3.20.6 \(\int \frac {1}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1906]

Optimal. Leaf size=262 \[ -\frac {c d^2+a e^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7} \]

[Out]

1/3*(-2*c*d*e*x-a*e^2-c*d^2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3+5/3*c*d*e*(2*c*d*e*x+a*e^2+c
*d^2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2-10*c^2*d^2*e^2*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^
2)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)-20*c^3*d^3*e^3*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^7+20*c^3*d^3*e^3*ln(e*x+d)/
(-a*e^2+c*d^2)^7

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Rubi [A]
time = 0.07, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {628, 630, 31} \begin {gather*} -\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}-\frac {10 c^2 d^2 e^2 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}+\frac {5 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \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 )^2}-\frac {a e^2+c d^2+2 c d e x}{3 \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 )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-4),x]

[Out]

-1/3*(c*d^2 + a*e^2 + 2*c*d*e*x)/((c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3) + (5*c*d*e*(c*d
^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2) - (10*c^2*d^2*e^2*(c*
d^2 + a*e^2 + 2*c*d*e*x))/((c*d^2 - a*e^2)^6*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - (20*c^3*d^3*e^3*Log[a*
e + c*d*x])/(c*d^2 - a*e^2)^7 + (20*c^3*d^3*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^7

Rule 31

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

Rule 628

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 630

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 )^4} \, dx &=-\frac {c d^2+a e^2+2 c d e x}{3 \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 )^3}-\frac {(10 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 )^3} \, dx}{3 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \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 )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \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 )^2}+\frac {\left (10 c^2 d^2 e^2\right ) \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 )^4}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \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 )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \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 )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {\left (20 c^3 d^3 e^3\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 )^6}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \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 )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \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 )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac {\left (20 c^4 d^4 e^4\right ) \int \frac {1}{c d^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^7}-\frac {\left (20 c^4 d^4 e^4\right ) \int \frac {1}{a e^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^7}\\ &=-\frac {c d^2+a e^2+2 c d e x}{3 \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 )^3}+\frac {5 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \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 )^2}-\frac {10 c^2 d^2 e^2 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 234, normalized size = 0.89 \begin {gather*} \frac {\frac {c^3 d^3 \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}-\frac {6 c^3 d^3 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {30 c^3 d^3 e^2 \left (c d^2-a e^2\right )}{a e+c d x}+\frac {\left (c d^2 e-a e^3\right )^3}{(d+e x)^3}+\frac {6 c d e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+\frac {30 c^2 d^2 e^3 \left (c d^2-a e^2\right )}{d+e x}+60 c^3 d^3 e^3 \log (a e+c d x)-60 c^3 d^3 e^3 \log (d+e x)}{3 \left (-c d^2+a e^2\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-4),x]

[Out]

((c^3*d^3*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (6*c^3*d^3*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (30*c^3*d^3*e
^2*(c*d^2 - a*e^2))/(a*e + c*d*x) + (c*d^2*e - a*e^3)^3/(d + e*x)^3 + (6*c*d*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^
2 + (30*c^2*d^2*e^3*(c*d^2 - a*e^2))/(d + e*x) + 60*c^3*d^3*e^3*Log[a*e + c*d*x] - 60*c^3*d^3*e^3*Log[d + e*x]
)/(3*(-(c*d^2) + a*e^2)^7)

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Maple [A]
time = 0.78, size = 253, normalized size = 0.97 Too large to display

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)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*e^3/(a*e^2-c*d^2)^4/(e*x+d)^3-20*e^3/(a*e^2-c*d^2)^7*c^3*d^3*ln(e*x+d)-10*e^3/(a*e^2-c*d^2)^6*c^2*d^2/(e*
x+d)+2*e^3/(a*e^2-c*d^2)^5*c*d/(e*x+d)^2-1/3*c^3*d^3/(a*e^2-c*d^2)^4/(c*d*x+a*e)^3+20*e^3/(a*e^2-c*d^2)^7*c^3*
d^3*ln(c*d*x+a*e)-10*c^3*d^3/(a*e^2-c*d^2)^6*e^2/(c*d*x+a*e)-2*c^3*d^3/(a*e^2-c*d^2)^5*e/(c*d*x+a*e)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (256) = 512\).
time = 0.37, size = 1193, normalized size = 4.55 \begin {gather*} -\frac {20 \, c^{3} d^{3} e^{3} \log \left (c d x + a e\right )}{c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}} + \frac {20 \, c^{3} d^{3} e^{3} \log \left (x e + d\right )}{c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}} - \frac {60 \, c^{5} d^{5} x^{5} e^{5} + c^{5} d^{10} - 8 \, a c^{4} d^{8} e^{2} + 37 \, a^{2} c^{3} d^{6} e^{4} + 37 \, a^{3} c^{2} d^{4} e^{6} - 8 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + 150 \, {\left (c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \, {\left (11 \, c^{5} d^{7} e^{3} + 38 \, a c^{4} d^{5} e^{5} + 11 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 15 \, {\left (c^{5} d^{8} e^{2} + 19 \, a c^{4} d^{6} e^{4} + 19 \, a^{2} c^{3} d^{4} e^{6} + a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 3 \, {\left (c^{5} d^{9} e - 14 \, a c^{4} d^{7} e^{3} - 74 \, a^{2} c^{3} d^{5} e^{5} - 14 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{3 \, {\left (a^{3} c^{6} d^{15} e^{3} - 6 \, a^{4} c^{5} d^{13} e^{5} + 15 \, a^{5} c^{4} d^{11} e^{7} - 20 \, a^{6} c^{3} d^{9} e^{9} + 15 \, a^{7} c^{2} d^{7} e^{11} - 6 \, a^{8} c d^{5} e^{13} + a^{9} d^{3} e^{15} + {\left (c^{9} d^{15} e^{3} - 6 \, a c^{8} d^{13} e^{5} + 15 \, a^{2} c^{7} d^{11} e^{7} - 20 \, a^{3} c^{6} d^{9} e^{9} + 15 \, a^{4} c^{5} d^{7} e^{11} - 6 \, a^{5} c^{4} d^{5} e^{13} + a^{6} c^{3} d^{3} e^{15}\right )} x^{6} + 3 \, {\left (c^{9} d^{16} e^{2} - 5 \, a c^{8} d^{14} e^{4} + 9 \, a^{2} c^{7} d^{12} e^{6} - 5 \, a^{3} c^{6} d^{10} e^{8} - 5 \, a^{4} c^{5} d^{8} e^{10} + 9 \, a^{5} c^{4} d^{6} e^{12} - 5 \, a^{6} c^{3} d^{4} e^{14} + a^{7} c^{2} d^{2} e^{16}\right )} x^{5} + 3 \, {\left (c^{9} d^{17} e - 3 \, a c^{8} d^{15} e^{3} - 2 \, a^{2} c^{7} d^{13} e^{5} + 19 \, a^{3} c^{6} d^{11} e^{7} - 30 \, a^{4} c^{5} d^{9} e^{9} + 19 \, a^{5} c^{4} d^{7} e^{11} - 2 \, a^{6} c^{3} d^{5} e^{13} - 3 \, a^{7} c^{2} d^{3} e^{15} + a^{8} c d e^{17}\right )} x^{4} + {\left (c^{9} d^{18} + 3 \, a c^{8} d^{16} e^{2} - 30 \, a^{2} c^{7} d^{14} e^{4} + 62 \, a^{3} c^{6} d^{12} e^{6} - 36 \, a^{4} c^{5} d^{10} e^{8} - 36 \, a^{5} c^{4} d^{8} e^{10} + 62 \, a^{6} c^{3} d^{6} e^{12} - 30 \, a^{7} c^{2} d^{4} e^{14} + 3 \, a^{8} c d^{2} e^{16} + a^{9} e^{18}\right )} x^{3} + 3 \, {\left (a c^{8} d^{17} e - 3 \, a^{2} c^{7} d^{15} e^{3} - 2 \, a^{3} c^{6} d^{13} e^{5} + 19 \, a^{4} c^{5} d^{11} e^{7} - 30 \, a^{5} c^{4} d^{9} e^{9} + 19 \, a^{6} c^{3} d^{7} e^{11} - 2 \, a^{7} c^{2} d^{5} e^{13} - 3 \, a^{8} c d^{3} e^{15} + a^{9} d e^{17}\right )} x^{2} + 3 \, {\left (a^{2} c^{7} d^{16} e^{2} - 5 \, a^{3} c^{6} d^{14} e^{4} + 9 \, a^{4} c^{5} d^{12} e^{6} - 5 \, a^{5} c^{4} d^{10} e^{8} - 5 \, a^{6} c^{3} d^{8} e^{10} + 9 \, a^{7} c^{2} d^{6} e^{12} - 5 \, a^{8} c d^{4} e^{14} + a^{9} d^{2} e^{16}\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)^4,x, algorithm="maxima")

[Out]

-20*c^3*d^3*e^3*log(c*d*x + a*e)/(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*
a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14) + 20*c^3*d^3*e^3*log(x*e + d)/(c^7*d^14 -
7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^
6*c*d^2*e^12 - a^7*e^14) - 1/3*(60*c^5*d^5*x^5*e^5 + c^5*d^10 - 8*a*c^4*d^8*e^2 + 37*a^2*c^3*d^6*e^4 + 37*a^3*
c^2*d^4*e^6 - 8*a^4*c*d^2*e^8 + a^5*e^10 + 150*(c^5*d^6*e^4 + a*c^4*d^4*e^6)*x^4 + 10*(11*c^5*d^7*e^3 + 38*a*c
^4*d^5*e^5 + 11*a^2*c^3*d^3*e^7)*x^3 + 15*(c^5*d^8*e^2 + 19*a*c^4*d^6*e^4 + 19*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e
^8)*x^2 - 3*(c^5*d^9*e - 14*a*c^4*d^7*e^3 - 74*a^2*c^3*d^5*e^5 - 14*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x)/(a^3*c^6
*d^15*e^3 - 6*a^4*c^5*d^13*e^5 + 15*a^5*c^4*d^11*e^7 - 20*a^6*c^3*d^9*e^9 + 15*a^7*c^2*d^7*e^11 - 6*a^8*c*d^5*
e^13 + a^9*d^3*e^15 + (c^9*d^15*e^3 - 6*a*c^8*d^13*e^5 + 15*a^2*c^7*d^11*e^7 - 20*a^3*c^6*d^9*e^9 + 15*a^4*c^5
*d^7*e^11 - 6*a^5*c^4*d^5*e^13 + a^6*c^3*d^3*e^15)*x^6 + 3*(c^9*d^16*e^2 - 5*a*c^8*d^14*e^4 + 9*a^2*c^7*d^12*e
^6 - 5*a^3*c^6*d^10*e^8 - 5*a^4*c^5*d^8*e^10 + 9*a^5*c^4*d^6*e^12 - 5*a^6*c^3*d^4*e^14 + a^7*c^2*d^2*e^16)*x^5
+ 3*(c^9*d^17*e - 3*a*c^8*d^15*e^3 - 2*a^2*c^7*d^13*e^5 + 19*a^3*c^6*d^11*e^7 - 30*a^4*c^5*d^9*e^9 + 19*a^5*c
^4*d^7*e^11 - 2*a^6*c^3*d^5*e^13 - 3*a^7*c^2*d^3*e^15 + a^8*c*d*e^17)*x^4 + (c^9*d^18 + 3*a*c^8*d^16*e^2 - 30*
a^2*c^7*d^14*e^4 + 62*a^3*c^6*d^12*e^6 - 36*a^4*c^5*d^10*e^8 - 36*a^5*c^4*d^8*e^10 + 62*a^6*c^3*d^6*e^12 - 30*
a^7*c^2*d^4*e^14 + 3*a^8*c*d^2*e^16 + a^9*e^18)*x^3 + 3*(a*c^8*d^17*e - 3*a^2*c^7*d^15*e^3 - 2*a^3*c^6*d^13*e^
5 + 19*a^4*c^5*d^11*e^7 - 30*a^5*c^4*d^9*e^9 + 19*a^6*c^3*d^7*e^11 - 2*a^7*c^2*d^5*e^13 - 3*a^8*c*d^3*e^15 + a
^9*d*e^17)*x^2 + 3*(a^2*c^7*d^16*e^2 - 5*a^3*c^6*d^14*e^4 + 9*a^4*c^5*d^12*e^6 - 5*a^5*c^4*d^10*e^8 - 5*a^6*c^
3*d^8*e^10 + 9*a^7*c^2*d^6*e^12 - 5*a^8*c*d^4*e^14 + a^9*d^2*e^16)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1619 vs. \(2 (256) = 512\).
time = 2.51, size = 1619, normalized size = 6.18 \begin {gather*} \frac {3 \, c^{6} d^{11} x e - c^{6} d^{12} - 3 \, a^{5} c d x e^{11} + a^{6} e^{12} + 3 \, {\left (5 \, a^{4} c^{2} d^{2} x^{2} - 3 \, a^{5} c d^{2}\right )} e^{10} + 5 \, {\left (22 \, a^{3} c^{3} d^{3} x^{3} + 9 \, a^{4} c^{2} d^{3} x\right )} e^{9} + 15 \, {\left (10 \, a^{2} c^{4} d^{4} x^{4} + 18 \, a^{3} c^{3} d^{4} x^{2} + 3 \, a^{4} c^{2} d^{4}\right )} e^{8} + 30 \, {\left (2 \, a c^{5} d^{5} x^{5} + 9 \, a^{2} c^{4} d^{5} x^{3} + 6 \, a^{3} c^{3} d^{5} x\right )} e^{7} - 30 \, {\left (2 \, c^{6} d^{7} x^{5} + 9 \, a c^{5} d^{7} x^{3} + 6 \, a^{2} c^{4} d^{7} x\right )} e^{5} - 15 \, {\left (10 \, c^{6} d^{8} x^{4} + 18 \, a c^{5} d^{8} x^{2} + 3 \, a^{2} c^{4} d^{8}\right )} e^{4} - 5 \, {\left (22 \, c^{6} d^{9} x^{3} + 9 \, a c^{5} d^{9} x\right )} e^{3} - 3 \, {\left (5 \, c^{6} d^{10} x^{2} - 3 \, a c^{5} d^{10}\right )} e^{2} - 60 \, {\left (c^{6} d^{9} x^{3} e^{3} + a^{3} c^{3} d^{3} x^{3} e^{9} + 3 \, {\left (a^{2} c^{4} d^{4} x^{4} + a^{3} c^{3} d^{4} x^{2}\right )} e^{8} + 3 \, {\left (a c^{5} d^{5} x^{5} + 3 \, a^{2} c^{4} d^{5} x^{3} + a^{3} c^{3} d^{5} x\right )} e^{7} + {\left (c^{6} d^{6} x^{6} + 9 \, a c^{5} d^{6} x^{4} + 9 \, a^{2} c^{4} d^{6} x^{2} + a^{3} c^{3} d^{6}\right )} e^{6} + 3 \, {\left (c^{6} d^{7} x^{5} + 3 \, a c^{5} d^{7} x^{3} + a^{2} c^{4} d^{7} x\right )} e^{5} + 3 \, {\left (c^{6} d^{8} x^{4} + a c^{5} d^{8} x^{2}\right )} e^{4}\right )} \log \left (c d x + a e\right ) + 60 \, {\left (c^{6} d^{9} x^{3} e^{3} + a^{3} c^{3} d^{3} x^{3} e^{9} + 3 \, {\left (a^{2} c^{4} d^{4} x^{4} + a^{3} c^{3} d^{4} x^{2}\right )} e^{8} + 3 \, {\left (a c^{5} d^{5} x^{5} + 3 \, a^{2} c^{4} d^{5} x^{3} + a^{3} c^{3} d^{5} x\right )} e^{7} + {\left (c^{6} d^{6} x^{6} + 9 \, a c^{5} d^{6} x^{4} + 9 \, a^{2} c^{4} d^{6} x^{2} + a^{3} c^{3} d^{6}\right )} e^{6} + 3 \, {\left (c^{6} d^{7} x^{5} + 3 \, a c^{5} d^{7} x^{3} + a^{2} c^{4} d^{7} x\right )} e^{5} + 3 \, {\left (c^{6} d^{8} x^{4} + a c^{5} d^{8} x^{2}\right )} e^{4}\right )} \log \left (x e + d\right )}{3 \, {\left (c^{10} d^{20} x^{3} - a^{10} x^{3} e^{20} - 3 \, {\left (a^{9} c d x^{4} + a^{10} d x^{2}\right )} e^{19} - {\left (3 \, a^{8} c^{2} d^{2} x^{5} + 2 \, a^{9} c d^{2} x^{3} + 3 \, a^{10} d^{2} x\right )} e^{18} - {\left (a^{7} c^{3} d^{3} x^{6} - 12 \, a^{8} c^{2} d^{3} x^{4} - 12 \, a^{9} c d^{3} x^{2} + a^{10} d^{3}\right )} e^{17} + 3 \, {\left (6 \, a^{7} c^{3} d^{4} x^{5} + 11 \, a^{8} c^{2} d^{4} x^{3} + 6 \, a^{9} c d^{4} x\right )} e^{16} + {\left (7 \, a^{6} c^{4} d^{5} x^{6} - 3 \, a^{7} c^{3} d^{5} x^{4} - 3 \, a^{8} c^{2} d^{5} x^{2} + 7 \, a^{9} c d^{5}\right )} e^{15} - 2 \, {\left (21 \, a^{6} c^{4} d^{6} x^{5} + 46 \, a^{7} c^{3} d^{6} x^{3} + 21 \, a^{8} c^{2} d^{6} x\right )} e^{14} - 21 \, {\left (a^{5} c^{5} d^{7} x^{6} + 3 \, a^{6} c^{4} d^{7} x^{4} + 3 \, a^{7} c^{3} d^{7} x^{2} + a^{8} c^{2} d^{7}\right )} e^{13} + 14 \, {\left (3 \, a^{5} c^{5} d^{8} x^{5} + 7 \, a^{6} c^{4} d^{8} x^{3} + 3 \, a^{7} c^{3} d^{8} x\right )} e^{12} + 7 \, {\left (5 \, a^{4} c^{6} d^{9} x^{6} + 21 \, a^{5} c^{5} d^{9} x^{4} + 21 \, a^{6} c^{4} d^{9} x^{2} + 5 \, a^{7} c^{3} d^{9}\right )} e^{11} - 7 \, {\left (5 \, a^{3} c^{7} d^{11} x^{6} + 21 \, a^{4} c^{6} d^{11} x^{4} + 21 \, a^{5} c^{5} d^{11} x^{2} + 5 \, a^{6} c^{4} d^{11}\right )} e^{9} - 14 \, {\left (3 \, a^{3} c^{7} d^{12} x^{5} + 7 \, a^{4} c^{6} d^{12} x^{3} + 3 \, a^{5} c^{5} d^{12} x\right )} e^{8} + 21 \, {\left (a^{2} c^{8} d^{13} x^{6} + 3 \, a^{3} c^{7} d^{13} x^{4} + 3 \, a^{4} c^{6} d^{13} x^{2} + a^{5} c^{5} d^{13}\right )} e^{7} + 2 \, {\left (21 \, a^{2} c^{8} d^{14} x^{5} + 46 \, a^{3} c^{7} d^{14} x^{3} + 21 \, a^{4} c^{6} d^{14} x\right )} e^{6} - {\left (7 \, a c^{9} d^{15} x^{6} - 3 \, a^{2} c^{8} d^{15} x^{4} - 3 \, a^{3} c^{7} d^{15} x^{2} + 7 \, a^{4} c^{6} d^{15}\right )} e^{5} - 3 \, {\left (6 \, a c^{9} d^{16} x^{5} + 11 \, a^{2} c^{8} d^{16} x^{3} + 6 \, a^{3} c^{7} d^{16} x\right )} e^{4} + {\left (c^{10} d^{17} x^{6} - 12 \, a c^{9} d^{17} x^{4} - 12 \, a^{2} c^{8} d^{17} x^{2} + a^{3} c^{7} d^{17}\right )} e^{3} + {\left (3 \, c^{10} d^{18} x^{5} + 2 \, a c^{9} d^{18} x^{3} + 3 \, a^{2} c^{8} d^{18} x\right )} e^{2} + 3 \, {\left (c^{10} d^{19} x^{4} + a c^{9} d^{19} x^{2}\right )} 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)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^6*d^11*x*e - c^6*d^12 - 3*a^5*c*d*x*e^11 + a^6*e^12 + 3*(5*a^4*c^2*d^2*x^2 - 3*a^5*c*d^2)*e^10 + 5*(2
2*a^3*c^3*d^3*x^3 + 9*a^4*c^2*d^3*x)*e^9 + 15*(10*a^2*c^4*d^4*x^4 + 18*a^3*c^3*d^4*x^2 + 3*a^4*c^2*d^4)*e^8 +
30*(2*a*c^5*d^5*x^5 + 9*a^2*c^4*d^5*x^3 + 6*a^3*c^3*d^5*x)*e^7 - 30*(2*c^6*d^7*x^5 + 9*a*c^5*d^7*x^3 + 6*a^2*c
^4*d^7*x)*e^5 - 15*(10*c^6*d^8*x^4 + 18*a*c^5*d^8*x^2 + 3*a^2*c^4*d^8)*e^4 - 5*(22*c^6*d^9*x^3 + 9*a*c^5*d^9*x
)*e^3 - 3*(5*c^6*d^10*x^2 - 3*a*c^5*d^10)*e^2 - 60*(c^6*d^9*x^3*e^3 + a^3*c^3*d^3*x^3*e^9 + 3*(a^2*c^4*d^4*x^4
+ a^3*c^3*d^4*x^2)*e^8 + 3*(a*c^5*d^5*x^5 + 3*a^2*c^4*d^5*x^3 + a^3*c^3*d^5*x)*e^7 + (c^6*d^6*x^6 + 9*a*c^5*d
^6*x^4 + 9*a^2*c^4*d^6*x^2 + a^3*c^3*d^6)*e^6 + 3*(c^6*d^7*x^5 + 3*a*c^5*d^7*x^3 + a^2*c^4*d^7*x)*e^5 + 3*(c^6
*d^8*x^4 + a*c^5*d^8*x^2)*e^4)*log(c*d*x + a*e) + 60*(c^6*d^9*x^3*e^3 + a^3*c^3*d^3*x^3*e^9 + 3*(a^2*c^4*d^4*x
^4 + a^3*c^3*d^4*x^2)*e^8 + 3*(a*c^5*d^5*x^5 + 3*a^2*c^4*d^5*x^3 + a^3*c^3*d^5*x)*e^7 + (c^6*d^6*x^6 + 9*a*c^5
*d^6*x^4 + 9*a^2*c^4*d^6*x^2 + a^3*c^3*d^6)*e^6 + 3*(c^6*d^7*x^5 + 3*a*c^5*d^7*x^3 + a^2*c^4*d^7*x)*e^5 + 3*(c
^6*d^8*x^4 + a*c^5*d^8*x^2)*e^4)*log(x*e + d))/(c^10*d^20*x^3 - a^10*x^3*e^20 - 3*(a^9*c*d*x^4 + a^10*d*x^2)*e
^19 - (3*a^8*c^2*d^2*x^5 + 2*a^9*c*d^2*x^3 + 3*a^10*d^2*x)*e^18 - (a^7*c^3*d^3*x^6 - 12*a^8*c^2*d^3*x^4 - 12*a
^9*c*d^3*x^2 + a^10*d^3)*e^17 + 3*(6*a^7*c^3*d^4*x^5 + 11*a^8*c^2*d^4*x^3 + 6*a^9*c*d^4*x)*e^16 + (7*a^6*c^4*d
^5*x^6 - 3*a^7*c^3*d^5*x^4 - 3*a^8*c^2*d^5*x^2 + 7*a^9*c*d^5)*e^15 - 2*(21*a^6*c^4*d^6*x^5 + 46*a^7*c^3*d^6*x^
3 + 21*a^8*c^2*d^6*x)*e^14 - 21*(a^5*c^5*d^7*x^6 + 3*a^6*c^4*d^7*x^4 + 3*a^7*c^3*d^7*x^2 + a^8*c^2*d^7)*e^13 +
14*(3*a^5*c^5*d^8*x^5 + 7*a^6*c^4*d^8*x^3 + 3*a^7*c^3*d^8*x)*e^12 + 7*(5*a^4*c^6*d^9*x^6 + 21*a^5*c^5*d^9*x^4
+ 21*a^6*c^4*d^9*x^2 + 5*a^7*c^3*d^9)*e^11 - 7*(5*a^3*c^7*d^11*x^6 + 21*a^4*c^6*d^11*x^4 + 21*a^5*c^5*d^11*x^
2 + 5*a^6*c^4*d^11)*e^9 - 14*(3*a^3*c^7*d^12*x^5 + 7*a^4*c^6*d^12*x^3 + 3*a^5*c^5*d^12*x)*e^8 + 21*(a^2*c^8*d^
13*x^6 + 3*a^3*c^7*d^13*x^4 + 3*a^4*c^6*d^13*x^2 + a^5*c^5*d^13)*e^7 + 2*(21*a^2*c^8*d^14*x^5 + 46*a^3*c^7*d^1
4*x^3 + 21*a^4*c^6*d^14*x)*e^6 - (7*a*c^9*d^15*x^6 - 3*a^2*c^8*d^15*x^4 - 3*a^3*c^7*d^15*x^2 + 7*a^4*c^6*d^15)
*e^5 - 3*(6*a*c^9*d^16*x^5 + 11*a^2*c^8*d^16*x^3 + 6*a^3*c^7*d^16*x)*e^4 + (c^10*d^17*x^6 - 12*a*c^9*d^17*x^4
- 12*a^2*c^8*d^17*x^2 + a^3*c^7*d^17)*e^3 + (3*c^10*d^18*x^5 + 2*a*c^9*d^18*x^3 + 3*a^2*c^8*d^18*x)*e^2 + 3*(c
^10*d^19*x^4 + a*c^9*d^19*x^2)*e)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1748 vs. \(2 (262) = 524\).
time = 28.25, size = 1748, normalized size = 6.67 \begin {gather*} - \frac {20 c^{3} d^{3} e^{3} \log {\left (x + \frac {- \frac {20 a^{8} c^{3} d^{3} e^{19}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {160 a^{7} c^{4} d^{5} e^{17}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {560 a^{6} c^{5} d^{7} e^{15}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {1120 a^{5} c^{6} d^{9} e^{13}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {1400 a^{4} c^{7} d^{11} e^{11}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {1120 a^{3} c^{8} d^{13} e^{9}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {560 a^{2} c^{9} d^{15} e^{7}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {160 a c^{10} d^{17} e^{5}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 a c^{3} d^{3} e^{5} - \frac {20 c^{11} d^{19} e^{3}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 c^{4} d^{5} e^{3}}{40 c^{4} d^{4} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {20 c^{3} d^{3} e^{3} \log {\left (x + \frac {\frac {20 a^{8} c^{3} d^{3} e^{19}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {160 a^{7} c^{4} d^{5} e^{17}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {560 a^{6} c^{5} d^{7} e^{15}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {1120 a^{5} c^{6} d^{9} e^{13}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {1400 a^{4} c^{7} d^{11} e^{11}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {1120 a^{3} c^{8} d^{13} e^{9}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {560 a^{2} c^{9} d^{15} e^{7}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac {160 a c^{10} d^{17} e^{5}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 a c^{3} d^{3} e^{5} + \frac {20 c^{11} d^{19} e^{3}}{\left (a e^{2} - c d^{2}\right )^{7}} + 20 c^{4} d^{5} e^{3}}{40 c^{4} d^{4} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{7}} + \frac {- a^{5} e^{10} + 8 a^{4} c d^{2} e^{8} - 37 a^{3} c^{2} d^{4} e^{6} - 37 a^{2} c^{3} d^{6} e^{4} + 8 a c^{4} d^{8} e^{2} - c^{5} d^{10} - 60 c^{5} d^{5} e^{5} x^{5} + x^{4} \left (- 150 a c^{4} d^{4} e^{6} - 150 c^{5} d^{6} e^{4}\right ) + x^{3} \left (- 110 a^{2} c^{3} d^{3} e^{7} - 380 a c^{4} d^{5} e^{5} - 110 c^{5} d^{7} e^{3}\right ) + x^{2} \left (- 15 a^{3} c^{2} d^{2} e^{8} - 285 a^{2} c^{3} d^{4} e^{6} - 285 a c^{4} d^{6} e^{4} - 15 c^{5} d^{8} e^{2}\right ) + x \left (3 a^{4} c d e^{9} - 42 a^{3} c^{2} d^{3} e^{7} - 222 a^{2} c^{3} d^{5} e^{5} - 42 a c^{4} d^{7} e^{3} + 3 c^{5} d^{9} e\right )}{3 a^{9} d^{3} e^{15} - 18 a^{8} c d^{5} e^{13} + 45 a^{7} c^{2} d^{7} e^{11} - 60 a^{6} c^{3} d^{9} e^{9} + 45 a^{5} c^{4} d^{11} e^{7} - 18 a^{4} c^{5} d^{13} e^{5} + 3 a^{3} c^{6} d^{15} e^{3} + x^{6} \cdot \left (3 a^{6} c^{3} d^{3} e^{15} - 18 a^{5} c^{4} d^{5} e^{13} + 45 a^{4} c^{5} d^{7} e^{11} - 60 a^{3} c^{6} d^{9} e^{9} + 45 a^{2} c^{7} d^{11} e^{7} - 18 a c^{8} d^{13} e^{5} + 3 c^{9} d^{15} e^{3}\right ) + x^{5} \cdot \left (9 a^{7} c^{2} d^{2} e^{16} - 45 a^{6} c^{3} d^{4} e^{14} + 81 a^{5} c^{4} d^{6} e^{12} - 45 a^{4} c^{5} d^{8} e^{10} - 45 a^{3} c^{6} d^{10} e^{8} + 81 a^{2} c^{7} d^{12} e^{6} - 45 a c^{8} d^{14} e^{4} + 9 c^{9} d^{16} e^{2}\right ) + x^{4} \cdot \left (9 a^{8} c d e^{17} - 27 a^{7} c^{2} d^{3} e^{15} - 18 a^{6} c^{3} d^{5} e^{13} + 171 a^{5} c^{4} d^{7} e^{11} - 270 a^{4} c^{5} d^{9} e^{9} + 171 a^{3} c^{6} d^{11} e^{7} - 18 a^{2} c^{7} d^{13} e^{5} - 27 a c^{8} d^{15} e^{3} + 9 c^{9} d^{17} e\right ) + x^{3} \cdot \left (3 a^{9} e^{18} + 9 a^{8} c d^{2} e^{16} - 90 a^{7} c^{2} d^{4} e^{14} + 186 a^{6} c^{3} d^{6} e^{12} - 108 a^{5} c^{4} d^{8} e^{10} - 108 a^{4} c^{5} d^{10} e^{8} + 186 a^{3} c^{6} d^{12} e^{6} - 90 a^{2} c^{7} d^{14} e^{4} + 9 a c^{8} d^{16} e^{2} + 3 c^{9} d^{18}\right ) + x^{2} \cdot \left (9 a^{9} d e^{17} - 27 a^{8} c d^{3} e^{15} - 18 a^{7} c^{2} d^{5} e^{13} + 171 a^{6} c^{3} d^{7} e^{11} - 270 a^{5} c^{4} d^{9} e^{9} + 171 a^{4} c^{5} d^{11} e^{7} - 18 a^{3} c^{6} d^{13} e^{5} - 27 a^{2} c^{7} d^{15} e^{3} + 9 a c^{8} d^{17} e\right ) + x \left (9 a^{9} d^{2} e^{16} - 45 a^{8} c d^{4} e^{14} + 81 a^{7} c^{2} d^{6} e^{12} - 45 a^{6} c^{3} d^{8} e^{10} - 45 a^{5} c^{4} d^{10} e^{8} + 81 a^{4} c^{5} d^{12} e^{6} - 45 a^{3} c^{6} d^{14} e^{4} + 9 a^{2} c^{7} d^{16} e^{2}\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)**4,x)

[Out]

-20*c**3*d**3*e**3*log(x + (-20*a**8*c**3*d**3*e**19/(a*e**2 - c*d**2)**7 + 160*a**7*c**4*d**5*e**17/(a*e**2 -
c*d**2)**7 - 560*a**6*c**5*d**7*e**15/(a*e**2 - c*d**2)**7 + 1120*a**5*c**6*d**9*e**13/(a*e**2 - c*d**2)**7 -
1400*a**4*c**7*d**11*e**11/(a*e**2 - c*d**2)**7 + 1120*a**3*c**8*d**13*e**9/(a*e**2 - c*d**2)**7 - 560*a**2*c
**9*d**15*e**7/(a*e**2 - c*d**2)**7 + 160*a*c**10*d**17*e**5/(a*e**2 - c*d**2)**7 + 20*a*c**3*d**3*e**5 - 20*c
**11*d**19*e**3/(a*e**2 - c*d**2)**7 + 20*c**4*d**5*e**3)/(40*c**4*d**4*e**4))/(a*e**2 - c*d**2)**7 + 20*c**3*
d**3*e**3*log(x + (20*a**8*c**3*d**3*e**19/(a*e**2 - c*d**2)**7 - 160*a**7*c**4*d**5*e**17/(a*e**2 - c*d**2)**
7 + 560*a**6*c**5*d**7*e**15/(a*e**2 - c*d**2)**7 - 1120*a**5*c**6*d**9*e**13/(a*e**2 - c*d**2)**7 + 1400*a**4
*c**7*d**11*e**11/(a*e**2 - c*d**2)**7 - 1120*a**3*c**8*d**13*e**9/(a*e**2 - c*d**2)**7 + 560*a**2*c**9*d**15*
e**7/(a*e**2 - c*d**2)**7 - 160*a*c**10*d**17*e**5/(a*e**2 - c*d**2)**7 + 20*a*c**3*d**3*e**5 + 20*c**11*d**19
*e**3/(a*e**2 - c*d**2)**7 + 20*c**4*d**5*e**3)/(40*c**4*d**4*e**4))/(a*e**2 - c*d**2)**7 + (-a**5*e**10 + 8*a
**4*c*d**2*e**8 - 37*a**3*c**2*d**4*e**6 - 37*a**2*c**3*d**6*e**4 + 8*a*c**4*d**8*e**2 - c**5*d**10 - 60*c**5*
d**5*e**5*x**5 + x**4*(-150*a*c**4*d**4*e**6 - 150*c**5*d**6*e**4) + x**3*(-110*a**2*c**3*d**3*e**7 - 380*a*c*
*4*d**5*e**5 - 110*c**5*d**7*e**3) + x**2*(-15*a**3*c**2*d**2*e**8 - 285*a**2*c**3*d**4*e**6 - 285*a*c**4*d**6
*e**4 - 15*c**5*d**8*e**2) + x*(3*a**4*c*d*e**9 - 42*a**3*c**2*d**3*e**7 - 222*a**2*c**3*d**5*e**5 - 42*a*c**4
*d**7*e**3 + 3*c**5*d**9*e))/(3*a**9*d**3*e**15 - 18*a**8*c*d**5*e**13 + 45*a**7*c**2*d**7*e**11 - 60*a**6*c**
3*d**9*e**9 + 45*a**5*c**4*d**11*e**7 - 18*a**4*c**5*d**13*e**5 + 3*a**3*c**6*d**15*e**3 + x**6*(3*a**6*c**3*d
**3*e**15 - 18*a**5*c**4*d**5*e**13 + 45*a**4*c**5*d**7*e**11 - 60*a**3*c**6*d**9*e**9 + 45*a**2*c**7*d**11*e*
*7 - 18*a*c**8*d**13*e**5 + 3*c**9*d**15*e**3) + x**5*(9*a**7*c**2*d**2*e**16 - 45*a**6*c**3*d**4*e**14 + 81*a
**5*c**4*d**6*e**12 - 45*a**4*c**5*d**8*e**10 - 45*a**3*c**6*d**10*e**8 + 81*a**2*c**7*d**12*e**6 - 45*a*c**8*
d**14*e**4 + 9*c**9*d**16*e**2) + x**4*(9*a**8*c*d*e**17 - 27*a**7*c**2*d**3*e**15 - 18*a**6*c**3*d**5*e**13 +
171*a**5*c**4*d**7*e**11 - 270*a**4*c**5*d**9*e**9 + 171*a**3*c**6*d**11*e**7 - 18*a**2*c**7*d**13*e**5 - 27*
a*c**8*d**15*e**3 + 9*c**9*d**17*e) + x**3*(3*a**9*e**18 + 9*a**8*c*d**2*e**16 - 90*a**7*c**2*d**4*e**14 + 186
*a**6*c**3*d**6*e**12 - 108*a**5*c**4*d**8*e**10 - 108*a**4*c**5*d**10*e**8 + 186*a**3*c**6*d**12*e**6 - 90*a*
*2*c**7*d**14*e**4 + 9*a*c**8*d**16*e**2 + 3*c**9*d**18) + x**2*(9*a**9*d*e**17 - 27*a**8*c*d**3*e**15 - 18*a*
*7*c**2*d**5*e**13 + 171*a**6*c**3*d**7*e**11 - 270*a**5*c**4*d**9*e**9 + 171*a**4*c**5*d**11*e**7 - 18*a**3*c
**6*d**13*e**5 - 27*a**2*c**7*d**15*e**3 + 9*a*c**8*d**17*e) + x*(9*a**9*d**2*e**16 - 45*a**8*c*d**4*e**14 + 8
1*a**7*c**2*d**6*e**12 - 45*a**6*c**3*d**8*e**10 - 45*a**5*c**4*d**10*e**8 + 81*a**4*c**5*d**12*e**6 - 45*a**3
*c**6*d**14*e**4 + 9*a**2*c**7*d**16*e**2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (256) = 512\).
time = 0.66, size = 599, normalized size = 2.29 \begin {gather*} -\frac {20 \, c^{4} d^{4} e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{8} d^{15} - 7 \, a c^{7} d^{13} e^{2} + 21 \, a^{2} c^{6} d^{11} e^{4} - 35 \, a^{3} c^{5} d^{9} e^{6} + 35 \, a^{4} c^{4} d^{7} e^{8} - 21 \, a^{5} c^{3} d^{5} e^{10} + 7 \, a^{6} c^{2} d^{3} e^{12} - a^{7} c d e^{14}} + \frac {20 \, c^{3} d^{3} e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{7} d^{14} e - 7 \, a c^{6} d^{12} e^{3} + 21 \, a^{2} c^{5} d^{10} e^{5} - 35 \, a^{3} c^{4} d^{8} e^{7} + 35 \, a^{4} c^{3} d^{6} e^{9} - 21 \, a^{5} c^{2} d^{4} e^{11} + 7 \, a^{6} c d^{2} e^{13} - a^{7} e^{15}} - \frac {60 \, c^{5} d^{5} x^{5} e^{5} + 150 \, c^{5} d^{6} x^{4} e^{4} + 110 \, c^{5} d^{7} x^{3} e^{3} + 15 \, c^{5} d^{8} x^{2} e^{2} - 3 \, c^{5} d^{9} x e + c^{5} d^{10} + 150 \, a c^{4} d^{4} x^{4} e^{6} + 380 \, a c^{4} d^{5} x^{3} e^{5} + 285 \, a c^{4} d^{6} x^{2} e^{4} + 42 \, a c^{4} d^{7} x e^{3} - 8 \, a c^{4} d^{8} e^{2} + 110 \, a^{2} c^{3} d^{3} x^{3} e^{7} + 285 \, a^{2} c^{3} d^{4} x^{2} e^{6} + 222 \, a^{2} c^{3} d^{5} x e^{5} + 37 \, a^{2} c^{3} d^{6} e^{4} + 15 \, a^{3} c^{2} d^{2} x^{2} e^{8} + 42 \, a^{3} c^{2} d^{3} x e^{7} + 37 \, a^{3} c^{2} d^{4} e^{6} - 3 \, a^{4} c d x e^{9} - 8 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}}{3 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

-20*c^4*d^4*e^3*log(abs(c*d*x + a*e))/(c^8*d^15 - 7*a*c^7*d^13*e^2 + 21*a^2*c^6*d^11*e^4 - 35*a^3*c^5*d^9*e^6
+ 35*a^4*c^4*d^7*e^8 - 21*a^5*c^3*d^5*e^10 + 7*a^6*c^2*d^3*e^12 - a^7*c*d*e^14) + 20*c^3*d^3*e^4*log(abs(x*e +
d))/(c^7*d^14*e - 7*a*c^6*d^12*e^3 + 21*a^2*c^5*d^10*e^5 - 35*a^3*c^4*d^8*e^7 + 35*a^4*c^3*d^6*e^9 - 21*a^5*c
^2*d^4*e^11 + 7*a^6*c*d^2*e^13 - a^7*e^15) - 1/3*(60*c^5*d^5*x^5*e^5 + 150*c^5*d^6*x^4*e^4 + 110*c^5*d^7*x^3*e
^3 + 15*c^5*d^8*x^2*e^2 - 3*c^5*d^9*x*e + c^5*d^10 + 150*a*c^4*d^4*x^4*e^6 + 380*a*c^4*d^5*x^3*e^5 + 285*a*c^4
*d^6*x^2*e^4 + 42*a*c^4*d^7*x*e^3 - 8*a*c^4*d^8*e^2 + 110*a^2*c^3*d^3*x^3*e^7 + 285*a^2*c^3*d^4*x^2*e^6 + 222*
a^2*c^3*d^5*x*e^5 + 37*a^2*c^3*d^6*e^4 + 15*a^3*c^2*d^2*x^2*e^8 + 42*a^3*c^2*d^3*x*e^7 + 37*a^3*c^2*d^4*e^6 -
3*a^4*c*d*x*e^9 - 8*a^4*c*d^2*e^8 + a^5*e^10)/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*
d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \left \{\begin {array}{cl} -\frac {20\,c^3\,d^3\,e^3\,\ln \left (\frac {\frac {a\,e^2}{2}+\frac {c\,d^2}{2}-\sqrt {\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2}+c\,d\,e\,x}{\frac {a\,e^2}{2}+\frac {c\,d^2}{2}+\sqrt {\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2}+c\,d\,e\,x}\right )}{{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^{7/2}}-\frac {20\,\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,\left (\frac {c\,d\,e}{30\,\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^3}-\frac {c^2\,d^2\,e^2}{6\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^2}+\frac {c^3\,d^3\,e^3}{{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^3\,\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}\right )}{c\,d\,e} & \text {\ if\ \ }0<{\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\\ -\frac {20\,\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,\left (\frac {c\,d\,e}{30\,\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^3}-\frac {c^2\,d^2\,e^2}{6\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^2}+\frac {c^3\,d^3\,e^3}{{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^3\,\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}\right )}{c\,d\,e}-\frac {20\,c^3\,d^3\,e^3\,\mathrm {atan}\left (\frac {\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}}{\sqrt {a\,c\,d^2\,e^2-\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}}}\right )}{\sqrt {a\,c\,d^2\,e^2-\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}}\,{\left ({\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\right )}^3} & \text {\ if\ \ }{\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2<0\\ \int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^4} \,d x & \text {\ if\ \ }{\left (c\,d^2+a\,e^2\right )}^2-4\,a\,c\,d^2\,e^2\notin \mathbb {R}\vee {\left (c\,d^2+a\,e^2\right )}^2=4\,a\,c\,d^2\,e^2 \end {array}\right . \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)^4,x)

[Out]

piecewise(0 < (a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2, - (20*c^3*d^3*e^3*log(((a*e^2)/2 + (c*d^2)/2 - ((a*e^2 + c*d^
2)^2/4 - a*c*d^2*e^2)^(1/2) + c*d*e*x)/((a*e^2)/2 + (c*d^2)/2 + ((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2)^(1/2) + c*
d*e*x)))/((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^(7/2) - (20*((a*e^2)/2 + (c*d^2)/2 + c*d*e*x)*((c*d*e)/(30*((a*e^
2 + c*d^2)^2 - 4*a*c*d^2*e^2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3) - (c^2*d^2*e^2)/(6*((a*e^2 + c*d^2)^2
- 4*a*c*d^2*e^2)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2) + (c^3*d^3*e^3)/(((a*e^2 + c*d^2)^2 - 4*a*c*d^2
*e^2)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2))))/(c*d*e), (a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2 < 0, - (20*((a*e
^2)/2 + (c*d^2)/2 + c*d*e*x)*((c*d*e)/(30*((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d
*e*x^2)^3) - (c^2*d^2*e^2)/(6*((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2)
+ (c^3*d^3*e^3)/(((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2))))/(c*d*e) - (
20*c^3*d^3*e^3*atan(((a*e^2)/2 + (c*d^2)/2 + c*d*e*x)/(- (a*e^2 + c*d^2)^2/4 + a*c*d^2*e^2)^(1/2)))/((- (a*e^2
+ c*d^2)^2/4 + a*c*d^2*e^2)^(1/2)*((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^3), ~in((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e
^2, 'real') | (a*e^2 + c*d^2)^2 == 4*a*c*d^2*e^2, int(1/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4, x))

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