3.1883 \(\int \frac{1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=146 \[ -\frac{c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4}-\frac{2 c d e}{(d+e x) \left (c d^2-a e^2\right )^3}-\frac{e}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.109691, antiderivative size = 146, 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} \[ -\frac{c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4}-\frac{2 c d e}{(d+e x) \left (c d^2-a e^2\right )^3}-\frac{e}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2} \]

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)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0953845, size = 130, normalized size = 0.89 \[ \frac{\frac{2 c^2 d^2 \left (a e^2-c d^2\right )}{a e+c d x}-6 c^2 d^2 e \log (a e+c d x)+\frac{4 c d e \left (a e^2-c d^2\right )}{d+e x}-\frac{e \left (c d^2-a e^2\right )^2}{(d+e x)^2}+6 c^2 d^2 e \log (d+e x)}{2 \left (c d^2-a e^2\right )^4} \]

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)^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 144, normalized size = 1. \begin{align*} -{\frac{e}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{2}{d}^{2}e\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}+2\,{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) }}+{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) }}-3\,{\frac{{c}^{2}{d}^{2}e\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}} \end{align*}

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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.12091, size = 571, normalized size = 3.91 \begin{align*} -\frac{3 \, c^{2} d^{2} e \log \left (c d x + a e\right )}{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}} + \frac{3 \, c^{2} d^{2} e \log \left (e x + d\right )}{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}} - \frac{6 \, c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{4} + 5 \, a c d^{2} e^{2} - a^{2} e^{4} + 3 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \,{\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} +{\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} +{\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} +{\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.00916, size = 1077, normalized size = 7.38 \begin{align*} -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + a c^{2} d^{4} e^{2} +{\left (2 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + a c^{2} d^{4} e^{2} +{\left (2 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a c^{4} d^{10} e - 4 \, a^{2} c^{3} d^{8} e^{3} + 6 \, a^{3} c^{2} d^{6} e^{5} - 4 \, a^{4} c d^{4} e^{7} + a^{5} d^{2} e^{9} +{\left (c^{5} d^{9} e^{2} - 4 \, a c^{4} d^{7} e^{4} + 6 \, a^{2} c^{3} d^{5} e^{6} - 4 \, a^{3} c^{2} d^{3} e^{8} + a^{4} c d e^{10}\right )} x^{3} +{\left (2 \, c^{5} d^{10} e - 7 \, a c^{4} d^{8} e^{3} + 8 \, a^{2} c^{3} d^{6} e^{5} - 2 \, a^{3} c^{2} d^{4} e^{7} - 2 \, a^{4} c d^{2} e^{9} + a^{5} e^{11}\right )} x^{2} +{\left (c^{5} d^{11} - 2 \, a c^{4} d^{9} e^{2} - 2 \, a^{2} c^{3} d^{7} e^{4} + 8 \, a^{3} c^{2} d^{5} e^{6} - 7 \, a^{4} c d^{3} e^{8} + 2 \, a^{5} d e^{10}\right )} x\right )}} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.54374, size = 734, normalized size = 5.03 \begin{align*} \frac{3 c^{2} d^{2} e \log{\left (x + \frac{- \frac{3 a^{5} c^{2} d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{15 a^{4} c^{3} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{30 a^{3} c^{4} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{30 a^{2} c^{5} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{15 a c^{6} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c^{2} d^{2} e^{3} + \frac{3 c^{7} d^{12} e}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{3} d^{4} e}{6 c^{3} d^{3} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{3 c^{2} d^{2} e \log{\left (x + \frac{\frac{3 a^{5} c^{2} d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{15 a^{4} c^{3} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{30 a^{3} c^{4} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{30 a^{2} c^{5} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{15 a c^{6} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c^{2} d^{2} e^{3} - \frac{3 c^{7} d^{12} e}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{3} d^{4} e}{6 c^{3} d^{3} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{- a^{2} e^{4} + 5 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 9 c^{2} d^{3} e\right )}{2 a^{4} d^{2} e^{7} - 6 a^{3} c d^{4} e^{5} + 6 a^{2} c^{2} d^{6} e^{3} - 2 a c^{3} d^{8} e + x^{3} \left (2 a^{3} c d e^{8} - 6 a^{2} c^{2} d^{3} e^{6} + 6 a c^{3} d^{5} e^{4} - 2 c^{4} d^{7} e^{2}\right ) + x^{2} \left (2 a^{4} e^{9} - 2 a^{3} c d^{2} e^{7} - 6 a^{2} c^{2} d^{4} e^{5} + 10 a c^{3} d^{6} e^{3} - 4 c^{4} d^{8} e\right ) + x \left (4 a^{4} d e^{8} - 10 a^{3} c d^{3} e^{6} + 6 a^{2} c^{2} d^{5} e^{4} + 2 a c^{3} d^{7} e^{2} - 2 c^{4} d^{9}\right )} \end{align*}

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)**2,x)

[Out]

3*c**2*d**2*e*log(x + (-3*a**5*c**2*d**2*e**11/(a*e**2 - c*d**2)**4 + 15*a**4*c**3*d**4*e**9/(a*e**2 - c*d**2)
**4 - 30*a**3*c**4*d**6*e**7/(a*e**2 - c*d**2)**4 + 30*a**2*c**5*d**8*e**5/(a*e**2 - c*d**2)**4 - 15*a*c**6*d*
*10*e**3/(a*e**2 - c*d**2)**4 + 3*a*c**2*d**2*e**3 + 3*c**7*d**12*e/(a*e**2 - c*d**2)**4 + 3*c**3*d**4*e)/(6*c
**3*d**3*e**2))/(a*e**2 - c*d**2)**4 - 3*c**2*d**2*e*log(x + (3*a**5*c**2*d**2*e**11/(a*e**2 - c*d**2)**4 - 15
*a**4*c**3*d**4*e**9/(a*e**2 - c*d**2)**4 + 30*a**3*c**4*d**6*e**7/(a*e**2 - c*d**2)**4 - 30*a**2*c**5*d**8*e*
*5/(a*e**2 - c*d**2)**4 + 15*a*c**6*d**10*e**3/(a*e**2 - c*d**2)**4 + 3*a*c**2*d**2*e**3 - 3*c**7*d**12*e/(a*e
**2 - c*d**2)**4 + 3*c**3*d**4*e)/(6*c**3*d**3*e**2))/(a*e**2 - c*d**2)**4 + (-a**2*e**4 + 5*a*c*d**2*e**2 + 2
*c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x*(3*a*c*d*e**3 + 9*c**2*d**3*e))/(2*a**4*d**2*e**7 - 6*a**3*c*d**4*e**5
+ 6*a**2*c**2*d**6*e**3 - 2*a*c**3*d**8*e + x**3*(2*a**3*c*d*e**8 - 6*a**2*c**2*d**3*e**6 + 6*a*c**3*d**5*e**4
- 2*c**4*d**7*e**2) + x**2*(2*a**4*e**9 - 2*a**3*c*d**2*e**7 - 6*a**2*c**2*d**4*e**5 + 10*a*c**3*d**6*e**3 -
4*c**4*d**8*e) + x*(4*a**4*d*e**8 - 10*a**3*c*d**3*e**6 + 6*a**2*c**2*d**5*e**4 + 2*a*c**3*d**7*e**2 - 2*c**4*
d**9))

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError