Optimal. Leaf size=107 \[ \frac {e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac {e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
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Rubi [A] time = 0.07, antiderivative size = 107, 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} \[ \frac {e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac {e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3 (d+e x)} \, dx\\ &=\int \left (\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac {c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac {e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.78 \[ -\frac {\frac {\left (c d^2-a e^2\right ) \left (c d (d-2 e x)-3 a e^2\right )}{(a e+c d x)^2}-2 e^2 \log (a e+c d x)+2 e^2 \log (d+e x)}{2 \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 280, normalized size = 2.62 \[ -\frac {c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 2 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x - 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (c d x + a e\right ) + 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} c^{3} d^{6} e^{2} - 3 \, a^{3} c^{2} d^{4} e^{4} + 3 \, a^{4} c d^{2} e^{6} - a^{5} e^{8} + {\left (c^{5} d^{8} - 3 \, a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (a c^{4} d^{7} e - 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} - a^{4} c d e^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 396, normalized size = 3.70 \[ \frac {2 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {2 \, c^{3} d^{5} x^{3} e^{3} + 3 \, c^{3} d^{6} x^{2} e^{2} - c^{3} d^{8} - 4 \, a c^{2} d^{3} x^{3} e^{5} - 3 \, a c^{2} d^{4} x^{2} e^{4} + 6 \, a c^{2} d^{5} x e^{3} + 5 \, a c^{2} d^{6} e^{2} + 2 \, a^{2} c d x^{3} e^{7} - 3 \, a^{2} c d^{2} x^{2} e^{6} - 12 \, a^{2} c d^{3} x e^{5} - 7 \, a^{2} c d^{4} e^{4} + 3 \, a^{3} x^{2} e^{8} + 6 \, a^{3} d x e^{7} + 3 \, a^{3} d^{2} e^{6}}{2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 106, normalized size = 0.99 \[ \frac {e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}-\frac {e^{2} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}+\frac {e}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c d x +a e \right )}+\frac {1}{2 \left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.07, size = 238, normalized size = 2.22 \[ \frac {e^{2} \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac {e^{2} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac {2 \, c d e x - c d^{2} + 3 \, a e^{2}}{2 \, {\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 225, normalized size = 2.10 \[ \frac {\frac {3\,a\,e^2-c\,d^2}{2\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {c\,d\,e\,x}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}}{a^2\,e^2+2\,a\,c\,d\,e\,x+c^2\,d^2\,x^2}-\frac {2\,e^2\,\mathrm {atanh}\left (\frac {a^3\,e^6-a^2\,c\,d^2\,e^4-a\,c^2\,d^4\,e^2+c^3\,d^6}{{\left (a\,e^2-c\,d^2\right )}^3}+\frac {2\,c\,d\,e\,x\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3}\right )}{{\left (a\,e^2-c\,d^2\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.26, size = 457, normalized size = 4.27 \[ \frac {e^{2} \log {\left (x + \frac {- \frac {a^{4} e^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {4 a^{3} c d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {6 a^{2} c^{2} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {4 a c^{3} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + a e^{4} - \frac {c^{4} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + c d^{2} e^{2}}{2 c d e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {e^{2} \log {\left (x + \frac {\frac {a^{4} e^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {4 a^{3} c d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {6 a^{2} c^{2} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac {4 a c^{3} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + a e^{4} + \frac {c^{4} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + c d^{2} e^{2}}{2 c d e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {3 a e^{2} - c d^{2} + 2 c d e x}{2 a^{4} e^{6} - 4 a^{3} c d^{2} e^{4} + 2 a^{2} c^{2} d^{4} e^{2} + x^{2} \left (2 a^{2} c^{2} d^{2} e^{4} - 4 a c^{3} d^{4} e^{2} + 2 c^{4} d^{6}\right ) + x \left (4 a^{3} c d e^{5} - 8 a^{2} c^{2} d^{3} e^{3} + 4 a c^{3} d^{5} e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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