3.1906 \(\int \frac {1}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

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

[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 = 0.10, 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 = {614, 616, 31} \[ -\frac {10 c^2 d^2 e^2 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}+\frac {5 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}-\frac {a e^2+c d^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*d^2 + a*e^2 + 2*c*d*e*x)/(3*(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 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 )^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 = 0.25, size = 234, normalized size = 0.89 \[ \frac {60 c^3 d^3 e^3 \log (a e+c d x)+\frac {30 c^3 d^3 e^2 \left (c d^2-a e^2\right )}{a e+c d x}-\frac {6 c^3 d^3 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {c^3 d^3 \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}+\frac {30 c^2 d^2 e^3 \left (c d^2-a e^2\right )}{d+e x}+\frac {\left (c d^2 e-a e^3\right )^3}{(d+e x)^3}+\frac {6 c d e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}-60 c^3 d^3 e^3 \log (d+e x)}{3 \left (a e^2-c d^2\right )^7} \]

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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fricas [B]  time = 1.53, size = 1618, normalized size = 6.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/3*(c^6*d^12 - 9*a*c^5*d^10*e^2 + 45*a^2*c^4*d^8*e^4 - 45*a^4*c^2*d^4*e^8 + 9*a^5*c*d^2*e^10 - a^6*e^12 + 60
*(c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 150*(c^6*d^8*e^4 - a^2*c^4*d^4*e^8)*x^4 + 10*(11*c^6*d^9*e^3 + 27*a*c^5*d
^7*e^5 - 27*a^2*c^4*d^5*e^7 - 11*a^3*c^3*d^3*e^9)*x^3 + 15*(c^6*d^10*e^2 + 18*a*c^5*d^8*e^4 - 18*a^3*c^3*d^4*e
^8 - a^4*c^2*d^2*e^10)*x^2 - 3*(c^6*d^11*e - 15*a*c^5*d^9*e^3 - 60*a^2*c^4*d^7*e^5 + 60*a^3*c^3*d^5*e^7 + 15*a
^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x + 60*(c^6*d^6*e^6*x^6 + a^3*c^3*d^6*e^6 + 3*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^5
+ 3*(c^6*d^8*e^4 + 3*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + (c^6*d^9*e^3 + 9*a*c^5*d^7*e^5 + 9*a^2*c^4*d^5*e^
7 + a^3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^4 + 3*a^2*c^4*d^6*e^6 + a^3*c^3*d^4*e^8)*x^2 + 3*(a^2*c^4*d^7*e^5 +
a^3*c^3*d^5*e^7)*x)*log(c*d*x + a*e) - 60*(c^6*d^6*e^6*x^6 + a^3*c^3*d^6*e^6 + 3*(c^6*d^7*e^5 + a*c^5*d^5*e^7)
*x^5 + 3*(c^6*d^8*e^4 + 3*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + (c^6*d^9*e^3 + 9*a*c^5*d^7*e^5 + 9*a^2*c^4*d^
5*e^7 + a^3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^4 + 3*a^2*c^4*d^6*e^6 + a^3*c^3*d^4*e^8)*x^2 + 3*(a^2*c^4*d^7*e^
5 + a^3*c^3*d^5*e^7)*x)*log(e*x + d))/(a^3*c^7*d^17*e^3 - 7*a^4*c^6*d^15*e^5 + 21*a^5*c^5*d^13*e^7 - 35*a^6*c^
4*d^11*e^9 + 35*a^7*c^3*d^9*e^11 - 21*a^8*c^2*d^7*e^13 + 7*a^9*c*d^5*e^15 - a^10*d^3*e^17 + (c^10*d^17*e^3 - 7
*a*c^9*d^15*e^5 + 21*a^2*c^8*d^13*e^7 - 35*a^3*c^7*d^11*e^9 + 35*a^4*c^6*d^9*e^11 - 21*a^5*c^5*d^7*e^13 + 7*a^
6*c^4*d^5*e^15 - a^7*c^3*d^3*e^17)*x^6 + 3*(c^10*d^18*e^2 - 6*a*c^9*d^16*e^4 + 14*a^2*c^8*d^14*e^6 - 14*a^3*c^
7*d^12*e^8 + 14*a^5*c^5*d^8*e^12 - 14*a^6*c^4*d^6*e^14 + 6*a^7*c^3*d^4*e^16 - a^8*c^2*d^2*e^18)*x^5 + 3*(c^10*
d^19*e - 4*a*c^9*d^17*e^3 + a^2*c^8*d^15*e^5 + 21*a^3*c^7*d^13*e^7 - 49*a^4*c^6*d^11*e^9 + 49*a^5*c^5*d^9*e^11
- 21*a^6*c^4*d^7*e^13 - a^7*c^3*d^5*e^15 + 4*a^8*c^2*d^3*e^17 - a^9*c*d*e^19)*x^4 + (c^10*d^20 + 2*a*c^9*d^18
*e^2 - 33*a^2*c^8*d^16*e^4 + 92*a^3*c^7*d^14*e^6 - 98*a^4*c^6*d^12*e^8 + 98*a^6*c^4*d^8*e^12 - 92*a^7*c^3*d^6*
e^14 + 33*a^8*c^2*d^4*e^16 - 2*a^9*c*d^2*e^18 - a^10*e^20)*x^3 + 3*(a*c^9*d^19*e - 4*a^2*c^8*d^17*e^3 + a^3*c^
7*d^15*e^5 + 21*a^4*c^6*d^13*e^7 - 49*a^5*c^5*d^11*e^9 + 49*a^6*c^4*d^9*e^11 - 21*a^7*c^3*d^7*e^13 - a^8*c^2*d
^5*e^15 + 4*a^9*c*d^3*e^17 - a^10*d*e^19)*x^2 + 3*(a^2*c^8*d^18*e^2 - 6*a^3*c^7*d^16*e^4 + 14*a^4*c^6*d^14*e^6
- 14*a^5*c^5*d^12*e^8 + 14*a^7*c^3*d^8*e^12 - 14*a^8*c^2*d^6*e^14 + 6*a^9*c*d^4*e^16 - a^10*d^2*e^18)*x)

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giac [B]  time = 0.22, size = 532, normalized size = 2.03 \[ -\frac {40 \, c^{3} d^{3} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right ) e^{3}}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac {60 \, c^{5} d^{5} x^{5} e^{5} + 150 \, c^{5} d^{6} x^{4} e^{4} + 110 \, c^{5} d^{7} x^{3} e^{3} + 15 \, c^{5} d^{8} x^{2} e^{2} - 3 \, c^{5} d^{9} x e + c^{5} d^{10} + 150 \, a c^{4} d^{4} x^{4} e^{6} + 380 \, a c^{4} d^{5} x^{3} e^{5} + 285 \, a c^{4} d^{6} x^{2} e^{4} + 42 \, a c^{4} d^{7} x e^{3} - 8 \, a c^{4} d^{8} e^{2} + 110 \, a^{2} c^{3} d^{3} x^{3} e^{7} + 285 \, a^{2} c^{3} d^{4} x^{2} e^{6} + 222 \, a^{2} c^{3} d^{5} x e^{5} + 37 \, a^{2} c^{3} d^{6} e^{4} + 15 \, a^{3} c^{2} d^{2} x^{2} e^{8} + 42 \, a^{3} c^{2} d^{3} x e^{7} + 37 \, a^{3} c^{2} d^{4} e^{6} - 3 \, a^{4} c d x e^{9} - 8 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}}{3 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \]

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]

-40*c^3*d^3*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^3/((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)*sqr
t(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) - 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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maple [A]  time = 0.05, size = 253, normalized size = 0.97 \[ -\frac {20 c^{3} d^{3} e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{7}}+\frac {20 c^{3} d^{3} e^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{7}}-\frac {10 c^{3} d^{3} e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{6} \left (c d x +a e \right )}-\frac {2 c^{3} d^{3} e}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (c d x +a e \right )^{2}}-\frac {10 c^{2} d^{2} e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{6} \left (e x +d \right )}-\frac {c^{3} d^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )^{3}}+\frac {2 c d \,e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (e x +d \right )^{2}}-\frac {e^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )^{3}} \]

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)

[Out]

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

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maxima [B]  time = 1.62, size = 1278, normalized size = 4.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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(e*x + 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*e^5*x^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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \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 . \]

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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sympy [B]  time = 6.63, size = 1748, normalized size = 6.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**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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