### 3.1894 $$\int \frac{1}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx$$

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]

-(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2) + (3*c*d*e*(c*d^2
+ a*e^2 + 2*c*d*e*x))/((c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (6*c^2*d^2*e^2*Log[a*e +
c*d*x])/(c*d^2 - a*e^2)^5 - (6*c^2*d^2*e^2*Log[d + e*x])/(c*d^2 - a*e^2)^5

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Rubi [A]  time = 0.0648698, 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, Rules used = {614, 616, 31} $\frac{6 c^2 d^2 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}-\frac{6 c^2 d^2 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^5}+\frac{3 c d e \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac{a e^2+c d^2+2 c d e x}{2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-3),x]

[Out]

-(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2) + (3*c*d*e*(c*d^2
+ a*e^2 + 2*c*d*e*x))/((c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (6*c^2*d^2*e^2*Log[a*e +
c*d*x])/(c*d^2 - a*e^2)^5 - (6*c^2*d^2*e^2*Log[d + e*x])/(c*d^2 - a*e^2)^5

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 )^3} \, dx &=-\frac{c d^2+a e^2+2 c d e x}{2 \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 )^2}-\frac{(3 c d e) \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=-\frac{c d^2+a e^2+2 c d e x}{2 \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 )^2}+\frac{3 c d e \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac{\left (6 c^2 d^2 e^2\right ) \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 )^4}\\ &=-\frac{c d^2+a e^2+2 c d e x}{2 \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 )^2}+\frac{3 c d e \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac{\left (6 c^3 d^3 e^3\right ) \int \frac{1}{c d^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^5}+\frac{\left (6 c^3 d^3 e^3\right ) \int \frac{1}{a e^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^5}\\ &=-\frac{c d^2+a e^2+2 c d e x}{2 \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 )^2}+\frac{3 c d e \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\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}\\ \end{align*}

Mathematica [A]  time = 0.151466, size = 168, normalized size = 0.88 $\frac{\frac{6 c^2 d^2 e \left (a e^2-c d^2\right )}{a e+c d x}+\frac{c^2 d^2 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}-12 c^2 d^2 e^2 \log (a e+c d x)+\frac{6 c d e^2 \left (a e^2-c d^2\right )}{d+e x}-\frac{\left (c d^2 e-a e^3\right )^2}{(d+e x)^2}+12 c^2 d^2 e^2 \log (d+e x)}{2 \left (a e^2-c d^2\right )^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-3),x]

[Out]

((c^2*d^2*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (6*c^2*d^2*e*(-(c*d^2) + a*e^2))/(a*e + c*d*x) - (c*d^2*e - a*e
^3)^2/(d + e*x)^2 + (6*c*d*e^2*(-(c*d^2) + a*e^2))/(d + e*x) - 12*c^2*d^2*e^2*Log[a*e + c*d*x] + 12*c^2*d^2*e^
2*Log[d + e*x])/(2*(-(c*d^2) + a*e^2)^5)

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Maple [A]  time = 0.055, size = 186, normalized size = 1. \begin{align*} -{\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 ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

-1/2*e^2/(a*e^2-c*d^2)^3/(e*x+d)^2+6*e^2/(a*e^2-c*d^2)^5*c^2*d^2*ln(e*x+d)+3*e^2/(a*e^2-c*d^2)^4*c*d/(e*x+d)+1
/2*c^2*d^2/(a*e^2-c*d^2)^3/(c*d*x+a*e)^2-6*e^2/(a*e^2-c*d^2)^5*c^2*d^2*ln(c*d*x+a*e)+3*c^2*d^2/(a*e^2-c*d^2)^4
*e/(c*d*x+a*e)

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Maxima [B]  time = 1.16293, size = 867, normalized size = 4.54 \begin{align*} \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 )}} \end{align*}

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, algorithm="maxima")

[Out]

6*c^2*d^2*e^2*log(c*d*x + a*e)/(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) - 6*c^2*d^2*e^2*log(e*x + d)/(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) + 1/2*(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*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 4*(c^3*d^5*e + 7*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(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 + (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)*x^4 + 2*(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)*x^3 + (c^6*d^12 - 9*a^2*c^4*d^8*e^4 + 1
6*a^3*c^3*d^6*e^6 - 9*a^4*c^2*d^4*e^8 + a^6*e^12)*x^2 + 2*(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)*x)

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Fricas [B]  time = 2.09057, size = 1628, normalized size = 8.52 \begin{align*} -\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 )}} \end{align*}

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, algorithm="fricas")

[Out]

-1/2*(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*(c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 - 18*(c^4*d
^6*e^2 - a^2*c^2*d^2*e^6)*x^2 - 4*(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)*x - 12*(c^4*
d^4*e^4*x^4 + a^2*c^2*d^4*e^4 + 2*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + (c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 + a^2*c^2
*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 + a^2*c^2*d^3*e^5)*x)*log(c*d*x + a*e) + 12*(c^4*d^4*e^4*x^4 + a^2*c^2*d^4*e^
4 + 2*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + (c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5
*e^3 + a^2*c^2*d^3*e^5)*x)*log(e*x + d))/(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 + (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)*x^4 + 2*(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)*x^3 + (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)*x^2 + 2*(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)*x)

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Sympy [B]  time = 3.74097, size = 1001, normalized size = 5.24 \begin{align*} \text{result too large to display} \end{align*}

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]

6*c**2*d**2*e**2*log(x + (-6*a**6*c**2*d**2*e**14/(a*e**2 - c*d**2)**5 + 36*a**5*c**3*d**4*e**12/(a*e**2 - c*d
**2)**5 - 90*a**4*c**4*d**6*e**10/(a*e**2 - c*d**2)**5 + 120*a**3*c**5*d**8*e**8/(a*e**2 - c*d**2)**5 - 90*a**
2*c**6*d**10*e**6/(a*e**2 - c*d**2)**5 + 36*a*c**7*d**12*e**4/(a*e**2 - c*d**2)**5 + 6*a*c**2*d**2*e**4 - 6*c*
*8*d**14*e**2/(a*e**2 - c*d**2)**5 + 6*c**3*d**4*e**2)/(12*c**3*d**3*e**3))/(a*e**2 - c*d**2)**5 - 6*c**2*d**2
*e**2*log(x + (6*a**6*c**2*d**2*e**14/(a*e**2 - c*d**2)**5 - 36*a**5*c**3*d**4*e**12/(a*e**2 - c*d**2)**5 + 90
*a**4*c**4*d**6*e**10/(a*e**2 - c*d**2)**5 - 120*a**3*c**5*d**8*e**8/(a*e**2 - c*d**2)**5 + 90*a**2*c**6*d**10
*e**6/(a*e**2 - c*d**2)**5 - 36*a*c**7*d**12*e**4/(a*e**2 - c*d**2)**5 + 6*a*c**2*d**2*e**4 + 6*c**8*d**14*e**
2/(a*e**2 - c*d**2)**5 + 6*c**3*d**4*e**2)/(12*c**3*d**3*e**3))/(a*e**2 - c*d**2)**5 + (-a**3*e**6 + 7*a**2*c*
d**2*e**4 + 7*a*c**2*d**4*e**2 - c**3*d**6 + 12*c**3*d**3*e**3*x**3 + x**2*(18*a*c**2*d**2*e**4 + 18*c**3*d**4
*e**2) + x*(4*a**2*c*d*e**5 + 28*a*c**2*d**3*e**3 + 4*c**3*d**5*e))/(2*a**6*d**2*e**10 - 8*a**5*c*d**4*e**8 +
12*a**4*c**2*d**6*e**6 - 8*a**3*c**3*d**8*e**4 + 2*a**2*c**4*d**10*e**2 + x**4*(2*a**4*c**2*d**2*e**10 - 8*a**
3*c**3*d**4*e**8 + 12*a**2*c**4*d**6*e**6 - 8*a*c**5*d**8*e**4 + 2*c**6*d**10*e**2) + x**3*(4*a**5*c*d*e**11 -
12*a**4*c**2*d**3*e**9 + 8*a**3*c**3*d**5*e**7 + 8*a**2*c**4*d**7*e**5 - 12*a*c**5*d**9*e**3 + 4*c**6*d**11*e
) + x**2*(2*a**6*e**12 - 18*a**4*c**2*d**4*e**8 + 32*a**3*c**3*d**6*e**6 - 18*a**2*c**4*d**8*e**4 + 2*c**6*d**
12) + x*(4*a**6*d*e**11 - 12*a**5*c*d**3*e**9 + 8*a**4*c**2*d**5*e**7 + 8*a**3*c**3*d**7*e**5 - 12*a**2*c**4*d
**9*e**3 + 4*a*c**5*d**11*e))

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Giac [A]  time = 1.22823, size = 440, normalized size = 2.3 \begin{align*} \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}} \end{align*}

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, algorithm="giac")

[Out]

12*c^2*d^2*arctan((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))*e^2/((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)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 1/2*(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)/((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)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2)