e \right ) d^{2}+2 \left (4 \left (a \,e^{2}+c \,d^{2}\right ) a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) d e +\left (a^{2} c \,d^{3} e^{3}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) a d e +2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (3 \left (a \,e^{2}+c \,d^{2}\right ) a^{2} d^{2} e^{4}+\left (4 \left (a \,e^{2}+c \,d^{2}\right ) a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) d^{2}+2 \left (a^{2} c \,d^{3} e^{3}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) a d e +2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e \right ) d e \right ) x^{4}}{4}+\frac {\left (a^{3} d^{3} e^{5}+6 \left (a \,e^{2}+c \,d^{2}\right ) a^{2} d^{3} e^{3}+\left (a^{2} c \,d^{3} e^{3}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) a d e +2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} d^{4} e^{4}+3 \left (a \,e^{2}+c \,d^{2}\right ) a^{2} d^{4} e^{2}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 303, normalized size = 2.73 \[ \frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} + {\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 295, normalized size = 2.66 \[ x^4\,\left (\frac {5\,a^3\,d^2\,e^6}{2}+\frac {15\,a^2\,c\,d^4\,e^4}{2}+\frac {15\,a\,c^2\,d^6\,e^2}{4}+\frac {c^3\,d^8}{4}\right )+x^6\,\left (\frac {a^3\,e^8}{6}+\frac {5\,a^2\,c\,d^2\,e^6}{2}+5\,a\,c^2\,d^4\,e^4+\frac {5\,c^3\,d^6\,e^2}{3}\right )+x^5\,\left (a^3\,d\,e^7+6\,a^2\,c\,d^3\,e^5+6\,a\,c^2\,d^5\,e^3+c^3\,d^7\,e\right )+a^3\,d^5\,e^3\,x+\frac {c^3\,d^3\,e^5\,x^9}{9}+\frac {a\,d^3\,e\,x^3\,\left (10\,a^2\,e^4+15\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{3}+\frac {c\,d\,e^3\,x^7\,\left (3\,a^2\,e^4+15\,a\,c\,d^2\,e^2+10\,c^2\,d^4\right )}{7}+\frac {a^2\,d^4\,e^2\,x^2\,\left (3\,c\,d^2+5\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^4\,x^8\,\left (5\,c\,d^2+3\,a\,e^2\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 335, normalized size = 3.02 \[ a^{3} d^{5} e^{3} x + \frac {c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac {3 a c^{2} d^{2} e^{6}}{8} + \frac {5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \left (\frac {3 a^{2} c d e^{7}}{7} + \frac {15 a c^{2} d^{3} e^{5}}{7} + \frac {10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{8}}{6} + \frac {5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac {5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \left (\frac {5 a^{3} d^{2} e^{6}}{2} + \frac {15 a^{2} c d^{4} e^{4}}{2} + \frac {15 a c^{2} d^{6} e^{2}}{4} + \frac {c^{3} d^{8}}{4}\right ) + x^{3} \left (\frac {10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \left (\frac {5 a^{3} d^{4} e^{4}}{2} + \frac {3 a^{2} c d^{6} e^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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