\(\int \frac {d+e x}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1893]

Optimal result

Integrand size = 33, antiderivative size = 142 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c d}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac {2 c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac {e^2}{\left (c d^2-a e^2\right )^3 (d+e x)}+\frac {3 c d e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {3 c d e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 46} \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac {2 c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {c d}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac {3 c d e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac {3 c d e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {-\frac {c d \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {4 c d e \left (c d^2-a e^2\right )}{a e+c d x}+\frac {2 c d^2 e^2-2 a e^4}{d+e x}+6 c d e^2 \log (a e+c d x)-6 c d e^2 \log (d+e x)}{2 \left (c d^2-a e^2\right )^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {c d}{2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{2}}+\frac {3 c d \,e^{2} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}-\frac {2 c d e}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )}-\frac {e^{2}}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )}-\frac {3 c d \,e^{2} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\)	\(142\)
risch	\(\frac {-\frac {3 c^{2} d^{2} e^{2} x^{2}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}-\frac {3 \left (3 e^{2} a +c \,d^{2}\right ) c d e x}{2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}-\frac {2 a^{2} e^{4}+5 a c \,d^{2} e^{2}-c^{2} d^{4}}{2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}}{\left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )}-\frac {3 c d \,e^{2} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {3 c d \,e^{2} \ln \left (-c d x -a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\)	\(368\)
norman	\(\frac {-\frac {3 c^{2} d^{2} e^{3} x^{3}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {\left (-d \,e^{6} a^{2} c^{2}-7 d^{3} e^{4} c^{3} a -d^{5} e^{2} c^{4}\right ) x}{e d \,c^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {-2 d \,e^{4} a^{2} c^{2}-5 d^{3} e^{2} c^{3} a +c^{4} d^{5}}{2 c^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {\left (-9 a \,c^{3} d^{3} e^{6}-9 c^{4} d^{5} e^{4}\right ) x^{2}}{2 e^{2} d^{2} c^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}-\frac {3 c d \,e^{2} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {3 c d \,e^{2} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\)	\(464\)
parallelrisch	\(-\frac {-6 a \,c^{4} d^{6} e^{3}-12 \ln \left (c d x +a e \right ) x a \,c^{4} d^{5} e^{4}+c^{5} d^{8} e -3 x \,c^{5} d^{7} e^{2}-6 x^{2} c^{5} d^{6} e^{3}+2 a^{3} c^{2} d^{2} e^{7}+3 a^{2} c^{3} d^{4} e^{5}+12 \ln \left (e x +d \right ) x a \,c^{4} d^{5} e^{4}-6 \ln \left (c d x +a e \right ) x \,a^{2} c^{3} d^{3} e^{6}+12 \ln \left (e x +d \right ) x^{2} a \,c^{4} d^{4} e^{5}-12 \ln \left (c d x +a e \right ) x^{2} a \,c^{4} d^{4} e^{5}+6 \ln \left (e x +d \right ) x \,a^{2} c^{3} d^{3} e^{6}+6 x^{2} a \,c^{4} d^{4} e^{5}+9 x \,a^{2} c^{3} d^{3} e^{6}-6 x a \,c^{4} d^{5} e^{4}+6 \ln \left (e x +d \right ) x^{3} c^{5} d^{5} e^{4}-6 \ln \left (c d x +a e \right ) x^{3} c^{5} d^{5} e^{4}+6 \ln \left (e x +d \right ) x^{2} c^{5} d^{6} e^{3}-6 \ln \left (c d x +a e \right ) x^{2} c^{5} d^{6} e^{3}+6 \ln \left (e x +d \right ) a^{2} c^{3} d^{4} e^{5}-6 \ln \left (c d x +a e \right ) a^{2} c^{3} d^{4} e^{5}}{2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right ) \left (c d x +a e \right ) e \,c^{2} d^{2}}\)	\(481\)

[In]

int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (140) = 280\).

Time = 0.30 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.91 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} - 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + a^{2} c d^{2} e^{4} + {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right ) + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + a^{2} c d^{2} e^{4} + {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} c^{4} d^{9} e^{2} - 4 \, a^{3} c^{3} d^{7} e^{4} + 6 \, a^{4} c^{2} d^{5} e^{6} - 4 \, a^{5} c d^{3} e^{8} + a^{6} d e^{10} + {\left (c^{6} d^{10} e - 4 \, a c^{5} d^{8} e^{3} + 6 \, a^{2} c^{4} d^{6} e^{5} - 4 \, a^{3} c^{3} d^{4} e^{7} + a^{4} c^{2} d^{2} e^{9}\right )} x^{3} + {\left (c^{6} d^{11} - 2 \, a c^{5} d^{9} e^{2} - 2 \, a^{2} c^{4} d^{7} e^{4} + 8 \, a^{3} c^{3} d^{5} e^{6} - 7 \, a^{4} c^{2} d^{3} e^{8} + 2 \, a^{5} c d e^{10}\right )} x^{2} + {\left (2 \, a c^{5} d^{10} e - 7 \, a^{2} c^{4} d^{8} e^{3} + 8 \, a^{3} c^{3} d^{6} e^{5} - 2 \, a^{4} c^{2} d^{4} e^{7} - 2 \, a^{5} c d^{2} e^{9} + a^{6} e^{11}\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (131) = 262\).

Time = 1.14 (sec) , antiderivative size = 736, normalized size of antiderivative = 5.18 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=- \frac {3 c d e^{2} \log {\left (x + \frac {- \frac {3 a^{5} c d e^{12}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {15 a^{4} c^{2} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {30 a^{3} c^{3} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {30 a^{2} c^{4} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {15 a c^{5} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c d e^{4} + \frac {3 c^{6} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{2} d^{3} e^{2}}{6 c^{2} d^{2} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {3 c d e^{2} \log {\left (x + \frac {\frac {3 a^{5} c d e^{12}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {15 a^{4} c^{2} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {30 a^{3} c^{3} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {30 a^{2} c^{4} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {15 a c^{5} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c d e^{4} - \frac {3 c^{6} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{2} d^{3} e^{2}}{6 c^{2} d^{2} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {- 2 a^{2} e^{4} - 5 a c d^{2} e^{2} + c^{2} d^{4} - 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 9 a c d e^{3} - 3 c^{2} d^{3} e\right )}{2 a^{5} d e^{8} - 6 a^{4} c d^{3} e^{6} + 6 a^{3} c^{2} d^{5} e^{4} - 2 a^{2} c^{3} d^{7} e^{2} + x^{3} \cdot \left (2 a^{3} c^{2} d^{2} e^{7} - 6 a^{2} c^{3} d^{4} e^{5} + 6 a c^{4} d^{6} e^{3} - 2 c^{5} d^{8} e\right ) + x^{2} \cdot \left (4 a^{4} c d e^{8} - 10 a^{3} c^{2} d^{3} e^{6} + 6 a^{2} c^{3} d^{5} e^{4} + 2 a c^{4} d^{7} e^{2} - 2 c^{5} d^{9}\right ) + x \left (2 a^{5} e^{9} - 2 a^{4} c d^{2} e^{7} - 6 a^{3} c^{2} d^{4} e^{5} + 10 a^{2} c^{3} d^{6} e^{3} - 4 a c^{4} d^{8} e\right )} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (140) = 280\).

Time = 0.20 (sec) , antiderivative size = 429, normalized size of antiderivative = 3.02 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {3 \, c d e^{2} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac {3 \, c d e^{2} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {6 \, c^{2} d^{2} e^{2} x^{2} - c^{2} d^{4} + 5 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{2 \, {\left (a^{2} c^{3} d^{7} e^{2} - 3 \, a^{3} c^{2} d^{5} e^{4} + 3 \, a^{4} c d^{3} e^{6} - a^{5} d e^{8} + {\left (c^{5} d^{8} e - 3 \, a c^{4} d^{6} e^{3} + 3 \, a^{2} c^{3} d^{4} e^{5} - a^{3} c^{2} d^{2} e^{7}\right )} x^{3} + {\left (c^{5} d^{9} - a c^{4} d^{7} e^{2} - 3 \, a^{2} c^{3} d^{5} e^{4} + 5 \, a^{3} c^{2} d^{3} e^{6} - 2 \, a^{4} c d e^{8}\right )} x^{2} + {\left (2 \, a c^{4} d^{8} e - 5 \, a^{2} c^{3} d^{6} e^{3} + 3 \, a^{3} c^{2} d^{4} e^{5} + a^{4} c d^{2} e^{7} - a^{5} e^{9}\right )} x\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (140) = 280\).

Time = 0.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.01 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {3 \, c^{2} d^{2} e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} - \frac {3 \, c d e^{3} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} - 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{2 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (c d x + a e\right )}^{2} {\left (e x + d\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.99 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.76 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {6\,c\,d\,e^2\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4}-\frac {\frac {2\,a^2\,e^4+5\,a\,c\,d^2\,e^2-c^2\,d^4}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {3\,e\,x\,\left (c^2\,d^3+3\,a\,c\,d\,e^2\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {3\,c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{x\,\left (a^2\,e^3+2\,c\,a\,d^2\,e\right )+x^2\,\left (c^2\,d^3+2\,a\,c\,d\,e^2\right )+a^2\,d\,e^2+c^2\,d^2\,e\,x^3} \]

[In]

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

[Out]

(6*c*d*e^2*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 - ((2*a^2*e^4 - c^2*d
^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*e*x*(c^2*d^3 + 3*a*c*d*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^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*(a^2*e^3 + 2*a*c*d^2*e) + x^2*(c^2*d^3 + 2*a*c*d*e^2) + a^2*d*e^2 + c^2*d^
2*e*x^3)