3.19.94 \(\int \frac {1}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1894]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {628, 630, 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]

-1/2*(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)^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 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 )^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*}

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Mathematica [A]
time = 0.10, size = 168, normalized size = 0.88 \begin {gather*} \frac {\frac {c^2 d^2 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {6 c^2 d^2 e \left (-c d^2+a e^2\right )}{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 (-c d^2+a e^2\right )}{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 \left (-c d^2+a e^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)

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Maple [A]
time = 0.73, size = 186, normalized size = 0.97

method	result	size
default	\(-\frac {e^{2}}{2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{2}}+\frac {6 e^{2} c^{2} d^{2} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}+\frac {3 e^{2} c d}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )}+\frac {c^{2} d^{2}}{2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}-\frac {6 e^{2} c^{2} d^{2} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}+\frac {3 c^{2} d^{2} e}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (c d x +a e \right )}\)	\(186\)
risch	\(\frac {\frac {6 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {9 c^{2} d^{2} e^{2} \left (e^{2} a +c \,d^{2}\right ) x^{2}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {2 \left (a^{2} e^{4}+7 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d e x}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {e^{6} a^{3}-7 e^{4} d^{2} a^{2} c -7 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{2}}-\frac {6 c^{2} d^{2} e^{2} \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {6 c^{2} d^{2} e^{2} \ln \left (-e x -d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\)	\(541\)
norman	\(\frac {\frac {\left (9 a \,c^{4} d^{4} e^{6}+9 c^{5} d^{6} e^{4}\right ) x^{2}}{e^{2} d^{2} c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {-a^{3} c^{2} e^{6}+7 c^{3} a^{2} d^{2} e^{4}+7 c^{4} a \,d^{4} e^{2}-c^{5} d^{6}}{2 c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {6 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {2 \left (a^{2} c^{3} d^{2} e^{6}+7 a \,c^{4} d^{4} e^{4}+c^{5} d^{6} e^{2}\right ) x}{e d \,c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}+\frac {6 c^{2} d^{2} e^{2} \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}-\frac {6 c^{2} d^{2} e^{2} \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\)	\(568\)

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,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (188) = 376\).
time = 0.30, size = 603, normalized size = 3.16 \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 (x e + 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} x^{3} e^{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(x*e + 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*x^3*e^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] Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (188) = 376\).
time = 3.85, size = 818, normalized size = 4.28 \begin {gather*} \frac {4 \, c^{4} d^{7} x e - c^{4} d^{8} - 4 \, a^{3} c d x e^{7} + a^{4} e^{8} - 2 \, {\left (9 \, a^{2} c^{2} d^{2} x^{2} + 4 \, a^{3} c d^{2}\right )} e^{6} - 12 \, {\left (a c^{3} d^{3} x^{3} + 2 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 12 \, {\left (c^{4} d^{5} x^{3} + 2 \, a c^{3} d^{5} x\right )} e^{3} + 2 \, {\left (9 \, c^{4} d^{6} x^{2} + 4 \, a c^{3} d^{6}\right )} e^{2} + 12 \, {\left (c^{4} d^{6} x^{2} e^{2} + a^{2} c^{2} d^{2} x^{2} e^{6} + 2 \, {\left (a c^{3} d^{3} x^{3} + a^{2} c^{2} d^{3} x\right )} e^{5} + {\left (c^{4} d^{4} x^{4} + 4 \, a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (c^{4} d^{5} x^{3} + a c^{3} d^{5} x\right )} e^{3}\right )} \log \left (c d x + a e\right ) - 12 \, {\left (c^{4} d^{6} x^{2} e^{2} + a^{2} c^{2} d^{2} x^{2} e^{6} + 2 \, {\left (a c^{3} d^{3} x^{3} + a^{2} c^{2} d^{3} x\right )} e^{5} + {\left (c^{4} d^{4} x^{4} + 4 \, a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (c^{4} d^{5} x^{3} + a c^{3} d^{5} x\right )} e^{3}\right )} \log \left (x e + d\right )}{2 \, {\left (c^{7} d^{14} x^{2} - a^{7} x^{2} e^{14} - 2 \, {\left (a^{6} c d x^{3} + a^{7} d x\right )} e^{13} - {\left (a^{5} c^{2} d^{2} x^{4} - a^{6} c d^{2} x^{2} + a^{7} d^{2}\right )} e^{12} + 8 \, {\left (a^{5} c^{2} d^{3} x^{3} + a^{6} c d^{3} x\right )} e^{11} + {\left (5 \, a^{4} c^{3} d^{4} x^{4} + 9 \, a^{5} c^{2} d^{4} x^{2} + 5 \, a^{6} c d^{4}\right )} e^{10} - 10 \, {\left (a^{4} c^{3} d^{5} x^{3} + a^{5} c^{2} d^{5} x\right )} e^{9} - 5 \, {\left (2 \, a^{3} c^{4} d^{6} x^{4} + 5 \, a^{4} c^{3} d^{6} x^{2} + 2 \, a^{5} c^{2} d^{6}\right )} e^{8} + 5 \, {\left (2 \, a^{2} c^{5} d^{8} x^{4} + 5 \, a^{3} c^{4} d^{8} x^{2} + 2 \, a^{4} c^{3} d^{8}\right )} e^{6} + 10 \, {\left (a^{2} c^{5} d^{9} x^{3} + a^{3} c^{4} d^{9} x\right )} e^{5} - {\left (5 \, a c^{6} d^{10} x^{4} + 9 \, a^{2} c^{5} d^{10} x^{2} + 5 \, a^{3} c^{4} d^{10}\right )} e^{4} - 8 \, {\left (a c^{6} d^{11} x^{3} + a^{2} c^{5} d^{11} x\right )} e^{3} + {\left (c^{7} d^{12} x^{4} - a c^{6} d^{12} x^{2} + a^{2} c^{5} d^{12}\right )} e^{2} + 2 \, {\left (c^{7} d^{13} x^{3} + a c^{6} d^{13} x\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)^3,x, algorithm="fricas")

[Out]

1/2*(4*c^4*d^7*x*e - c^4*d^8 - 4*a^3*c*d*x*e^7 + a^4*e^8 - 2*(9*a^2*c^2*d^2*x^2 + 4*a^3*c*d^2)*e^6 - 12*(a*c^3
*d^3*x^3 + 2*a^2*c^2*d^3*x)*e^5 + 12*(c^4*d^5*x^3 + 2*a*c^3*d^5*x)*e^3 + 2*(9*c^4*d^6*x^2 + 4*a*c^3*d^6)*e^2 +
12*(c^4*d^6*x^2*e^2 + a^2*c^2*d^2*x^2*e^6 + 2*(a*c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^5 + (c^4*d^4*x^4 + 4*a*c^3*d^
4*x^2 + a^2*c^2*d^4)*e^4 + 2*(c^4*d^5*x^3 + a*c^3*d^5*x)*e^3)*log(c*d*x + a*e) - 12*(c^4*d^6*x^2*e^2 + a^2*c^2
*d^2*x^2*e^6 + 2*(a*c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^5 + (c^4*d^4*x^4 + 4*a*c^3*d^4*x^2 + a^2*c^2*d^4)*e^4 + 2*(
c^4*d^5*x^3 + a*c^3*d^5*x)*e^3)*log(x*e + d))/(c^7*d^14*x^2 - a^7*x^2*e^14 - 2*(a^6*c*d*x^3 + a^7*d*x)*e^13 -
(a^5*c^2*d^2*x^4 - a^6*c*d^2*x^2 + a^7*d^2)*e^12 + 8*(a^5*c^2*d^3*x^3 + a^6*c*d^3*x)*e^11 + (5*a^4*c^3*d^4*x^4
+ 9*a^5*c^2*d^4*x^2 + 5*a^6*c*d^4)*e^10 - 10*(a^4*c^3*d^5*x^3 + a^5*c^2*d^5*x)*e^9 - 5*(2*a^3*c^4*d^6*x^4 + 5
*a^4*c^3*d^6*x^2 + 2*a^5*c^2*d^6)*e^8 + 5*(2*a^2*c^5*d^8*x^4 + 5*a^3*c^4*d^8*x^2 + 2*a^4*c^3*d^8)*e^6 + 10*(a^
2*c^5*d^9*x^3 + a^3*c^4*d^9*x)*e^5 - (5*a*c^6*d^10*x^4 + 9*a^2*c^5*d^10*x^2 + 5*a^3*c^4*d^10)*e^4 - 8*(a*c^6*d
^11*x^3 + a^2*c^5*d^11*x)*e^3 + (c^7*d^12*x^4 - a*c^6*d^12*x^2 + a^2*c^5*d^12)*e^2 + 2*(c^7*d^13*x^3 + a*c^6*d
^13*x)*e)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (187) = 374\).
time = 2.35, 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} \cdot \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} \cdot \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} \cdot \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} \cdot \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))

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Giac [A]
time = 1.94, size = 367, normalized size = 1.92 \begin {gather*} \frac {6 \, c^{3} d^{3} e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{11} - 5 \, a c^{5} d^{9} e^{2} + 10 \, a^{2} c^{4} d^{7} e^{4} - 10 \, a^{3} c^{3} d^{5} e^{6} + 5 \, a^{4} c^{2} d^{3} e^{8} - a^{5} c d e^{10}} - \frac {6 \, c^{2} d^{2} e^{3} \log \left ({\left | x e + d \right |}\right )}{c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}} + \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]

6*c^3*d^3*e^2*log(abs(c*d*x + a*e))/(c^6*d^11 - 5*a*c^5*d^9*e^2 + 10*a^2*c^4*d^7*e^4 - 10*a^3*c^3*d^5*e^6 + 5*
a^4*c^2*d^3*e^8 - a^5*c*d*e^10) - 6*c^2*d^2*e^3*log(abs(x*e + d))/(c^5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d
^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 - a^5*e^11) + 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)

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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

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