Optimal. Leaf size=191 \[ \frac{6 c^2 d^2 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}-\frac{6 c^2 d^2 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^5}+\frac{3 c d e \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac{a e^2+c d^2+2 c d e x}{2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.177423, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{6 c^2 d^2 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}-\frac{6 c^2 d^2 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^5}+\frac{3 c d e \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac{a e^2+c d^2+2 c d e x}{2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-3),x]
[Out]
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Rubi in Sympy [A] time = 21.1084, size = 175, normalized size = 0.92 \[ - \frac{12 c^{2} d^{2} e^{2} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{a e^{2} - c d^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac{3 c d e \left (a e^{2} + c d^{2} + 2 c d e x\right )}{\left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )} - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.315802, size = 168, normalized size = 0.88 \[ \frac{\frac{6 c^2 d^2 e \left (a e^2-c d^2\right )}{a e+c d x}+\frac{c^2 d^2 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}-12 c^2 d^2 e^2 \log (a e+c d x)-\frac{\left (c d^2 e-a e^3\right )^2}{(d+e x)^2}+\frac{6 c d e^2 \left (a e^2-c d^2\right )}{d+e x}+12 c^2 d^2 e^2 \log (d+e x)}{2 \left (a e^2-c d^2\right )^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-3),x]
[Out]
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Maple [A] time = 0.02, size = 186, normalized size = 1. \[ -{\frac{{e}^{2}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{e}^{2}{c}^{2}{d}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}+3\,{\frac{cd{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) }}+{\frac{{c}^{2}{d}^{2}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) ^{2}}}-6\,{\frac{{e}^{2}{c}^{2}{d}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}+3\,{\frac{{c}^{2}{d}^{2}e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
[Out]
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Maxima [A] time = 0.748427, size = 867, normalized size = 4.54 \[ \frac{6 \, c^{2} d^{2} e^{2} \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac{6 \, c^{2} d^{2} e^{2} \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac{12 \, c^{3} d^{3} e^{3} x^{3} - c^{3} d^{6} + 7 \, a c^{2} d^{4} e^{2} + 7 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c^{3} d^{5} e + 7 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (a^{2} c^{4} d^{10} e^{2} - 4 \, a^{3} c^{3} d^{8} e^{4} + 6 \, a^{4} c^{2} d^{6} e^{6} - 4 \, a^{5} c d^{4} e^{8} + a^{6} d^{2} e^{10} +{\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{4} + 2 \,{\left (c^{6} d^{11} e - 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} + 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x^{3} +{\left (c^{6} d^{12} - 9 \, a^{2} c^{4} d^{8} e^{4} + 16 \, a^{3} c^{3} d^{6} e^{6} - 9 \, a^{4} c^{2} d^{4} e^{8} + a^{6} e^{12}\right )} x^{2} + 2 \,{\left (a c^{5} d^{11} e - 3 \, a^{2} c^{4} d^{9} e^{3} + 2 \, a^{3} c^{3} d^{7} e^{5} + 2 \, a^{4} c^{2} d^{5} e^{7} - 3 \, a^{5} c d^{3} e^{9} + a^{6} d e^{11}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2312, size = 1118, normalized size = 5.85 \[ -\frac{c^{4} d^{8} - 8 \, a c^{3} d^{6} e^{2} + 8 \, 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} - 18 \,{\left (c^{4} d^{6} e^{2} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 4 \,{\left (c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} - 6 \, 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^{2} c^{2} d^{4} e^{4} + 2 \,{\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} +{\left (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 (a c^{3} d^{5} e^{3} + a^{2} c^{2} d^{3} e^{5}\right )} x\right )} \log \left (c d x + a e\right ) + 12 \,{\left (c^{4} d^{4} e^{4} x^{4} + a^{2} c^{2} d^{4} e^{4} + 2 \,{\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} +{\left (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 (a c^{3} d^{5} e^{3} + a^{2} c^{2} d^{3} e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} c^{5} d^{12} e^{2} - 5 \, a^{3} c^{4} d^{10} e^{4} + 10 \, a^{4} c^{3} d^{8} e^{6} - 10 \, a^{5} c^{2} d^{6} e^{8} + 5 \, a^{6} c d^{4} e^{10} - a^{7} d^{2} e^{12} +{\left (c^{7} d^{12} e^{2} - 5 \, a c^{6} d^{10} e^{4} + 10 \, a^{2} c^{5} d^{8} e^{6} - 10 \, a^{3} c^{4} d^{6} e^{8} + 5 \, a^{4} c^{3} d^{4} e^{10} - a^{5} c^{2} d^{2} e^{12}\right )} x^{4} + 2 \,{\left (c^{7} d^{13} e - 4 \, a c^{6} d^{11} e^{3} + 5 \, a^{2} c^{5} d^{9} e^{5} - 5 \, a^{4} c^{3} d^{5} e^{9} + 4 \, a^{5} c^{2} d^{3} e^{11} - a^{6} c d e^{13}\right )} x^{3} +{\left (c^{7} d^{14} - a c^{6} d^{12} e^{2} - 9 \, a^{2} c^{5} d^{10} e^{4} + 25 \, a^{3} c^{4} d^{8} e^{6} - 25 \, a^{4} c^{3} d^{6} e^{8} + 9 \, a^{5} c^{2} d^{4} e^{10} + a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} x^{2} + 2 \,{\left (a c^{6} d^{13} e - 4 \, a^{2} c^{5} d^{11} e^{3} + 5 \, a^{3} c^{4} d^{9} e^{5} - 5 \, a^{5} c^{2} d^{5} e^{9} + 4 \, a^{6} c d^{3} e^{11} - a^{7} d e^{13}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.1268, size = 1001, normalized size = 5.24 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.215998, size = 440, normalized size = 2.3 \[ \frac{12 \, c^{2} d^{2} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right ) e^{2}}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{12 \, c^{3} d^{3} x^{3} e^{3} + 18 \, c^{3} d^{4} x^{2} e^{2} + 4 \, c^{3} d^{5} x e - c^{3} d^{6} + 18 \, a c^{2} d^{2} x^{2} e^{4} + 28 \, a c^{2} d^{3} x e^{3} + 7 \, a c^{2} d^{4} e^{2} + 4 \, a^{2} c d x e^{5} + 7 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{2 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-3),x, algorithm="giac")
[Out]