Optimal. Leaf size=77 \[ \frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 77, 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 {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{3 e^3 (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 )^2}{(d+e x)^8} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^6} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^6}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^5}+\frac {c^2 d^2}{e^2 (d+e x)^4}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}+\frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {c^2 d^2}{3 e^3 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.79 \begin {gather*} -\frac {6 a^2 e^4+3 a c d e^2 (d+5 e x)+c^2 d^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \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 )^2}{(d+e x)^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 119, normalized size = 1.55 \begin {gather*} -\frac {10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 140, normalized size = 1.82 \begin {gather*} -\frac {{\left (10 \, c^{2} d^{2} x^{4} e^{4} + 25 \, c^{2} d^{3} x^{3} e^{3} + 21 \, c^{2} d^{4} x^{2} e^{2} + 7 \, c^{2} d^{5} x e + c^{2} d^{6} + 15 \, a c d x^{3} e^{5} + 33 \, a c d^{2} x^{2} e^{4} + 21 \, a c d^{3} x e^{3} + 3 \, a c d^{4} e^{2} + 6 \, a^{2} x^{2} e^{6} + 12 \, a^{2} d x e^{5} + 6 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{30 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 83, normalized size = 1.08 \begin {gather*} -\frac {c^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) c d}{2 \left (e x +d \right )^{4} e^{3}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 119, normalized size = 1.55 \begin {gather*} -\frac {10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 119, normalized size = 1.55 \begin {gather*} -\frac {\frac {6\,a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4}{30\,e^3}+\frac {c^2\,d^2\,x^2}{3\,e}+\frac {c\,d\,x\,\left (c\,d^2+3\,a\,e^2\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.42, size = 126, normalized size = 1.64 \begin {gather*} \frac {- 6 a^{2} e^{4} - 3 a c d^{2} e^{2} - c^{2} d^{4} - 10 c^{2} d^{2} e^{2} x^{2} + x \left (- 15 a c d e^{3} - 5 c^{2} d^{3} e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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