Optimal. Leaf size=73 \[ \frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {640, 47, 37}
\begin {gather*} \frac {c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 640
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)^9} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^6} \, dx\\ &=\frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {(c d) \int \frac {(a e+c d x)^3}{(d+e x)^5} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 103, normalized size = 1.41 \begin {gather*} -\frac {4 a^3 e^6+3 a^2 c d e^4 (d+5 e x)+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs.
\(2(69)=138\).
time = 0.69, size = 141, normalized size = 1.93
method | result | size |
risch | \(\frac {-\frac {c^{3} d^{3} x^{3}}{2 e}-\frac {d^{2} c^{2} \left (2 e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{2}}-\frac {d c \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{4 e^{3}}-\frac {4 e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +2 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{20 e^{4}}}{\left (e x +d \right )^{5}}\) | \(129\) |
gosper | \(-\frac {10 c^{3} d^{3} e^{3} x^{3}+20 a \,c^{2} d^{2} e^{4} x^{2}+10 c^{3} d^{4} e^{2} x^{2}+15 a^{2} c d \,e^{5} x +10 a \,c^{2} d^{3} e^{3} x +5 c^{3} d^{5} e x +4 e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +2 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{20 e^{4} \left (e x +d \right )^{5}}\) | \(130\) |
default | \(-\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 e^{4} \left (e x +d \right )^{4}}-\frac {c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{e^{4} \left (e x +d \right )^{3}}-\frac {c^{3} d^{3}}{2 e^{4} \left (e x +d \right )^{2}}-\frac {e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}{5 e^{4} \left (e x +d \right )^{5}}\) | \(141\) |
norman | \(\frac {-\frac {d^{3} \left (4 a^{3} e^{10}+3 a^{2} c \,d^{2} e^{8}+2 d^{4} c^{2} a \,e^{6}+c^{3} d^{6} e^{4}\right )}{20 e^{8}}-\frac {\left (a^{3} e^{10}+12 a^{2} c \,d^{2} e^{8}+23 d^{4} c^{2} a \,e^{6}+14 c^{3} d^{6} e^{4}\right ) x^{3}}{5 e^{5}}-\frac {d \left (3 a^{2} c \,e^{8}+14 c^{2} d^{2} a \,e^{6}+13 c^{3} d^{4} e^{4}\right ) x^{4}}{4 e^{4}}-\frac {d \left (6 a^{3} e^{10}+27 a^{2} c \,d^{2} e^{8}+28 d^{4} c^{2} a \,e^{6}+14 c^{3} d^{6} e^{4}\right ) x^{2}}{10 e^{6}}-\frac {e^{2} c^{3} d^{3} x^{6}}{2}-\frac {d^{2} \left (a \,c^{2} e^{6}+2 c^{3} d^{2} e^{4}\right ) x^{5}}{e^{3}}-\frac {d^{2} \left (3 a^{3} e^{10}+6 a^{2} c \,d^{2} e^{8}+4 d^{4} c^{2} a \,e^{6}+2 c^{3} d^{6} e^{4}\right ) x}{5 e^{7}}}{\left (e x +d \right )^{8}}\) | \(305\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (71) = 142\).
time = 0.30, size = 162, normalized size = 2.22 \begin {gather*} -\frac {10 \, c^{3} d^{3} x^{3} e^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (x^{5} e^{9} + 5 \, d x^{4} e^{8} + 10 \, d^{2} x^{3} e^{7} + 10 \, d^{3} x^{2} e^{6} + 5 \, d^{4} x e^{5} + d^{5} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (71) = 142\).
time = 2.58, size = 162, normalized size = 2.22 \begin {gather*} -\frac {5 \, c^{3} d^{5} x e + c^{3} d^{6} + 15 \, a^{2} c d x e^{5} + 4 \, a^{3} e^{6} + {\left (20 \, a c^{2} d^{2} x^{2} + 3 \, a^{2} c d^{2}\right )} e^{4} + 10 \, {\left (c^{3} d^{3} x^{3} + a c^{2} d^{3} x\right )} e^{3} + 2 \, {\left (5 \, c^{3} d^{4} x^{2} + a c^{2} d^{4}\right )} e^{2}}{20 \, {\left (x^{5} e^{9} + 5 \, d x^{4} e^{8} + 10 \, d^{2} x^{3} e^{7} + 10 \, d^{3} x^{2} e^{6} + 5 \, d^{4} x e^{5} + d^{5} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs.
\(2 (61) = 122\).
time = 109.82, size = 187, normalized size = 2.56 \begin {gather*} \frac {- 4 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 2 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 20 a c^{2} d^{2} e^{4} - 10 c^{3} d^{4} e^{2}\right ) + x \left (- 15 a^{2} c d e^{5} - 10 a c^{2} d^{3} e^{3} - 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 122, normalized size = 1.67 \begin {gather*} -\frac {{\left (10 \, c^{3} d^{3} x^{3} e^{3} + 10 \, c^{3} d^{4} x^{2} e^{2} + 5 \, c^{3} d^{5} x e + c^{3} d^{6} + 20 \, a c^{2} d^{2} x^{2} e^{4} + 10 \, a c^{2} d^{3} x e^{3} + 2 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d x e^{5} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6}\right )} e^{\left (-4\right )}}{20 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 135, normalized size = 1.85 \begin {gather*} -\frac {d^2\,\left (\frac {3\,a^2\,c}{20}+a\,c^2\,x^2-\frac {c^3\,x^4}{4}\right )-d\,\left (\frac {c^3\,e\,x^5}{20}-\frac {3\,a^2\,c\,e\,x}{4}\right )+\frac {a^3\,e^2}{5}+\frac {a\,c^2\,d^4}{10\,e^2}+\frac {a\,c^2\,d^3\,x}{2\,e}}{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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