Optimal. Leaf size=146 \[ -\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4}-\frac {2 c d e}{(d+e x) \left (c d^2-a e^2\right )^3}-\frac {e}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 146, 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} \begin {gather*} -\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4}-\frac {2 c d e}{(d+e x) \left (c d^2-a e^2\right )^3}-\frac {e}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 626
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \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)^3} \, dx\\ &=\int \left (\frac {c^3 d^3}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {3 c^3 d^3 e}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {e^2}{\left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {2 c d e^2}{\left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx\\ &=-\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e}{2 \left (c d^2-a e^2\right )^2 (d+e x)^2}-\frac {2 c d e}{\left (c d^2-a e^2\right )^3 (d+e x)}-\frac {3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 130, normalized size = 0.89 \begin {gather*} \frac {\frac {2 c^2 d^2 \left (a e^2-c d^2\right )}{a e+c d x}-6 c^2 d^2 e \log (a e+c d x)+\frac {4 c d e \left (a e^2-c d^2\right )}{d+e x}-\frac {e \left (c d^2-a e^2\right )^2}{(d+e x)^2}+6 c^2 d^2 e \log (d+e x)}{2 \left (c d^2-a e^2\right )^4} \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 {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 544, normalized size = 3.73 \begin {gather*} -\frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a c^{4} d^{10} e - 4 \, a^{2} c^{3} d^{8} e^{3} + 6 \, a^{3} c^{2} d^{6} e^{5} - 4 \, a^{4} c d^{4} e^{7} + a^{5} d^{2} e^{9} + {\left (c^{5} d^{9} e^{2} - 4 \, a c^{4} d^{7} e^{4} + 6 \, a^{2} c^{3} d^{5} e^{6} - 4 \, a^{3} c^{2} d^{3} e^{8} + a^{4} c d e^{10}\right )} x^{3} + {\left (2 \, c^{5} d^{10} e - 7 \, a c^{4} d^{8} e^{3} + 8 \, a^{2} c^{3} d^{6} e^{5} - 2 \, a^{3} c^{2} d^{4} e^{7} - 2 \, a^{4} c d^{2} e^{9} + a^{5} e^{11}\right )} x^{2} + {\left (c^{5} d^{11} - 2 \, a c^{4} d^{9} e^{2} - 2 \, a^{2} c^{3} d^{7} e^{4} + 8 \, a^{3} c^{2} d^{5} e^{6} - 7 \, a^{4} c d^{3} e^{8} + 2 \, a^{5} d e^{10}\right )} x\right )}} \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 144, normalized size = 0.99 \begin {gather*} \frac {3 c^{2} d^{2} e \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4}}-\frac {3 c^{2} d^{2} e \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4}}+\frac {c^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )}+\frac {2 c d e}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )}-\frac {e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.31, size = 423, normalized size = 2.90 \begin {gather*} -\frac {3 \, c^{2} d^{2} e \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 {3 \, c^{2} d^{2} e \log \left (e x + 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} e^{2} x^{2} + 2 \, c^{2} d^{4} + 5 \, a c d^{2} e^{2} - a^{2} e^{4} + 3 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} + {\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} + {\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} + {\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 381, normalized size = 2.61 \begin {gather*} \frac {\frac {-a^2\,e^4+5\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{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 {3\,c\,d\,x\,\left (3\,c\,d^2\,e+a\,e^3\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 {3\,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}}{x^2\,\left (2\,c\,d^2\,e+a\,e^3\right )+x\,\left (c\,d^3+2\,a\,d\,e^2\right )+a\,d^2\,e+c\,d\,e^2\,x^3}-\frac {6\,c^2\,d^2\,e\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.94, size = 734, normalized size = 5.03 \begin {gather*} \frac {3 c^{2} d^{2} e \log {\left (x + \frac {- \frac {3 a^{5} c^{2} d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {15 a^{4} c^{3} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {30 a^{3} c^{4} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {30 a^{2} c^{5} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {15 a c^{6} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c^{2} d^{2} e^{3} + \frac {3 c^{7} d^{12} e}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{3} d^{4} e}{6 c^{3} d^{3} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {3 c^{2} d^{2} e \log {\left (x + \frac {\frac {3 a^{5} c^{2} d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {15 a^{4} c^{3} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {30 a^{3} c^{4} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {30 a^{2} c^{5} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {15 a c^{6} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c^{2} d^{2} e^{3} - \frac {3 c^{7} d^{12} e}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{3} d^{4} e}{6 c^{3} d^{3} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {- a^{2} e^{4} + 5 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 9 c^{2} d^{3} e\right )}{2 a^{4} d^{2} e^{7} - 6 a^{3} c d^{4} e^{5} + 6 a^{2} c^{2} d^{6} e^{3} - 2 a c^{3} d^{8} e + x^{3} \left (2 a^{3} c d e^{8} - 6 a^{2} c^{2} d^{3} e^{6} + 6 a c^{3} d^{5} e^{4} - 2 c^{4} d^{7} e^{2}\right ) + x^{2} \left (2 a^{4} e^{9} - 2 a^{3} c d^{2} e^{7} - 6 a^{2} c^{2} d^{4} e^{5} + 10 a c^{3} d^{6} e^{3} - 4 c^{4} d^{8} e\right ) + x \left (4 a^{4} d e^{8} - 10 a^{3} c d^{3} e^{6} + 6 a^{2} c^{2} d^{5} e^{4} + 2 a c^{3} d^{7} e^{2} - 2 c^{4} d^{9}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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