\(\int \frac {1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [1883]

Optimal result

Integrand size = 35, antiderivative size = 146 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e}{2 \left (c d^2-a e^2\right )^2 (d+e x)^2}-\frac {2 c d e}{\left (c d^2-a e^2\right )^3 (d+e x)}-\frac {3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 46} \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^2 d^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {3 c^2 d^2 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {3 c^2 d^2 e \log (d+e x)}{\left (c d^2-a e^2\right )^4}-\frac {2 c d e}{(d+e x) \left (c d^2-a e^2\right )^3}-\frac {e}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2} \]

[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 46

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] && Lt
Q[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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*} \text {integral}& = \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] (verified)

Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\frac {2 c^2 d^2 \left (-c d^2+a e^2\right )}{a e+c d x}-\frac {e \left (c d^2-a e^2\right )^2}{(d+e x)^2}+\frac {4 c d e \left (-c d^2+a e^2\right )}{d+e x}-6 c^2 d^2 e \log (a e+c d x)+6 c^2 d^2 e \log (d+e x)}{2 \left (c d^2-a e^2\right )^4} \]

[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] (verified)

Time = 2.67 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )}-\frac {3 c^{2} d^{2} e \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}-\frac {e}{2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}+\frac {3 c^{2} d^{2} e \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}+\frac {2 e c d}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )}\)	\(144\)
risch	\(\frac {\frac {3 c^{2} d^{2} e^{2} x^{2}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {3 \left (e^{2} a +3 c \,d^{2}\right ) c d e x}{2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}-\frac {a^{2} e^{4}-5 a c \,d^{2} e^{2}-2 c^{2} d^{4}}{2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}}{\left (e x +d \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )}+\frac {3 e \,c^{2} d^{2} \ln \left (-e x -d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {3 e \,c^{2} d^{2} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\)	\(369\)
norman	\(\frac {\frac {3 c^{2} d^{2} e^{2} x^{2}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {-a^{2} c \,e^{5}+5 e^{3} a \,c^{2} d^{2}+2 d^{4} e \,c^{3}}{2 e c \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {\left (3 a \,c^{2} d \,e^{5}+9 c^{3} d^{3} e^{3}\right ) x}{2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) e^{2} c}}{\left (e x +d \right )^{2} \left (c d x +a e \right )}+\frac {3 e \,c^{2} d^{2} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {3 e \,c^{2} d^{2} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\)	\(374\)
parallelrisch	\(\frac {6 \ln \left (e x +d \right ) x^{2} a \,c^{3} d^{3} e^{6}-6 \ln \left (c d x +a e \right ) x^{2} a \,c^{3} d^{3} e^{6}+12 \ln \left (e x +d \right ) x a \,c^{3} d^{4} e^{5}-12 \ln \left (c d x +a e \right ) x a \,c^{3} d^{4} e^{5}+6 x^{2} a \,c^{3} d^{3} e^{6}+3 x \,a^{2} c^{2} d^{2} e^{7}+6 x a \,c^{3} d^{4} e^{5}+6 \ln \left (e x +d \right ) x^{3} c^{4} d^{4} e^{5}-6 \ln \left (c d x +a e \right ) x^{3} c^{4} d^{4} e^{5}+12 \ln \left (e x +d \right ) x^{2} c^{4} d^{5} e^{4}-12 \ln \left (c d x +a e \right ) x^{2} c^{4} d^{5} e^{4}+6 \ln \left (e x +d \right ) x \,c^{4} d^{6} e^{3}-2 c^{4} e^{2} d^{7}-3 a \,c^{3} e^{4} d^{5}+6 a^{2} c^{2} e^{6} d^{3}-a^{3} c \,e^{8} d -6 \ln \left (c d x +a e \right ) x \,c^{4} d^{6} e^{3}+6 \ln \left (e x +d \right ) a \,c^{3} d^{5} e^{4}-6 \ln \left (c d x +a e \right ) a \,c^{3} d^{5} e^{4}-6 x^{2} c^{4} d^{5} e^{4}-9 x \,c^{4} d^{6} e^{3}}{2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right ) \left (e x +d \right ) c d \,e^{2}}\)	\(467\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (144) = 288\).

Time = 0.28 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.73 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a c^{4} d^{10} e - 4 \, a^{2} c^{3} d^{8} e^{3} + 6 \, a^{3} c^{2} d^{6} e^{5} - 4 \, a^{4} c d^{4} e^{7} + a^{5} d^{2} e^{9} + {\left (c^{5} d^{9} e^{2} - 4 \, a c^{4} d^{7} e^{4} + 6 \, a^{2} c^{3} d^{5} e^{6} - 4 \, a^{3} c^{2} d^{3} e^{8} + a^{4} c d e^{10}\right )} x^{3} + {\left (2 \, c^{5} d^{10} e - 7 \, a c^{4} d^{8} e^{3} + 8 \, a^{2} c^{3} d^{6} e^{5} - 2 \, a^{3} c^{2} d^{4} e^{7} - 2 \, a^{4} c d^{2} e^{9} + a^{5} e^{11}\right )} x^{2} + {\left (c^{5} d^{11} - 2 \, a c^{4} d^{9} e^{2} - 2 \, a^{2} c^{3} d^{7} e^{4} + 8 \, a^{3} c^{2} d^{5} e^{6} - 7 \, a^{4} c d^{3} e^{8} + 2 \, a^{5} d e^{10}\right )} x\right )}} \]

[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] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 734 vs. \(2 (133) = 266\).

Time = 1.06 (sec) , antiderivative size = 734, normalized size of antiderivative = 5.03 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {3 c^{2} d^{2} e \log {\left (x + \frac {- \frac {3 a^{5} c^{2} d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {15 a^{4} c^{3} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {30 a^{3} c^{4} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {30 a^{2} c^{5} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {15 a c^{6} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c^{2} d^{2} e^{3} + \frac {3 c^{7} d^{12} e}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{3} d^{4} e}{6 c^{3} d^{3} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {3 c^{2} d^{2} e \log {\left (x + \frac {\frac {3 a^{5} c^{2} d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {15 a^{4} c^{3} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {30 a^{3} c^{4} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {30 a^{2} c^{5} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {15 a c^{6} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c^{2} d^{2} e^{3} - \frac {3 c^{7} d^{12} e}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{3} d^{4} e}{6 c^{3} d^{3} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {- a^{2} e^{4} + 5 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 9 c^{2} d^{3} e\right )}{2 a^{4} d^{2} e^{7} - 6 a^{3} c d^{4} e^{5} + 6 a^{2} c^{2} d^{6} e^{3} - 2 a c^{3} d^{8} e + x^{3} \cdot \left (2 a^{3} c d e^{8} - 6 a^{2} c^{2} d^{3} e^{6} + 6 a c^{3} d^{5} e^{4} - 2 c^{4} d^{7} e^{2}\right ) + x^{2} \cdot \left (2 a^{4} e^{9} - 2 a^{3} c d^{2} e^{7} - 6 a^{2} c^{2} d^{4} e^{5} + 10 a c^{3} d^{6} e^{3} - 4 c^{4} d^{8} e\right ) + x \left (4 a^{4} d e^{8} - 10 a^{3} c d^{3} e^{6} + 6 a^{2} c^{2} d^{5} e^{4} + 2 a c^{3} d^{7} e^{2} - 2 c^{4} d^{9}\right )} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (144) = 288\).

Time = 0.22 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.90 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {3 \, c^{2} d^{2} e \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {3 \, c^{2} d^{2} e \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac {6 \, c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{4} + 5 \, a c d^{2} e^{2} - a^{2} e^{4} + 3 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} + {\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} + {\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} + {\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (144) = 288\).

Time = 0.28 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {3 \, c^{3} d^{3} e \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} + \frac {3 \, c^{2} d^{2} e^{2} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{2 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (c d x + a e\right )} {\left (e x + d\right )}^{2}} \]

[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]

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

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.61 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\frac {-a^2\,e^4+5\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {3\,c\,d\,x\,\left (3\,c\,d^2\,e+a\,e^3\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {3\,c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{x^2\,\left (2\,c\,d^2\,e+a\,e^3\right )+x\,\left (c\,d^3+2\,a\,d\,e^2\right )+a\,d^2\,e+c\,d\,e^2\,x^3}-\frac {6\,c^2\,d^2\,e\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4} \]

[In]

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

[Out]

((2*c^2*d^4 - a^2*e^4 + 5*a*c*d^2*e^2)/(2*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4)) + (3*c*d*x*
(a*e^3 + 3*c*d^2*e))/(2*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4)) + (3*c^2*d^2*e^2*x^2)/(a^3*e^
6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4))/(x^2*(a*e^3 + 2*c*d^2*e) + x*(c*d^3 + 2*a*d*e^2) + a*d^2*e +
c*d*e^2*x^3) - (6*c^2*d^2*e*atanh((a^4*e^8 - c^4*d^8 + 2*a*c^3*d^6*e^2 - 2*a^3*c*d^2*e^6)/(a*e^2 - c*d^2)^4 +
(2*c*d*e*x*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4))/(a*e^2 - c*d^2)^4))/(a*e^2 - c*d^2)^4