Optimal. Leaf size=77 \[ \frac {2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac {\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac {c^2 d^2}{4 e^3 (d+e x)^4} \]
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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 {2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac {\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac {c^2 d^2}{4 e^3 (d+e x)^4} \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)^9} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^7} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^7}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^6}+\frac {c^2 d^2}{e^2 (d+e x)^5}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}+\frac {2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{4 e^3 (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.79 \begin {gather*} -\frac {10 a^2 e^4+4 a c d e^2 (d+6 e x)+c^2 d^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (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 )^2}{(d+e x)^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 130, normalized size = 1.69 \begin {gather*} -\frac {15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} 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 (15 \, c^{2} d^{2} x^{4} e^{4} + 36 \, c^{2} d^{3} x^{3} e^{3} + 28 \, c^{2} d^{4} x^{2} e^{2} + 8 \, c^{2} d^{5} x e + c^{2} d^{6} + 24 \, a c d x^{3} e^{5} + 52 \, a c d^{2} x^{2} e^{4} + 32 \, a c d^{3} x e^{3} + 4 \, a c d^{4} e^{2} + 10 \, a^{2} x^{2} e^{6} + 20 \, a^{2} d x e^{5} + 10 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 1.08 \begin {gather*} -\frac {c^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}-\frac {2 \left (a \,e^{2}-c \,d^{2}\right ) c d}{5 \left (e x +d \right )^{5} e^{3}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{6 \left (e x +d \right )^{6} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 130, normalized size = 1.69 \begin {gather*} -\frac {15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 130, normalized size = 1.69 \begin {gather*} -\frac {\frac {10\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4}{60\,e^3}+\frac {c^2\,d^2\,x^2}{4\,e}+\frac {c\,d\,x\,\left (c\,d^2+4\,a\,e^2\right )}{10\,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 [B] time = 2.36, size = 138, normalized size = 1.79 \begin {gather*} \frac {- 10 a^{2} e^{4} - 4 a c d^{2} e^{2} - c^{2} d^{4} - 15 c^{2} d^{2} e^{2} x^{2} + x \left (- 24 a c d e^{3} - 6 c^{2} d^{3} e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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