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

Optimal. Leaf size=223 \[ \frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac {6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac {3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac {e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]

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Rubi [A]  time = 0.22, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 44} \begin {gather*} \frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac {3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac {e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]

[Out]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 201, normalized size = 0.90 \begin {gather*} \frac {60 c^3 d^3 e^2 \log (a e+c d x)+\frac {24 c^3 d^3 e \left (c d^2-a e^2\right )}{a e+c d x}-\frac {3 c^3 d^3 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {36 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{d+e x}+\frac {9 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^2}-\frac {2 e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^3}-60 c^3 d^3 e^2 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]

[Out]

((-3*c^3*d^3*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (24*c^3*d^3*e*(c*d^2 - a*e^2))/(a*e + c*d*x) - (2*e^2*(-(c*d
^2) + a*e^2)^3)/(d + e*x)^3 + (9*c*d*(c*d^2*e - a*e^3)^2)/(d + e*x)^2 + (36*c^2*d^2*e^2*(c*d^2 - a*e^2))/(d +
e*x) + 60*c^3*d^3*e^2*Log[a*e + c*d*x] - 60*c^3*d^3*e^2*Log[d + e*x])/(6*(c*d^2 - a*e^2)^6)

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

[Out]

IntegrateAlgebraic[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3), x]

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fricas [B]  time = 0.44, size = 1222, normalized size = 5.48

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

-1/6*(3*c^5*d^10 - 30*a*c^4*d^8*e^2 - 20*a^2*c^3*d^6*e^4 + 60*a^3*c^2*d^4*e^6 - 15*a^4*c*d^2*e^8 + 2*a^5*e^10
- 60*(c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 - 30*(5*c^5*d^7*e^3 - 2*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^3 - 10*(11
*c^5*d^8*e^2 + 12*a*c^4*d^6*e^4 - 21*a^2*c^3*d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 - 5*(3*c^5*d^9*e + 32*a*c^4*d^7*
e^3 - 24*a^2*c^3*d^5*e^5 - 12*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x - 60*(c^5*d^5*e^5*x^5 + a^2*c^3*d^6*e^4 + (3*c^
5*d^6*e^4 + 2*a*c^4*d^4*e^6)*x^4 + (3*c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + (c^5*d^8*e^2 + 6*
a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6)*x^2 + (2*a*c^4*d^7*e^3 + 3*a^2*c^3*d^5*e^5)*x)*log(c*d*x + a*e) + 60*(c^5*d
^5*e^5*x^5 + a^2*c^3*d^6*e^4 + (3*c^5*d^6*e^4 + 2*a*c^4*d^4*e^6)*x^4 + (3*c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + a^2*
c^3*d^3*e^7)*x^3 + (c^5*d^8*e^2 + 6*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6)*x^2 + (2*a*c^4*d^7*e^3 + 3*a^2*c^3*d^5*
e^5)*x)*log(e*x + d))/(a^2*c^6*d^15*e^2 - 6*a^3*c^5*d^13*e^4 + 15*a^4*c^4*d^11*e^6 - 20*a^5*c^3*d^9*e^8 + 15*a
^6*c^2*d^7*e^10 - 6*a^7*c*d^5*e^12 + a^8*d^3*e^14 + (c^8*d^14*e^3 - 6*a*c^7*d^12*e^5 + 15*a^2*c^6*d^10*e^7 - 2
0*a^3*c^5*d^8*e^9 + 15*a^4*c^4*d^6*e^11 - 6*a^5*c^3*d^4*e^13 + a^6*c^2*d^2*e^15)*x^5 + (3*c^8*d^15*e^2 - 16*a*
c^7*d^13*e^4 + 33*a^2*c^6*d^11*e^6 - 30*a^3*c^5*d^9*e^8 + 5*a^4*c^4*d^7*e^10 + 12*a^5*c^3*d^5*e^12 - 9*a^6*c^2
*d^3*e^14 + 2*a^7*c*d*e^16)*x^4 + (3*c^8*d^16*e - 12*a*c^7*d^14*e^3 + 10*a^2*c^6*d^12*e^5 + 24*a^3*c^5*d^10*e^
7 - 60*a^4*c^4*d^8*e^9 + 52*a^5*c^3*d^6*e^11 - 18*a^6*c^2*d^4*e^13 + a^8*e^17)*x^3 + (c^8*d^17 - 18*a^2*c^6*d^
13*e^4 + 52*a^3*c^5*d^11*e^6 - 60*a^4*c^4*d^9*e^8 + 24*a^5*c^3*d^7*e^10 + 10*a^6*c^2*d^5*e^12 - 12*a^7*c*d^3*e
^14 + 3*a^8*d*e^16)*x^2 + (2*a*c^7*d^16*e - 9*a^2*c^6*d^14*e^3 + 12*a^3*c^5*d^12*e^5 + 5*a^4*c^4*d^10*e^7 - 30
*a^5*c^3*d^8*e^9 + 33*a^6*c^2*d^6*e^11 - 16*a^7*c*d^4*e^13 + 3*a^8*d^2*e^15)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: exp(1)^3/(d^3*exp(1)^7*a^3-3*d^3*exp(1)^
5*a^3*exp(2)+3*d^3*exp(1)^3*a^3*exp(2)^2-d^3*exp(1)*a^3*exp(2)^3)*ln(abs(x*exp(1)+d))-exp(1)^2/(2*d^3*exp(1)^6
*a^3-6*d^3*exp(1)^4*a^3*exp(2)+6*d^3*exp(1)^2*a^3*exp(2)^2-2*d^3*a^3*exp(2)^3)*ln(x^2*c*d*exp(1)+x*c*d^2+x*a*e
xp(2)+d*exp(1)*a)+(c^5*d^10*exp(1)^2-10*c^4*d^8*exp(1)^4*a+5*c^4*d^8*exp(1)^2*a*exp(2)+30*c^3*d^6*exp(1)^6*a^2
-30*c^3*d^6*exp(1)^4*a^2*exp(2)+10*c^3*d^6*exp(1)^2*a^2*exp(2)^2-30*c^2*d^4*exp(1)^6*a^3*exp(2)+30*c^2*d^4*exp
(1)^4*a^3*exp(2)^2-10*c^2*d^4*exp(1)^2*a^3*exp(2)^3+10*c*d^2*exp(1)^4*a^4*exp(2)^3-5*c*d^2*exp(1)^2*a^4*exp(2)
^4-exp(1)^2*a^5*exp(2)^5)/(c^4*d^11*exp(1)^6*a^3-3*c^4*d^11*exp(1)^4*a^3*exp(2)+3*c^4*d^11*exp(1)^2*a^3*exp(2)
^2-c^4*d^11*a^3*exp(2)^3-8*c^3*d^9*exp(1)^8*a^4+28*c^3*d^9*exp(1)^6*a^4*exp(2)-36*c^3*d^9*exp(1)^4*a^4*exp(2)^
2+20*c^3*d^9*exp(1)^2*a^4*exp(2)^3-4*c^3*d^9*a^4*exp(2)^4+16*c^2*d^7*exp(1)^10*a^5-64*c^2*d^7*exp(1)^8*a^5*exp
(2)+102*c^2*d^7*exp(1)^6*a^5*exp(2)^2-82*c^2*d^7*exp(1)^4*a^5*exp(2)^3+34*c^2*d^7*exp(1)^2*a^5*exp(2)^4-6*c^2*
d^7*a^5*exp(2)^5-8*c*d^5*exp(1)^8*a^6*exp(2)^2+28*c*d^5*exp(1)^6*a^6*exp(2)^3-36*c*d^5*exp(1)^4*a^6*exp(2)^4+2
0*c*d^5*exp(1)^2*a^6*exp(2)^5-4*c*d^5*a^6*exp(2)^6+d^3*exp(1)^6*a^7*exp(2)^4-3*d^3*exp(1)^4*a^7*exp(2)^5+3*d^3
*exp(1)^2*a^7*exp(2)^6-d^3*a^7*exp(2)^7)/sqrt(-c^2*d^4+4*a*c*d^2*exp(1)^2-a^2*exp(2)^2-2*a*c*d^2*exp(2))*atan(
(a*exp(2)+c*d^2+2*c*d*x*exp(1))/sqrt(-c^2*d^4+4*a*c*d^2*exp(1)^2-a^2*exp(2)^2-2*a*c*d^2*exp(2)))-((-4*c^5*d^9*
exp(1)^5*a+4*c^5*d^9*exp(1)^3*a*exp(2)+28*c^4*d^7*exp(1)^7*a^2-44*c^4*d^7*exp(1)^5*a^2*exp(2)+16*c^4*d^7*exp(1
)^3*a^2*exp(2)^2-28*c^3*d^5*exp(1)^7*a^3*exp(2)+44*c^3*d^5*exp(1)^5*a^3*exp(2)^2-16*c^3*d^5*exp(1)^3*a^3*exp(2
)^3+4*c^2*d^3*exp(1)^5*a^4*exp(2)^3-4*c^2*d^3*exp(1)^3*a^4*exp(2)^4)*x^3+(-4*c^5*d^10*exp(1)^4*a+4*c^5*d^10*ex
p(1)^2*a*exp(2)+26*c^4*d^8*exp(1)^6*a^2-48*c^4*d^8*exp(1)^4*a^2*exp(2)+22*c^4*d^8*exp(1)^2*a^2*exp(2)^2+32*c^3
*d^6*exp(1)^8*a^3-64*c^3*d^6*exp(1)^6*a^3*exp(2)+44*c^3*d^6*exp(1)^4*a^3*exp(2)^2-12*c^3*d^6*exp(1)^2*a^3*exp(
2)^3-58*c^2*d^4*exp(1)^6*a^4*exp(2)^2+96*c^2*d^4*exp(1)^4*a^4*exp(2)^3-38*c^2*d^4*exp(1)^2*a^4*exp(2)^4+8*c*d^
2*exp(1)^4*a^5*exp(2)^4-8*c*d^2*exp(1)^2*a^5*exp(2)^5)*x^2+(-8*c^4*d^9*exp(1)^5*a^2+8*c^4*d^9*exp(1)^3*a^2*exp
(2)+68*c^3*d^7*exp(1)^7*a^3-128*c^3*d^7*exp(1)^5*a^3*exp(2)+72*c^3*d^7*exp(1)^3*a^3*exp(2)^2-12*c^3*d^7*exp(1)
*a^3*exp(2)^3-4*c^2*d^5*exp(1)^7*a^4*exp(2)-32*c^2*d^5*exp(1)^5*a^4*exp(2)^2+64*c^2*d^5*exp(1)^3*a^4*exp(2)^3-
28*c^2*d^5*exp(1)*a^4*exp(2)^4-24*c*d^3*exp(1)^5*a^5*exp(2)^3+44*c*d^3*exp(1)^3*a^5*exp(2)^4-20*c*d^3*exp(1)*a
^5*exp(2)^5+4*d*exp(1)^3*a^6*exp(2)^5-4*d*exp(1)*a^6*exp(2)^6)*x-6*c^3*d^8*exp(1)^6*a^3+10*c^3*d^8*exp(1)^4*a^
3*exp(2)-6*c^3*d^8*exp(1)^2*a^3*exp(2)^2+2*c^3*d^8*a^3*exp(2)^3+48*c^2*d^6*exp(1)^8*a^4-112*c^2*d^6*exp(1)^6*a
^4*exp(2)+98*c^2*d^6*exp(1)^4*a^4*exp(2)^2-40*c^2*d^6*exp(1)^2*a^4*exp(2)^3+6*c^2*d^6*a^4*exp(2)^4-42*c*d^4*ex
p(1)^6*a^5*exp(2)^2+78*c*d^4*exp(1)^4*a^5*exp(2)^3-42*c*d^4*exp(1)^2*a^5*exp(2)^4+6*c*d^4*a^5*exp(2)^5+6*d^2*e
xp(1)^4*a^6*exp(2)^4-8*d^2*exp(1)^2*a^6*exp(2)^5+2*d^2*a^6*exp(2)^6)/4/(exp(2)-exp(1)^2)^3/a^3/d^3/(c^2*d^4-4*
c*d^2*exp(1)^2*a+2*c*d^2*a*exp(2)+a^2*exp(2)^2)^2/(x^2*c*d*exp(1)+x*c*d^2+x*a*exp(2)+d*exp(1)*a)^2

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maple [A]  time = 0.06, size = 218, normalized size = 0.98 \begin {gather*} -\frac {10 c^{3} d^{3} e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}+\frac {10 c^{3} d^{3} e^{2} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}-\frac {4 c^{3} d^{3} e}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (c d x +a e \right )}-\frac {c^{3} d^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )^{2}}-\frac {6 c^{2} d^{2} e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (e x +d \right )}+\frac {3 c d \,e^{2}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )^{2}}-\frac {e^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.72, size = 947, normalized size = 4.25 \begin {gather*} \frac {10 \, c^{3} d^{3} e^{2} \log \left (c d x + a e\right )}{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}} - \frac {10 \, c^{3} d^{3} e^{2} \log \left (e x + d\right )}{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}} + \frac {60 \, c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 27 \, a c^{3} d^{6} e^{2} + 47 \, a^{2} c^{2} d^{4} e^{4} - 13 \, a^{3} c d^{2} e^{6} + 2 \, a^{4} e^{8} + 30 \, {\left (5 \, c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 10 \, {\left (11 \, c^{4} d^{6} e^{2} + 23 \, a c^{3} d^{4} e^{4} + 2 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, c^{4} d^{7} e + 35 \, a c^{3} d^{5} e^{3} + 11 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{6 \, {\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} + {\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} + {\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} + {\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} + {\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} + {\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

10*c^3*d^3*e^2*log(c*d*x + a*e)/(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) - 10*c^3*d^3*e^2*log(e*x + d)/(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) + 1/6*(60*c^4*d^4*e^4*
x^4 - 3*c^4*d^8 + 27*a*c^3*d^6*e^2 + 47*a^2*c^2*d^4*e^4 - 13*a^3*c*d^2*e^6 + 2*a^4*e^8 + 30*(5*c^4*d^5*e^3 + 3
*a*c^3*d^3*e^5)*x^3 + 10*(11*c^4*d^6*e^2 + 23*a*c^3*d^4*e^4 + 2*a^2*c^2*d^2*e^6)*x^2 + 5*(3*c^4*d^7*e + 35*a*c
^3*d^5*e^3 + 11*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/(a^2*c^5*d^13*e^2 - 5*a^3*c^4*d^11*e^4 + 10*a^4*c^3*d^9*e^6
- 10*a^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d^3*e^12 + (c^7*d^12*e^3 - 5*a*c^6*d^10*e^5 + 10*a^2*c^5*d^8*e^7
- 10*a^3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - a^5*c^2*d^2*e^13)*x^5 + (3*c^7*d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*
a^2*c^5*d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c^3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 - 2*a^6*c*d*e^14)*x^4 + (3*c^7*
d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 + 25*a^3*c^4*d^8*e^7 - 35*a^4*c^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 -
a^6*c*d^2*e^13 - a^7*e^15)*x^3 + (c^7*d^15 + a*c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 35*a^3*c^4*d^9*e^6 - 25*a
^4*c^3*d^7*e^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3*e^12 - 3*a^7*d*e^14)*x^2 + (2*a*c^6*d^14*e - 7*a^2*c^5*d^12*e^
3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 20*a^5*c^2*d^6*e^9 + 13*a^6*c*d^4*e^11 - 3*a^7*d^2*e^13)*x)

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mupad [B]  time = 1.13, size = 878, normalized size = 3.94 \begin {gather*} \frac {20\,c^3\,d^3\,e^2\,\mathrm {atanh}\left (\frac {a^6\,e^{12}-4\,a^5\,c\,d^2\,e^{10}+5\,a^4\,c^2\,d^4\,e^8-5\,a^2\,c^4\,d^8\,e^4+4\,a\,c^5\,d^{10}\,e^2-c^6\,d^{12}}{{\left (a\,e^2-c\,d^2\right )}^6}+\frac {2\,c\,d\,e\,x\,\left (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}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}-\frac {\frac {2\,a^4\,e^8-13\,a^3\,c\,d^2\,e^6+47\,a^2\,c^2\,d^4\,e^4+27\,a\,c^3\,d^6\,e^2-3\,c^4\,d^8}{6\,\left (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}\right )}+\frac {5\,c^2\,d\,x^3\,\left (5\,c^2\,d^4\,e^3+3\,a\,c\,d^2\,e^5\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 {5\,c^2\,d^2\,x^2\,\left (2\,a^2\,e^6+23\,a\,c\,d^2\,e^4+11\,c^2\,d^4\,e^2\right )}{3\,\left (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}\right )}+\frac {10\,c^4\,d^4\,e^4\,x^4}{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 {5\,c\,d\,e\,x\,\left (-a^3\,e^6+11\,a^2\,c\,d^2\,e^4+35\,a\,c^2\,d^4\,e^2+3\,c^3\,d^6\right )}{6\,\left (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}\right )}}{x\,\left (3\,a^2\,d^2\,e^3+2\,c\,a\,d^4\,e\right )+x^2\,\left (3\,a^2\,d\,e^4+6\,a\,c\,d^3\,e^2+c^2\,d^5\right )+x^3\,\left (a^2\,e^5+6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )+x^4\,\left (3\,c^2\,d^3\,e^2+2\,a\,c\,d\,e^4\right )+a^2\,d^3\,e^2+c^2\,d^2\,e^3\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*c^3*d^3*e^2*atanh((a^6*e^12 - c^6*d^12 + 4*a*c^5*d^10*e^2 - 4*a^5*c*d^2*e^10 - 5*a^2*c^4*d^8*e^4 + 5*a^4*c
^2*d^4*e^8)/(a*e^2 - c*d^2)^6 + (2*c*d*e*x*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c
^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6))/(a*e^2 - c*d^2)^6))/(a*e^2 - c*d^2)^6 - ((2*a^4*e^8 - 3*c^4*d^8 + 27*a*c^3*d
^6*e^2 - 13*a^3*c*d^2*e^6 + 47*a^2*c^2*d^4*e^4)/(6*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 -
10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*c^2*d*x^3*(5*c^2*d^4*e^3 + 3*a*c*d^2*e^5))/(a^5*e^10 - c^5*d^10
+ 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6) + (5*c^2*d^2*x^2*(2*a^2*e^6 +
11*c^2*d^4*e^2 + 23*a*c*d^2*e^4))/(3*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6
*e^4 + 10*a^3*c^2*d^4*e^6)) + (10*c^4*d^4*e^4*x^4)/(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 -
10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6) + (5*c*d*e*x*(3*c^3*d^6 - a^3*e^6 + 35*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e
^4))/(6*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)))/
(x*(3*a^2*d^2*e^3 + 2*a*c*d^4*e) + x^2*(c^2*d^5 + 3*a^2*d*e^4 + 6*a*c*d^3*e^2) + x^3*(a^2*e^5 + 3*c^2*d^4*e +
6*a*c*d^2*e^3) + x^4*(3*c^2*d^3*e^2 + 2*a*c*d*e^4) + a^2*d^3*e^2 + c^2*d^2*e^3*x^5)

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sympy [B]  time = 4.47, size = 1357, normalized size = 6.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*c**3*d**3*e**2*log(x + (-10*a**7*c**3*d**3*e**16/(a*e**2 - c*d**2)**6 + 70*a**6*c**4*d**5*e**14/(a*e**2 -
c*d**2)**6 - 210*a**5*c**5*d**7*e**12/(a*e**2 - c*d**2)**6 + 350*a**4*c**6*d**9*e**10/(a*e**2 - c*d**2)**6 - 3
50*a**3*c**7*d**11*e**8/(a*e**2 - c*d**2)**6 + 210*a**2*c**8*d**13*e**6/(a*e**2 - c*d**2)**6 - 70*a*c**9*d**15
*e**4/(a*e**2 - c*d**2)**6 + 10*a*c**3*d**3*e**4 + 10*c**10*d**17*e**2/(a*e**2 - c*d**2)**6 + 10*c**4*d**5*e**
2)/(20*c**4*d**4*e**3))/(a*e**2 - c*d**2)**6 + 10*c**3*d**3*e**2*log(x + (10*a**7*c**3*d**3*e**16/(a*e**2 - c*
d**2)**6 - 70*a**6*c**4*d**5*e**14/(a*e**2 - c*d**2)**6 + 210*a**5*c**5*d**7*e**12/(a*e**2 - c*d**2)**6 - 350*
a**4*c**6*d**9*e**10/(a*e**2 - c*d**2)**6 + 350*a**3*c**7*d**11*e**8/(a*e**2 - c*d**2)**6 - 210*a**2*c**8*d**1
3*e**6/(a*e**2 - c*d**2)**6 + 70*a*c**9*d**15*e**4/(a*e**2 - c*d**2)**6 + 10*a*c**3*d**3*e**4 - 10*c**10*d**17
*e**2/(a*e**2 - c*d**2)**6 + 10*c**4*d**5*e**2)/(20*c**4*d**4*e**3))/(a*e**2 - c*d**2)**6 + (-2*a**4*e**8 + 13
*a**3*c*d**2*e**6 - 47*a**2*c**2*d**4*e**4 - 27*a*c**3*d**6*e**2 + 3*c**4*d**8 - 60*c**4*d**4*e**4*x**4 + x**3
*(-90*a*c**3*d**3*e**5 - 150*c**4*d**5*e**3) + x**2*(-20*a**2*c**2*d**2*e**6 - 230*a*c**3*d**4*e**4 - 110*c**4
*d**6*e**2) + x*(5*a**3*c*d*e**7 - 55*a**2*c**2*d**3*e**5 - 175*a*c**3*d**5*e**3 - 15*c**4*d**7*e))/(6*a**7*d*
*3*e**12 - 30*a**6*c*d**5*e**10 + 60*a**5*c**2*d**7*e**8 - 60*a**4*c**3*d**9*e**6 + 30*a**3*c**4*d**11*e**4 -
6*a**2*c**5*d**13*e**2 + x**5*(6*a**5*c**2*d**2*e**13 - 30*a**4*c**3*d**4*e**11 + 60*a**3*c**4*d**6*e**9 - 60*
a**2*c**5*d**8*e**7 + 30*a*c**6*d**10*e**5 - 6*c**7*d**12*e**3) + x**4*(12*a**6*c*d*e**14 - 42*a**5*c**2*d**3*
e**12 + 30*a**4*c**3*d**5*e**10 + 60*a**3*c**4*d**7*e**8 - 120*a**2*c**5*d**9*e**6 + 78*a*c**6*d**11*e**4 - 18
*c**7*d**13*e**2) + x**3*(6*a**7*e**15 + 6*a**6*c*d**2*e**13 - 102*a**5*c**2*d**4*e**11 + 210*a**4*c**3*d**6*e
**9 - 150*a**3*c**4*d**8*e**7 - 6*a**2*c**5*d**10*e**5 + 54*a*c**6*d**12*e**3 - 18*c**7*d**14*e) + x**2*(18*a*
*7*d*e**14 - 54*a**6*c*d**3*e**12 + 6*a**5*c**2*d**5*e**10 + 150*a**4*c**3*d**7*e**8 - 210*a**3*c**4*d**9*e**6
+ 102*a**2*c**5*d**11*e**4 - 6*a*c**6*d**13*e**2 - 6*c**7*d**15) + x*(18*a**7*d**2*e**13 - 78*a**6*c*d**4*e**
11 + 120*a**5*c**2*d**6*e**9 - 60*a**4*c**3*d**8*e**7 - 30*a**3*c**4*d**10*e**5 + 42*a**2*c**5*d**12*e**3 - 12
*a*c**6*d**14*e))

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