Optimal. Leaf size=176 \[ -\frac {c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5}-\frac {3 c^2 d^2 e}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {c d e}{(d+e x)^2 \left (c d^2-a e^2\right )^3}-\frac {e}{3 (d+e x)^3 \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.15, antiderivative size = 176, 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, 44} \[ -\frac {c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {3 c^2 d^2 e}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5}-\frac {c d e}{(d+e x)^2 \left (c d^2-a e^2\right )^3}-\frac {e}{3 (d+e x)^3 \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 626
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 (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)^2}-\frac {4 c^4 d^4 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {e^2}{\left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {2 c d e^2}{\left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {4 c^3 d^3 e^2}{\left (c d^2-a e^2\right )^5 (d+e x)}\right ) \, dx\\ &=-\frac {c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^3}-\frac {c d e}{\left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac {3 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac {4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 160, normalized size = 0.91 \[ \frac {12 c^3 d^3 e \log (a e+c d x)+\frac {3 c^3 d^3 \left (c d^2-a e^2\right )}{a e+c d x}+\frac {9 c^2 d^2 e \left (c d^2-a e^2\right )}{d+e x}+\frac {3 c d e \left (c d^2-a e^2\right )^2}{(d+e x)^2}-\frac {e \left (a e^2-c d^2\right )^3}{(d+e x)^3}-12 c^3 d^3 e \log (d+e x)}{3 \left (a e^2-c d^2\right )^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 807, normalized size = 4.59 \[ -\frac {3 \, c^{4} d^{8} + 10 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 6 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (5 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (11 \, c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x + 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a c^{3} d^{6} e^{2} + {\left (3 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (c^{4} d^{6} e^{2} + a c^{3} d^{4} e^{4}\right )} x^{2} + {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a c^{3} d^{6} e^{2} + {\left (3 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (c^{4} d^{6} e^{2} + a c^{3} d^{4} e^{4}\right )} x^{2} + {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a c^{5} d^{13} e - 5 \, a^{2} c^{4} d^{11} e^{3} + 10 \, a^{3} c^{3} d^{9} e^{5} - 10 \, a^{4} c^{2} d^{7} e^{7} + 5 \, a^{5} c d^{5} e^{9} - a^{6} d^{3} e^{11} + {\left (c^{6} d^{11} e^{3} - 5 \, a c^{5} d^{9} e^{5} + 10 \, a^{2} c^{4} d^{7} e^{7} - 10 \, a^{3} c^{3} d^{5} e^{9} + 5 \, a^{4} c^{2} d^{3} e^{11} - a^{5} c d e^{13}\right )} x^{4} + {\left (3 \, c^{6} d^{12} e^{2} - 14 \, a c^{5} d^{10} e^{4} + 25 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 5 \, a^{4} c^{2} d^{4} e^{10} + 2 \, a^{5} c d^{2} e^{12} - a^{6} e^{14}\right )} x^{3} + 3 \, {\left (c^{6} d^{13} e - 4 \, a c^{5} d^{11} e^{3} + 5 \, a^{2} c^{4} d^{9} e^{5} - 5 \, a^{4} c^{2} d^{5} e^{9} + 4 \, a^{5} c d^{3} e^{11} - a^{6} d e^{13}\right )} x^{2} + {\left (c^{6} d^{14} - 2 \, a c^{5} d^{12} e^{2} - 5 \, a^{2} c^{4} d^{10} e^{4} + 20 \, a^{3} c^{3} d^{8} e^{6} - 25 \, a^{4} c^{2} d^{6} e^{8} + 14 \, a^{5} c d^{4} e^{10} - 3 \, a^{6} d^{2} e^{12}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 174, normalized size = 0.99 \[ -\frac {4 c^{3} d^{3} e \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}+\frac {4 c^{3} d^{3} e \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}-\frac {c^{3} d^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )}-\frac {3 c^{2} d^{2} e}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )}+\frac {c d e}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{2}}-\frac {e}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 641, normalized size = 3.64 \[ -\frac {4 \, c^{3} d^{3} e \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac {4 \, c^{3} d^{3} e \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac {12 \, c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{6} + 13 \, a c^{2} d^{4} e^{2} - 5 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (11 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{3 \, {\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} + {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} + {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} + {\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 595, normalized size = 3.38 \[ \frac {8\,c^3\,d^3\,e\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5}-\frac {\frac {a^3\,e^6-5\,a^2\,c\,d^2\,e^4+13\,a\,c^2\,d^4\,e^2+3\,c^3\,d^6}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {4\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {2\,c\,d\,x\,\left (-a^2\,e^5+8\,a\,c\,d^2\,e^3+11\,c^2\,d^4\,e\right )}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,c^2\,d^2\,x^2\,\left (5\,c\,d^2\,e^2+a\,e^4\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x\,\left (c\,d^4+3\,a\,d^2\,e^2\right )+x^3\,\left (3\,c\,d^2\,e^2+a\,e^4\right )+x^2\,\left (3\,c\,d^3\,e+3\,a\,d\,e^3\right )+a\,d^3\,e+c\,d\,e^3\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.84, size = 996, normalized size = 5.66 \[ - \frac {4 c^{3} d^{3} e \log {\left (x + \frac {- \frac {4 a^{6} c^{3} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a^{5} c^{4} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{4} c^{5} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {80 a^{3} c^{6} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{2} c^{7} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a c^{8} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c^{3} d^{3} e^{3} - \frac {4 c^{9} d^{15} e}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{4} d^{5} e}{8 c^{4} d^{4} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {4 c^{3} d^{3} e \log {\left (x + \frac {\frac {4 a^{6} c^{3} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a^{5} c^{4} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{4} c^{5} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {80 a^{3} c^{6} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{2} c^{7} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a c^{8} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c^{3} d^{3} e^{3} + \frac {4 c^{9} d^{15} e}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{4} d^{5} e}{8 c^{4} d^{4} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {- a^{3} e^{6} + 5 a^{2} c d^{2} e^{4} - 13 a c^{2} d^{4} e^{2} - 3 c^{3} d^{6} - 12 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 6 a c^{2} d^{2} e^{4} - 30 c^{3} d^{4} e^{2}\right ) + x \left (2 a^{2} c d e^{5} - 16 a c^{2} d^{3} e^{3} - 22 c^{3} d^{5} e\right )}{3 a^{5} d^{3} e^{9} - 12 a^{4} c d^{5} e^{7} + 18 a^{3} c^{2} d^{7} e^{5} - 12 a^{2} c^{3} d^{9} e^{3} + 3 a c^{4} d^{11} e + x^{4} \left (3 a^{4} c d e^{11} - 12 a^{3} c^{2} d^{3} e^{9} + 18 a^{2} c^{3} d^{5} e^{7} - 12 a c^{4} d^{7} e^{5} + 3 c^{5} d^{9} e^{3}\right ) + x^{3} \left (3 a^{5} e^{12} - 3 a^{4} c d^{2} e^{10} - 18 a^{3} c^{2} d^{4} e^{8} + 42 a^{2} c^{3} d^{6} e^{6} - 33 a c^{4} d^{8} e^{4} + 9 c^{5} d^{10} e^{2}\right ) + x^{2} \left (9 a^{5} d e^{11} - 27 a^{4} c d^{3} e^{9} + 18 a^{3} c^{2} d^{5} e^{7} + 18 a^{2} c^{3} d^{7} e^{5} - 27 a c^{4} d^{9} e^{3} + 9 c^{5} d^{11} e\right ) + x \left (9 a^{5} d^{2} e^{10} - 33 a^{4} c d^{4} e^{8} + 42 a^{3} c^{2} d^{6} e^{6} - 18 a^{2} c^{3} d^{8} e^{4} - 3 a c^{4} d^{10} e^{2} + 3 c^{5} d^{12}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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