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

Optimal. Leaf size=191 \[ \frac {6 c^2 d^2 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}-\frac {6 c^2 d^2 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^5}+\frac {3 c d e \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac {a e^2+c d^2+2 c d e x}{2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 191, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 31

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

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]

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 = 0.14, size = 168, normalized size = 0.88 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 828, normalized size = 4.34 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 326, normalized size = 1.71 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 186, normalized size = 0.97 \begin {gather*} \frac {6 c^{2} d^{2} e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}-\frac {6 c^{2} d^{2} e^{2} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}+\frac {3 c^{2} d^{2} e}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )}+\frac {c^{2} d^{2}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}+\frac {3 c d \,e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )}-\frac {e^{2}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{2}} \end {gather*}

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

________________________________________________________________________________________

maxima [B]  time = 1.29, size = 642, normalized size = 3.36 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.99, size = 616, normalized size = 3.23 \begin {gather*} \frac {\frac {9\,c\,x^2\,\left (c^2\,d^4\,e^2+a\,c\,d^2\,e^4\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}-\frac {a^3\,e^6-7\,a^2\,c\,d^2\,e^4-7\,a\,c^2\,d^4\,e^2+c^3\,d^6}{2\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {6\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {2\,c\,d\,e\,x\,\left (a^2\,e^4+7\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x^2\,\left (a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )+x^3\,\left (2\,c^2\,d^3\,e+2\,a\,c\,d\,e^3\right )+x\,\left (2\,a^2\,d\,e^3+2\,c\,a\,d^3\,e\right )+a^2\,d^2\,e^2+c^2\,d^2\,e^2\,x^4}-\frac {12\,c^2\,d^2\,e^2\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((9*c*x^2*(c^2*d^4*e^2 + a*c*d^2*e^4))/(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*
e^4) - (a^3*e^6 + c^3*d^6 - 7*a*c^2*d^4*e^2 - 7*a^2*c*d^2*e^4)/(2*(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4*a^3
*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (6*c^3*d^3*e^3*x^3)/(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4*a^3*c*d^2*e^6
+ 6*a^2*c^2*d^4*e^4) + (2*c*d*e*x*(a^2*e^4 + c^2*d^4 + 7*a*c*d^2*e^2))/(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 -
4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4))/(x^2*(a^2*e^4 + c^2*d^4 + 4*a*c*d^2*e^2) + x^3*(2*c^2*d^3*e + 2*a*c*d*e^
3) + x*(2*a^2*d*e^3 + 2*a*c*d^3*e) + a^2*d^2*e^2 + c^2*d^2*e^2*x^4) - (12*c^2*d^2*e^2*atanh((a^5*e^10 + c^5*d^
10 - 3*a*c^4*d^8*e^2 - 3*a^4*c*d^2*e^8 + 2*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6)/(a*e^2 - c*d^2)^5 + (2*c*d*e*x
*(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4))/(a*e^2 - c*d^2)^5))/(a*e^2 - c*d
^2)^5

________________________________________________________________________________________

sympy [B]  time = 2.73, size = 1001, normalized size = 5.24 \begin {gather*} \frac {6 c^{2} d^{2} e^{2} \log {\left (x + \frac {- \frac {6 a^{6} c^{2} d^{2} e^{14}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {36 a^{5} c^{3} d^{4} e^{12}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {90 a^{4} c^{4} d^{6} e^{10}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {120 a^{3} c^{5} d^{8} e^{8}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {90 a^{2} c^{6} d^{10} e^{6}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {36 a c^{7} d^{12} e^{4}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 a c^{2} d^{2} e^{4} - \frac {6 c^{8} d^{14} e^{2}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 c^{3} d^{4} e^{2}}{12 c^{3} d^{3} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {6 c^{2} d^{2} e^{2} \log {\left (x + \frac {\frac {6 a^{6} c^{2} d^{2} e^{14}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {36 a^{5} c^{3} d^{4} e^{12}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {90 a^{4} c^{4} d^{6} e^{10}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {120 a^{3} c^{5} d^{8} e^{8}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {90 a^{2} c^{6} d^{10} e^{6}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {36 a c^{7} d^{12} e^{4}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 a c^{2} d^{2} e^{4} + \frac {6 c^{8} d^{14} e^{2}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 c^{3} d^{4} e^{2}}{12 c^{3} d^{3} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {- 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} \left (18 a c^{2} d^{2} e^{4} + 18 c^{3} d^{4} e^{2}\right ) + x \left (4 a^{2} c d e^{5} + 28 a c^{2} d^{3} e^{3} + 4 c^{3} d^{5} e\right )}{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} \left (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}\right ) + x^{3} \left (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\right ) + x^{2} \left (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}\right ) + x \left (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\right )} \end {gather*}

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))

________________________________________________________________________________________