Optimal. Leaf size=91 \[ \frac {e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3} \]
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Rubi [A] time = 0.10, antiderivative size = 91, 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} \begin {gather*} \frac {e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{d+e x} \, dx &=\int (a e+c d x)^3 (d+e x)^2 \, dx\\ &=\int \left (\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^2 d^2}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^4}{c^2 d^2}+\frac {e^2 (a e+c d x)^5}{c^2 d^2}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3}+\frac {2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac {e^2 (a e+c d x)^6}{6 c^3 d^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 123, normalized size = 1.35 \begin {gather*} \frac {1}{60} x \left (20 a^3 e^3 \left (3 d^2+3 d e x+e^2 x^2\right )+15 a^2 c d e^2 x \left (6 d^2+8 d e x+3 e^2 x^2\right )+6 a c^2 d^2 e x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+c^3 d^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 176, normalized size = 1.93 \begin {gather*} \frac {(d+e x)^6 \left (\frac {20 a^3 e^6}{(d+e x)^3}-\frac {60 a^2 c d^2 e^4}{(d+e x)^3}+\frac {45 a^2 c d e^4}{(d+e x)^2}+\frac {60 a c^2 d^4 e^2}{(d+e x)^3}-\frac {90 a c^2 d^3 e^2}{(d+e x)^2}+\frac {36 a c^2 d^2 e^2}{d+e x}-\frac {20 c^3 d^6}{(d+e x)^3}+\frac {45 c^3 d^5}{(d+e x)^2}-\frac {36 c^3 d^4}{d+e x}+10 c^3 d^3\right )}{60 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 150, normalized size = 1.65 \begin {gather*} \frac {1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac {1}{5} \, {\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 158, normalized size = 1.74 \begin {gather*} \frac {1}{60} \, {\left (10 \, c^{3} d^{3} x^{6} e^{8} + 24 \, c^{3} d^{4} x^{5} e^{7} + 15 \, c^{3} d^{5} x^{4} e^{6} + 36 \, a c^{2} d^{2} x^{5} e^{9} + 90 \, a c^{2} d^{3} x^{4} e^{8} + 60 \, a c^{2} d^{4} x^{3} e^{7} + 45 \, a^{2} c d x^{4} e^{10} + 120 \, a^{2} c d^{2} x^{3} e^{9} + 90 \, a^{2} c d^{3} x^{2} e^{8} + 20 \, a^{3} x^{3} e^{11} + 60 \, a^{3} d x^{2} e^{10} + 60 \, a^{3} d^{2} x e^{9}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 205, normalized size = 2.25 \begin {gather*} \frac {c^{3} d^{3} e^{2} x^{6}}{6}+a^{3} d^{2} e^{3} x +\frac {\left (a \,c^{2} d^{2} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{2} e \right ) x^{5}}{5}+\frac {\left (2 \left (a \,e^{2}+c \,d^{2}\right ) a c d \,e^{2}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) c d \right ) x^{4}}{4}+\frac {\left (2 \left (a \,e^{2}+c \,d^{2}\right ) a c \,d^{2} e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) a e \right ) x^{3}}{3}+\frac {\left (a^{2} c \,d^{3} e^{2}+2 \left (a \,e^{2}+c \,d^{2}\right ) a^{2} d \,e^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 150, normalized size = 1.65 \begin {gather*} \frac {1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac {1}{5} \, {\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 145, normalized size = 1.59 \begin {gather*} x^3\,\left (\frac {a^3\,e^5}{3}+2\,a^2\,c\,d^2\,e^3+a\,c^2\,d^4\,e\right )+x^4\,\left (\frac {3\,a^2\,c\,d\,e^4}{4}+\frac {3\,a\,c^2\,d^3\,e^2}{2}+\frac {c^3\,d^5}{4}\right )+a^3\,d^2\,e^3\,x+\frac {c^3\,d^3\,e^2\,x^6}{6}+\frac {a^2\,d\,e^2\,x^2\,\left (3\,c\,d^2+2\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e\,x^5\,\left (2\,c\,d^2+3\,a\,e^2\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 160, normalized size = 1.76 \begin {gather*} a^{3} d^{2} e^{3} x + \frac {c^{3} d^{3} e^{2} x^{6}}{6} + x^{5} \left (\frac {3 a c^{2} d^{2} e^{3}}{5} + \frac {2 c^{3} d^{4} e}{5}\right ) + x^{4} \left (\frac {3 a^{2} c d e^{4}}{4} + \frac {3 a c^{2} d^{3} e^{2}}{2} + \frac {c^{3} d^{5}}{4}\right ) + x^{3} \left (\frac {a^{3} e^{5}}{3} + 2 a^{2} c d^{2} e^{3} + a c^{2} d^{4} e\right ) + x^{2} \left (a^{3} d e^{4} + \frac {3 a^{2} c d^{3} e^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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