Optimal. Leaf size=114 \[ -\frac {a e^2+c d^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac {2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac {2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
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Rubi [A] time = 0.03, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {614, 616, 31} \[ -\frac {a e^2+c d^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac {2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac {2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 614
Rule 616
Rubi steps
\begin {align*} \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=-\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {(2 c d e) \int \frac {1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=-\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac {\left (2 c^2 d^2 e^2\right ) \int \frac {1}{c d^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^3}-\frac {\left (2 c^2 d^2 e^2\right ) \int \frac {1}{a e^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^3}\\ &=-\frac {c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac {2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 86, normalized size = 0.75 \[ \frac {\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d (d+2 e x)\right )}{(d+e x) (a e+c d x)}+2 c d e \log (a e+c d x)-2 c d e \log (d+e x)}{\left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.19, size = 277, normalized size = 2.43 \[ -\frac {c^{2} d^{4} - a^{2} e^{4} + 2 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{a c^{3} d^{7} e - 3 \, a^{2} c^{2} d^{5} e^{3} + 3 \, a^{3} c d^{3} e^{5} - a^{4} d e^{7} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x^{2} + {\left (c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} + 2 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 176, normalized size = 1.54 \[ -\frac {4 \, c d \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac {2 \, c d x e + c d^{2} + a e^{2}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 0.94 \[ -\frac {2 c d e \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}+\frac {2 c d e \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}-\frac {c d}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c d x +a e \right )}-\frac {e}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 236, normalized size = 2.07 \[ -\frac {2 \, c d e \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac {2 \, c d e \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac {2 \, c d e x + c d^{2} + a e^{2}}{a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} + {\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 223, normalized size = 1.96 \[ \frac {4\,c\,d\,e\,\mathrm {atanh}\left (\frac {a^3\,e^6-a^2\,c\,d^2\,e^4-a\,c^2\,d^4\,e^2+c^3\,d^6}{{\left (a\,e^2-c\,d^2\right )}^3}+\frac {2\,c\,d\,e\,x\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3}\right )}{{\left (a\,e^2-c\,d^2\right )}^3}-\frac {\frac {c\,d^2+a\,e^2}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}+\frac {2\,c\,d\,e\,x}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}}{c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.27, size = 486, normalized size = 4.26 \[ - \frac {2 c d e \log {\left (x + \frac {- \frac {2 a^{4} c d e^{9}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {8 a^{3} c^{2} d^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {12 a^{2} c^{3} d^{5} e^{5}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {8 a c^{4} d^{7} e^{3}}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 a c d e^{3} - \frac {2 c^{5} d^{9} e}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 c^{2} d^{3} e}{4 c^{2} d^{2} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {2 c d e \log {\left (x + \frac {\frac {2 a^{4} c d e^{9}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {8 a^{3} c^{2} d^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {12 a^{2} c^{3} d^{5} e^{5}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {8 a c^{4} d^{7} e^{3}}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 a c d e^{3} + \frac {2 c^{5} d^{9} e}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 c^{2} d^{3} e}{4 c^{2} d^{2} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {- a e^{2} - c d^{2} - 2 c d e x}{a^{3} d e^{5} - 2 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e + x^{2} \left (a^{2} c d e^{5} - 2 a c^{2} d^{3} e^{3} + c^{3} d^{5} e\right ) + x \left (a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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