\(\int \frac {d+e x}{x (a+b x+c x^2)^2} \, dx\) [53]

Optimal result

Integrand size = 21, antiderivative size = 135 \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx=\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (b^3 d-6 a b c d+4 a^2 c e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {d \log (x)}{a^2}-\frac {d \log \left (a+b x+c x^2\right )}{2 a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 a \left (-b^2 d+b (a e-c d x)+2 a c (d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \left (b^3 d-6 a b c d+4 a^2 c e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+2 d \log (x)-d \log (a+x (b+c x))}{2 a^2} \] Input:

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

Output:

((-2*a*(-(b^2*d) + b*(a*e - c*d*x) + 2*a*c*(d + e*x)))/((b^2 - 4*a*c)*(a + 
 x*(b + c*x))) + (2*(b^3*d - 6*a*b*c*d + 4*a^2*c*e)*ArcTan[(b + 2*c*x)/Sqr 
t[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*d*Log[x] - d*Log[a + x*(b + c*x 
)])/(2*a^2)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.23, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1235, 25, 1200, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {\left (b^2-4 a c\right ) d+c (b d-2 a e) x}{x \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b^2-4 a c\right ) d+c (b d-2 a e) x}{x \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {\int \left (\frac {-d b^3+5 a c d b-2 a^2 c e-c \left (b^2-4 a c\right ) d x}{a \left (c x^2+b x+a\right )}-\frac {\left (4 a c-b^2\right ) d}{a x}\right )dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (4 a^2 c e-6 a b c d+b^3 d\right )}{a \sqrt {b^2-4 a c}}-\frac {d \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )}{2 a}+\frac {d \log (x) \left (b^2-4 a c\right )}{a}}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

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

Output:

(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x)/(a*(b^2 - 4*a*c)*(a + b*x + 
c*x^2)) + (((b^3*d - 6*a*b*c*d + 4*a^2*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 
 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)*d*Log[x])/a - ((b^2 - 4*a 
*c)*d*Log[a + b*x + c*x^2])/(2*a))/(a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {d \ln \left (x \right )}{a^{2}}+\frac {\frac {\frac {a c \left (2 a e -b d \right ) x}{4 a c -b^{2}}+\frac {a \left (a b e +2 a c d -b^{2} d \right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-4 a \,c^{2} d +c d \,b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (2 a^{2} c e -5 a b c d +b^{3} d -\frac {\left (-4 a \,c^{2} d +c d \,b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{2}}\)	\(200\)
risch	\(\frac {\frac {c \left (2 a e -b d \right ) x}{\left (4 a c -b^{2}\right ) a}+\frac {a b e +2 a c d -b^{2} d}{a \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {d \ln \left (x \right )}{a^{2}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3}-48 a^{4} b^{2} c^{2}+12 a^{3} b^{4} c -a^{2} b^{6}\right ) \textit {\_Z}^{2}+\left (64 a^{3} c^{3} d -48 a^{2} b^{2} c^{2} d +12 a \,b^{4} c d -b^{6} d \right ) \textit {\_Z} +4 a^{2} c^{2} e^{2}-12 a b \,c^{2} d e +16 a \,c^{3} d^{2}+2 b^{3} c d e -3 b^{2} c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (96 a^{5} c^{3}-80 a^{4} b^{2} c^{2}+22 a^{3} b^{4} c -2 a^{2} b^{6}\right ) \textit {\_R}^{2}+\left (-8 a^{3} b \,c^{2} e +48 a^{3} c^{3} d +2 a^{2} b^{3} c e -20 a^{2} b^{2} c^{2} d +2 a \,b^{4} c d \right ) \textit {\_R} +4 a^{2} c^{2} e^{2}-4 a b \,c^{2} d e +b^{2} c^{2} d^{2}\right ) x +\left (-16 a^{5} b \,c^{2}+8 a^{4} b^{3} c -a^{3} b^{5}\right ) \textit {\_R}^{2}+\left (-8 a^{4} c^{2} e +2 a^{3} b^{2} c e +20 a^{3} b \,c^{2} d -9 a^{2} b^{3} c d +a \,b^{5} d \right ) \textit {\_R} +8 a^{2} c^{2} d e -2 a \,b^{2} c d e -4 b \,c^{2} d^{2} a +b^{3} c \,d^{2}\right )\right )\)	\(460\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (129) = 258\).

Time = 0.32 (sec) , antiderivative size = 955, normalized size of antiderivative = 7.07 \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*((4*a^3*c*e + (4*a^2*c^2*e + (b^3*c - 6*a*b*c^2)*d)*x^2 + (a*b^3 - 6 
*a^2*b*c)*d + (4*a^2*b*c*e + (b^4 - 6*a*b^2*c)*d)*x)*sqrt(b^2 - 4*a*c)*log 
((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^ 
2 + b*x + a)) - 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d + 2*(a^2*b^3 - 4*a^3 
*b*c)*e - 2*((a*b^3*c - 4*a^2*b*c^2)*d - 2*(a^2*b^2*c - 4*a^3*c^2)*e)*x + 
((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^ 
2)*d*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(c*x^2 + b*x + a) - 2*(( 
b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2) 
*d*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(x))/(a^3*b^4 - 8*a^4*b^2* 
c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^2 + (a^2*b^5 - 
 8*a^3*b^3*c + 16*a^4*b*c^2)*x), 1/2*(2*(4*a^3*c*e + (4*a^2*c^2*e + (b^3*c 
 - 6*a*b*c^2)*d)*x^2 + (a*b^3 - 6*a^2*b*c)*d + (4*a^2*b*c*e + (b^4 - 6*a*b 
^2*c)*d)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 
 - 4*a*c)) + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3 - 4*a^3*b* 
c)*e + 2*((a*b^3*c - 4*a^2*b*c^2)*d - 2*(a^2*b^2*c - 4*a^3*c^2)*e)*x - ((b 
^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)* 
d*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(c*x^2 + b*x + a) + 2*((b^4 
*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d* 
x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(x))/(a^3*b^4 - 8*a^4*b^2*c + 
 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^2 + (a^2*b^5 -...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx=-\frac {{\left (b^{3} d - 6 \, a b c d + 4 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {d \log \left (c x^{2} + b x + a\right )}{2 \, a^{2}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {a b^{2} d - 2 \, a^{2} c d - a^{2} b e + {\left (a b c d - 2 \, a^{2} c e\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2}} \] Input:

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

Output:

-(b^3*d - 6*a*b*c*d + 4*a^2*c*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(( 
a^2*b^2 - 4*a^3*c)*sqrt(-b^2 + 4*a*c)) - 1/2*d*log(c*x^2 + b*x + a)/a^2 + 
d*log(abs(x))/a^2 + (a*b^2*d - 2*a^2*c*d - a^2*b*e + (a*b*c*d - 2*a^2*c*e) 
*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*a^2)
 

Mupad [B] (verification not implemented)

Time = 12.20 (sec) , antiderivative size = 920, normalized size of antiderivative = 6.81 \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {-d\,b^2+a\,e\,b+2\,a\,c\,d}{a\,\left (4\,a\,c-b^2\right )}+\frac {c\,x\,\left (2\,a\,e-b\,d\right )}{a\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\ln \left (96\,a^4\,c^3\,d-2\,a\,b^6\,d-2\,b^7\,d\,x-84\,a^3\,b^2\,c^2\,d+2\,a\,b^3\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+23\,a^2\,b^4\,c\,d-2\,a^3\,b^3\,c\,e+8\,a^4\,b\,c^2\,e+2\,a^3\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+2\,b^4\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-16\,a^4\,c^3\,e\,x-9\,a^2\,b\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+120\,a^3\,b\,c^3\,d\,x-2\,a^2\,b^4\,c\,e\,x-94\,a^2\,b^3\,c^2\,d\,x+12\,a^2\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^3\,b^2\,c^2\,e\,x+24\,a\,b^5\,c\,d\,x-12\,a\,b^2\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+2\,a^2\,b\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {d}{2\,a^2}-\frac {\frac {b^3\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}+2\,a^2\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-3\,a\,b\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^5\,c^3+48\,a^4\,b^2\,c^2-12\,a^3\,b^4\,c+a^2\,b^6}\right )-\ln \left (2\,a\,b^6\,d-96\,a^4\,c^3\,d+2\,b^7\,d\,x+84\,a^3\,b^2\,c^2\,d+2\,a\,b^3\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-23\,a^2\,b^4\,c\,d+2\,a^3\,b^3\,c\,e-8\,a^4\,b\,c^2\,e+2\,a^3\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+2\,b^4\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+16\,a^4\,c^3\,e\,x-9\,a^2\,b\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-120\,a^3\,b\,c^3\,d\,x+2\,a^2\,b^4\,c\,e\,x+94\,a^2\,b^3\,c^2\,d\,x+12\,a^2\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-12\,a^3\,b^2\,c^2\,e\,x-24\,a\,b^5\,c\,d\,x-12\,a\,b^2\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+2\,a^2\,b\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {d}{2\,a^2}+\frac {\frac {b^3\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}+2\,a^2\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-3\,a\,b\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^5\,c^3+48\,a^4\,b^2\,c^2-12\,a^3\,b^4\,c+a^2\,b^6}\right )+\frac {d\,\ln \left (x\right )}{a^2} \] Input:

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

Output:

((a*b*e - b^2*d + 2*a*c*d)/(a*(4*a*c - b^2)) + (c*x*(2*a*e - b*d))/(a*(4*a 
*c - b^2)))/(a + b*x + c*x^2) - log(96*a^4*c^3*d - 2*a*b^6*d - 2*b^7*d*x - 
 84*a^3*b^2*c^2*d + 2*a*b^3*d*(-(4*a*c - b^2)^3)^(1/2) + 23*a^2*b^4*c*d - 
2*a^3*b^3*c*e + 8*a^4*b*c^2*e + 2*a^3*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^4 
*d*x*(-(4*a*c - b^2)^3)^(1/2) - 16*a^4*c^3*e*x - 9*a^2*b*c*d*(-(4*a*c - b^ 
2)^3)^(1/2) + 120*a^3*b*c^3*d*x - 2*a^2*b^4*c*e*x - 94*a^2*b^3*c^2*d*x + 1 
2*a^2*c^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 12*a^3*b^2*c^2*e*x + 24*a*b^5*c*d 
*x - 12*a*b^2*c*d*x*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b*c*e*x*(-(4*a*c - b^ 
2)^3)^(1/2))*(d/(2*a^2) - ((b^3*d*(-(4*a*c - b^2)^3)^(1/2))/2 + 2*a^2*c*e* 
(-(4*a*c - b^2)^3)^(1/2) - 3*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(a^2*b^6 - 
64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)) - log(2*a*b^6*d - 96*a^4*c^3* 
d + 2*b^7*d*x + 84*a^3*b^2*c^2*d + 2*a*b^3*d*(-(4*a*c - b^2)^3)^(1/2) - 23 
*a^2*b^4*c*d + 2*a^3*b^3*c*e - 8*a^4*b*c^2*e + 2*a^3*c*e*(-(4*a*c - b^2)^3 
)^(1/2) + 2*b^4*d*x*(-(4*a*c - b^2)^3)^(1/2) + 16*a^4*c^3*e*x - 9*a^2*b*c* 
d*(-(4*a*c - b^2)^3)^(1/2) - 120*a^3*b*c^3*d*x + 2*a^2*b^4*c*e*x + 94*a^2* 
b^3*c^2*d*x + 12*a^2*c^2*d*x*(-(4*a*c - b^2)^3)^(1/2) - 12*a^3*b^2*c^2*e*x 
 - 24*a*b^5*c*d*x - 12*a*b^2*c*d*x*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b*c*e* 
x*(-(4*a*c - b^2)^3)^(1/2))*(d/(2*a^2) + ((b^3*d*(-(4*a*c - b^2)^3)^(1/2)) 
/2 + 2*a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b*c*d*(-(4*a*c - b^2)^3)^(1/ 
2))/(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)) + (d*log(x)...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 873, normalized size of antiderivative = 6.47 \[ \int \frac {d+e x}{x \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c*e - 12 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c*d + 8 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c*e*x + 
 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**2*e*x 
**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*d - 
 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c*d*x - 
 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*d* 
x**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*d*x 
+ 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c*d*x**2 
- 16*log(a + b*x + c*x**2)*a**3*b*c**2*d + 8*log(a + b*x + c*x**2)*a**2*b* 
*3*c*d - 16*log(a + b*x + c*x**2)*a**2*b**2*c**2*d*x - 16*log(a + b*x + c* 
x**2)*a**2*b*c**3*d*x**2 - log(a + b*x + c*x**2)*a*b**5*d + 8*log(a + b*x 
+ c*x**2)*a*b**4*c*d*x + 8*log(a + b*x + c*x**2)*a*b**3*c**2*d*x**2 - log( 
a + b*x + c*x**2)*b**6*d*x - log(a + b*x + c*x**2)*b**5*c*d*x**2 + 32*log( 
x)*a**3*b*c**2*d - 16*log(x)*a**2*b**3*c*d + 32*log(x)*a**2*b**2*c**2*d*x 
+ 32*log(x)*a**2*b*c**3*d*x**2 + 2*log(x)*a*b**5*d - 16*log(x)*a*b**4*c*d* 
x - 16*log(x)*a*b**3*c**2*d*x**2 + 2*log(x)*b**6*d*x + 2*log(x)*b**5*c*d*x 
**2 - 16*a**4*c**2*e + 12*a**3*b**2*c*e + 24*a**3*b*c**2*d - 16*a**3*c**3* 
e*x**2 - 2*a**2*b**4*e - 14*a**2*b**3*c*d + 4*a**2*b**2*c**2*e*x**2 + 8*a* 
*2*b*c**3*d*x**2 + 2*a*b**5*d - 2*a*b**3*c**2*d*x**2)/(2*a**2*b*(16*a**...