Optimal. Leaf size=139 \[ \frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {c d}{2 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (d+e x)}+\frac {c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 139, 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 = {640, 46}
\begin {gather*} \frac {c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac {c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac {c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {1}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 640
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac {1}{(a e+c d x) (d+e x)^4} \, dx\\ &=\int \left (\frac {c^4 d^4}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e}{\left (c d^2-a e^2\right ) (d+e x)^4}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (d+e x)^3}-\frac {c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac {c^3 d^3 e}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {c d}{2 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (d+e x)}+\frac {c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 135, normalized size = 0.97 \begin {gather*} \frac {\left (c d^2-a e^2\right ) \left (2 a^2 e^4-a c d e^2 (7 d+3 e x)+c^2 d^2 \left (11 d^2+15 d e x+6 e^2 x^2\right )\right )+6 c^3 d^3 (d+e x)^3 \log (a e+c d x)-6 c^3 d^3 (d+e x)^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 137, normalized size = 0.99
method | result | size |
default | \(-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{3}}-\frac {c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )}+\frac {c d}{2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}-\frac {c^{3} d^{3} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}+\frac {c^{3} d^{3} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\) | \(137\) |
risch | \(\frac {-\frac {c^{2} d^{2} e^{2} x^{2}}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}+\frac {\left (e^{2} a -5 c \,d^{2}\right ) c d e x}{2 e^{6} a^{3}-6 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a -2 d^{6} c^{3}}-\frac {2 a^{2} e^{4}-7 a c \,d^{2} e^{2}+11 c^{2} d^{4}}{6 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}}{\left (e x +d \right )^{3}}+\frac {c^{3} d^{3} \ln \left (-c d x -a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {c^{3} d^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(340\) |
norman | \(\frac {\frac {-a^{2} e^{5}+3 a c \,e^{3} d^{2}-3 c^{2} d^{4} e}{3 e \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}+\frac {e^{2} \left (-a c d \,e^{3}+5 c^{2} d^{3} e \right ) x^{3}}{6 d^{2} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}+\frac {\left (-a c d \,e^{3}+3 c^{2} d^{3} e \right ) e \,x^{2}}{2 d \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}}{\left (e x +d \right )^{3}}+\frac {c^{3} d^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {c^{3} d^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(366\) |
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. 371 vs.
\(2 (135) = 270\).
time = 0.34, size = 371, normalized size = 2.67 \begin {gather*} \frac {c^{3} d^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac {c^{3} d^{3} \log \left (x e + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {6 \, c^{2} d^{2} x^{2} e^{2} + 11 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{6 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\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. 463 vs.
\(2 (135) = 270\).
time = 3.78, size = 463, normalized size = 3.33 \begin {gather*} \frac {15 \, c^{3} d^{5} x e + 11 \, c^{3} d^{6} - 18 \, a c^{2} d^{3} x e^{3} + 3 \, a^{2} c d x e^{5} - 2 \, a^{3} e^{6} - 3 \, {\left (2 \, a c^{2} d^{2} x^{2} - 3 \, a^{2} c d^{2}\right )} e^{4} + 6 \, {\left (c^{3} d^{4} x^{2} - 3 \, a c^{2} d^{4}\right )} e^{2} + 6 \, {\left (c^{3} d^{3} x^{3} e^{3} + 3 \, c^{3} d^{4} x^{2} e^{2} + 3 \, c^{3} d^{5} x e + c^{3} d^{6}\right )} \log \left (c d x + a e\right ) - 6 \, {\left (c^{3} d^{3} x^{3} e^{3} + 3 \, c^{3} d^{4} x^{2} e^{2} + 3 \, c^{3} d^{5} x e + c^{3} d^{6}\right )} \log \left (x e + d\right )}{6 \, {\left (3 \, c^{4} d^{10} x e + c^{4} d^{11} + a^{4} x^{3} e^{11} + 3 \, a^{4} d x^{2} e^{10} - {\left (4 \, a^{3} c d^{2} x^{3} - 3 \, a^{4} d^{2} x\right )} e^{9} - {\left (12 \, a^{3} c d^{3} x^{2} - a^{4} d^{3}\right )} e^{8} + 6 \, {\left (a^{2} c^{2} d^{4} x^{3} - 2 \, a^{3} c d^{4} x\right )} e^{7} + 2 \, {\left (9 \, a^{2} c^{2} d^{5} x^{2} - 2 \, a^{3} c d^{5}\right )} e^{6} - 2 \, {\left (2 \, a c^{3} d^{6} x^{3} - 9 \, a^{2} c^{2} d^{6} x\right )} e^{5} - 6 \, {\left (2 \, a c^{3} d^{7} x^{2} - a^{2} c^{2} d^{7}\right )} e^{4} + {\left (c^{4} d^{8} x^{3} - 12 \, a c^{3} d^{8} x\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} - 4 \, a c^{3} d^{9}\right )} e^{2}\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. 672 vs.
\(2 (121) = 242\).
time = 0.93, size = 672, normalized size = 4.83 \begin {gather*} - \frac {c^{3} d^{3} \log {\left (x + \frac {- \frac {a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} + \frac {c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {c^{3} d^{3} \log {\left (x + \frac {\frac {a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} - \frac {c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {- 2 a^{2} e^{4} + 7 a c d^{2} e^{2} - 11 c^{2} d^{4} - 6 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} - 15 c^{2} d^{3} e\right )}{6 a^{3} d^{3} e^{6} - 18 a^{2} c d^{5} e^{4} + 18 a c^{2} d^{7} e^{2} - 6 c^{3} d^{9} + x^{3} \cdot \left (6 a^{3} e^{9} - 18 a^{2} c d^{2} e^{7} + 18 a c^{2} d^{4} e^{5} - 6 c^{3} d^{6} e^{3}\right ) + x^{2} \cdot \left (18 a^{3} d e^{8} - 54 a^{2} c d^{3} e^{6} + 54 a c^{2} d^{5} e^{4} - 18 c^{3} d^{7} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{7} - 54 a^{2} c d^{4} e^{5} + 54 a c^{2} d^{6} e^{3} - 18 c^{3} d^{8} e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.90, size = 265, normalized size = 1.91 \begin {gather*} \frac {c^{4} d^{4} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} - \frac {c^{3} d^{3} e \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac {11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.76, size = 359, normalized size = 2.58 \begin {gather*} \frac {2\,c^3\,d^3\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4}-\frac {\frac {2\,a^2\,e^4-7\,a\,c\,d^2\,e^2+11\,c^2\,d^4}{6\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}-\frac {c\,d\,x\,\left (a\,e^3-5\,c\,d^2\,e\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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