Optimal. Leaf size=111 \[ \frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{4 e^4 (d+e x)^4}-\frac {3 c d \left (c d^2-a e^2\right )^2}{5 e^4 (d+e x)^5}+\frac {\left (c d^2-a e^2\right )^3}{6 e^4 (d+e x)^6}-\frac {c^3 d^3}{3 e^4 (d+e x)^3} \]
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Rubi [A] time = 0.07, 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} \begin {gather*} \frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{4 e^4 (d+e x)^4}-\frac {3 c d \left (c d^2-a e^2\right )^2}{5 e^4 (d+e x)^5}+\frac {\left (c d^2-a e^2\right )^3}{6 e^4 (d+e x)^6}-\frac {c^3 d^3}{3 e^4 (d+e x)^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)^{10}} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^7} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^7}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^6}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^5}+\frac {c^3 d^3}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right )^3}{6 e^4 (d+e x)^6}-\frac {3 c d \left (c d^2-a e^2\right )^2}{5 e^4 (d+e x)^5}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{4 e^4 (d+e x)^4}-\frac {c^3 d^3}{3 e^4 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 103, normalized size = 0.93 \begin {gather*} -\frac {10 a^3 e^6+6 a^2 c d e^4 (d+6 e x)+3 a c^2 d^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^3 d^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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)^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 186, normalized size = 1.68 \begin {gather*} -\frac {20 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} + 10 \, a^{3} e^{6} + 15 \, {\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \, {\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 6 \, a^{2} c d e^{5}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 280, normalized size = 2.52 \begin {gather*} -\frac {{\left (20 \, c^{3} d^{3} x^{6} e^{6} + 75 \, c^{3} d^{4} x^{5} e^{5} + 111 \, c^{3} d^{5} x^{4} e^{4} + 84 \, c^{3} d^{6} x^{3} e^{3} + 36 \, c^{3} d^{7} x^{2} e^{2} + 9 \, c^{3} d^{8} x e + c^{3} d^{9} + 45 \, a c^{2} d^{2} x^{5} e^{7} + 153 \, a c^{2} d^{3} x^{4} e^{6} + 192 \, a c^{2} d^{4} x^{3} e^{5} + 108 \, a c^{2} d^{5} x^{2} e^{4} + 27 \, a c^{2} d^{6} x e^{3} + 3 \, a c^{2} d^{7} e^{2} + 36 \, a^{2} c d x^{4} e^{8} + 114 \, a^{2} c d^{2} x^{3} e^{7} + 126 \, a^{2} c d^{3} x^{2} e^{6} + 54 \, a^{2} c d^{4} x e^{5} + 6 \, a^{2} c d^{5} e^{4} + 10 \, a^{3} x^{3} e^{9} + 30 \, a^{3} d x^{2} e^{8} + 30 \, a^{3} d^{2} x e^{7} + 10 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 141, normalized size = 1.27 \begin {gather*} -\frac {c^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{5 \left (e x +d \right )^{5} e^{4}}-\frac {a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}}{6 \left (e x +d \right )^{6} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 186, normalized size = 1.68 \begin {gather*} -\frac {20 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} + 10 \, a^{3} e^{6} + 15 \, {\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \, {\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 6 \, a^{2} c d e^{5}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 184, normalized size = 1.66 \begin {gather*} -\frac {\frac {10\,a^3\,e^6+6\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6}{60\,e^4}+\frac {c^3\,d^3\,x^3}{3\,e}+\frac {c\,d\,x\,\left (6\,a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{10\,e^3}+\frac {c^2\,d^2\,x^2\,\left (c\,d^2+3\,a\,e^2\right )}{4\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.84, size = 199, normalized size = 1.79 \begin {gather*} \frac {- 10 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 20 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 45 a c^{2} d^{2} e^{4} - 15 c^{3} d^{4} e^{2}\right ) + x \left (- 36 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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