3.1884 \(\int \frac{1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=176 \[ -\frac{c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac{3 c^2 d^2 e}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac{4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5}-\frac{c d e}{(d+e x)^2 \left (c d^2-a e^2\right )^3}-\frac{e}{3 (d+e x)^3 \left (c d^2-a e^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.153476, antiderivative size = 176, normalized size of antiderivative = 1., 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} \[ -\frac{c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac{3 c^2 d^2 e}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac{4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5}-\frac{c d e}{(d+e x)^2 \left (c d^2-a e^2\right )^3}-\frac{e}{3 (d+e x)^3 \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.141354, size = 160, normalized size = 0.91 \[ \frac{\frac{3 c^3 d^3 \left (c d^2-a e^2\right )}{a e+c d x}+\frac{9 c^2 d^2 e \left (c d^2-a e^2\right )}{d+e x}+12 c^3 d^3 e \log (a e+c d x)+\frac{3 c d e \left (c d^2-a e^2\right )^2}{(d+e x)^2}-\frac{e \left (a e^2-c d^2\right )^3}{(d+e x)^3}-12 c^3 d^3 e \log (d+e x)}{3 \left (a e^2-c d^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*c^3*d^3*(c*d^2 - a*e^2))/(a*e + c*d*x) - (e*(-(c*d^2) + a*e^2)^3)/(d + e*x)^3 + (3*c*d*e*(c*d^2 - a*e^2)^2
)/(d + e*x)^2 + (9*c^2*d^2*e*(c*d^2 - a*e^2))/(d + e*x) + 12*c^3*d^3*e*Log[a*e + c*d*x] - 12*c^3*d^3*e*Log[d +
e*x])/(3*(-(c*d^2) + a*e^2)^5)

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Maple [A]  time = 0.056, size = 174, normalized size = 1. \begin{align*} -{\frac{e}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{c}^{3}e{d}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}-3\,{\frac{{c}^{2}{d}^{2}e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) }}+{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) }}+4\,{\frac{{c}^{3}e{d}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.14173, size = 865, normalized size = 4.91 \begin{align*} -\frac{4 \, c^{3} d^{3} e \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{4 \, c^{3} d^{3} e \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac{12 \, c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{6} + 13 \, a c^{2} d^{4} e^{2} - 5 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{3 \,{\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} +{\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} +{\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \,{\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} +{\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.06131, size = 1615, normalized size = 9.18 \begin{align*} -\frac{3 \, c^{4} d^{8} + 10 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 6 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + 12 \,{\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (11 \, c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x + 12 \,{\left (c^{4} d^{4} e^{4} x^{4} + a c^{3} d^{6} e^{2} +{\left (3 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (c^{4} d^{6} e^{2} + a c^{3} d^{4} e^{4}\right )} x^{2} +{\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 12 \,{\left (c^{4} d^{4} e^{4} x^{4} + a c^{3} d^{6} e^{2} +{\left (3 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (c^{4} d^{6} e^{2} + a c^{3} d^{4} e^{4}\right )} x^{2} +{\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a c^{5} d^{13} e - 5 \, a^{2} c^{4} d^{11} e^{3} + 10 \, a^{3} c^{3} d^{9} e^{5} - 10 \, a^{4} c^{2} d^{7} e^{7} + 5 \, a^{5} c d^{5} e^{9} - a^{6} d^{3} e^{11} +{\left (c^{6} d^{11} e^{3} - 5 \, a c^{5} d^{9} e^{5} + 10 \, a^{2} c^{4} d^{7} e^{7} - 10 \, a^{3} c^{3} d^{5} e^{9} + 5 \, a^{4} c^{2} d^{3} e^{11} - a^{5} c d e^{13}\right )} x^{4} +{\left (3 \, c^{6} d^{12} e^{2} - 14 \, a c^{5} d^{10} e^{4} + 25 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 5 \, a^{4} c^{2} d^{4} e^{10} + 2 \, a^{5} c d^{2} e^{12} - a^{6} e^{14}\right )} x^{3} + 3 \,{\left (c^{6} d^{13} e - 4 \, a c^{5} d^{11} e^{3} + 5 \, a^{2} c^{4} d^{9} e^{5} - 5 \, a^{4} c^{2} d^{5} e^{9} + 4 \, a^{5} c d^{3} e^{11} - a^{6} d e^{13}\right )} x^{2} +{\left (c^{6} d^{14} - 2 \, a c^{5} d^{12} e^{2} - 5 \, a^{2} c^{4} d^{10} e^{4} + 20 \, a^{3} c^{3} d^{8} e^{6} - 25 \, a^{4} c^{2} d^{6} e^{8} + 14 \, a^{5} c d^{4} e^{10} - 3 \, a^{6} d^{2} e^{12}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c^4*d^8 + 10*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 6*a^3*c*d^2*e^6 - a^4*e^8 + 12*(c^4*d^5*e^3 - a*c^3*
d^3*e^5)*x^3 + 6*(5*c^4*d^6*e^2 - 4*a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 2*(11*c^4*d^7*e - 3*a*c^3*d^5*e^3 -
9*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x + 12*(c^4*d^4*e^4*x^4 + a*c^3*d^6*e^2 + (3*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^
3 + 3*(c^4*d^6*e^2 + a*c^3*d^4*e^4)*x^2 + (c^4*d^7*e + 3*a*c^3*d^5*e^3)*x)*log(c*d*x + a*e) - 12*(c^4*d^4*e^4*
x^4 + a*c^3*d^6*e^2 + (3*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 3*(c^4*d^6*e^2 + a*c^3*d^4*e^4)*x^2 + (c^4*d^7*e +
3*a*c^3*d^5*e^3)*x)*log(e*x + d))/(a*c^5*d^13*e - 5*a^2*c^4*d^11*e^3 + 10*a^3*c^3*d^9*e^5 - 10*a^4*c^2*d^7*e^
7 + 5*a^5*c*d^5*e^9 - a^6*d^3*e^11 + (c^6*d^11*e^3 - 5*a*c^5*d^9*e^5 + 10*a^2*c^4*d^7*e^7 - 10*a^3*c^3*d^5*e^9
+ 5*a^4*c^2*d^3*e^11 - a^5*c*d*e^13)*x^4 + (3*c^6*d^12*e^2 - 14*a*c^5*d^10*e^4 + 25*a^2*c^4*d^8*e^6 - 20*a^3*
c^3*d^6*e^8 + 5*a^4*c^2*d^4*e^10 + 2*a^5*c*d^2*e^12 - a^6*e^14)*x^3 + 3*(c^6*d^13*e - 4*a*c^5*d^11*e^3 + 5*a^2
*c^4*d^9*e^5 - 5*a^4*c^2*d^5*e^9 + 4*a^5*c*d^3*e^11 - a^6*d*e^13)*x^2 + (c^6*d^14 - 2*a*c^5*d^12*e^2 - 5*a^2*c
^4*d^10*e^4 + 20*a^3*c^3*d^8*e^6 - 25*a^4*c^2*d^6*e^8 + 14*a^5*c*d^4*e^10 - 3*a^6*d^2*e^12)*x)

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Sympy [B]  time = 3.9785, size = 994, normalized size = 5.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c**3*d**3*e*log(x + (-4*a**6*c**3*d**3*e**13/(a*e**2 - c*d**2)**5 + 24*a**5*c**4*d**5*e**11/(a*e**2 - c*d**
2)**5 - 60*a**4*c**5*d**7*e**9/(a*e**2 - c*d**2)**5 + 80*a**3*c**6*d**9*e**7/(a*e**2 - c*d**2)**5 - 60*a**2*c*
*7*d**11*e**5/(a*e**2 - c*d**2)**5 + 24*a*c**8*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*a*c**3*d**3*e**3 - 4*c**9*d
**15*e/(a*e**2 - c*d**2)**5 + 4*c**4*d**5*e)/(8*c**4*d**4*e**2))/(a*e**2 - c*d**2)**5 + 4*c**3*d**3*e*log(x +
(4*a**6*c**3*d**3*e**13/(a*e**2 - c*d**2)**5 - 24*a**5*c**4*d**5*e**11/(a*e**2 - c*d**2)**5 + 60*a**4*c**5*d**
7*e**9/(a*e**2 - c*d**2)**5 - 80*a**3*c**6*d**9*e**7/(a*e**2 - c*d**2)**5 + 60*a**2*c**7*d**11*e**5/(a*e**2 -
c*d**2)**5 - 24*a*c**8*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*a*c**3*d**3*e**3 + 4*c**9*d**15*e/(a*e**2 - c*d**2)
**5 + 4*c**4*d**5*e)/(8*c**4*d**4*e**2))/(a*e**2 - c*d**2)**5 - (a**3*e**6 - 5*a**2*c*d**2*e**4 + 13*a*c**2*d*
*4*e**2 + 3*c**3*d**6 + 12*c**3*d**3*e**3*x**3 + x**2*(6*a*c**2*d**2*e**4 + 30*c**3*d**4*e**2) + x*(-2*a**2*c*
d*e**5 + 16*a*c**2*d**3*e**3 + 22*c**3*d**5*e))/(3*a**5*d**3*e**9 - 12*a**4*c*d**5*e**7 + 18*a**3*c**2*d**7*e*
*5 - 12*a**2*c**3*d**9*e**3 + 3*a*c**4*d**11*e + x**4*(3*a**4*c*d*e**11 - 12*a**3*c**2*d**3*e**9 + 18*a**2*c**
3*d**5*e**7 - 12*a*c**4*d**7*e**5 + 3*c**5*d**9*e**3) + x**3*(3*a**5*e**12 - 3*a**4*c*d**2*e**10 - 18*a**3*c**
2*d**4*e**8 + 42*a**2*c**3*d**6*e**6 - 33*a*c**4*d**8*e**4 + 9*c**5*d**10*e**2) + x**2*(9*a**5*d*e**11 - 27*a*
*4*c*d**3*e**9 + 18*a**3*c**2*d**5*e**7 + 18*a**2*c**3*d**7*e**5 - 27*a*c**4*d**9*e**3 + 9*c**5*d**11*e) + x*(
9*a**5*d**2*e**10 - 33*a**4*c*d**4*e**8 + 42*a**3*c**2*d**6*e**6 - 18*a**2*c**3*d**8*e**4 - 3*a*c**4*d**10*e**
2 + 3*c**5*d**12))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError