\(\int \frac {A+B x}{(d+e x) (a+c x^2)^2} \, dx\) [93]

Optimal result

Integrand size = 22, antiderivative size = 195 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx=-\frac {a (B d-A e)-(A c d+a B e) x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (a B e \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^2}-\frac {e^2 (B d-A e) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {e^2 (B d-A e) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx=\frac {\frac {\left (c d^2+a e^2\right ) (A c d x+a (-B d+A e+B e x))}{a \left (a+c x^2\right )}+\frac {\left (a B e \left (-c d^2+a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {c}}+2 e^2 (-B d+A e) \log (d+e x)+e^2 (B d-A e) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \] Input:

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

Output:

(((c*d^2 + a*e^2)*(A*c*d*x + a*(-(B*d) + A*e + B*e*x)))/(a*(a + c*x^2)) + 
((a*B*e*(-(c*d^2) + a*e^2) + A*c*d*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/S 
qrt[a]])/(a^(3/2)*Sqrt[c]) + 2*e^2*(-(B*d) + A*e)*Log[d + e*x] + e^2*(B*d 
- A*e)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {686, 27, 25, 657, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 657

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{\sqrt {a} \sqrt {c} \left (a e^2+c d^2\right )}+\frac {a e^2 \log \left (a+c x^2\right ) (B d-A e)}{a e^2+c d^2}-\frac {2 a e^2 (B d-A e) \log (d+e x)}{a e^2+c d^2}}{2 a \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}\)

Input:

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

Output:

-1/2*(a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(a*(c*d^2 + a*e^2)*(a + c*x^2)) + 
 (-(((a*B*e*(c*d^2 - a*e^2) - A*c*d*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/ 
Sqrt[a]])/(Sqrt[a]*Sqrt[c]*(c*d^2 + a*e^2))) - (2*a*e^2*(B*d - A*e)*Log[d 
+ e*x])/(c*d^2 + a*e^2) + (a*e^2*(B*d - A*e)*Log[a + c*x^2])/(c*d^2 + a*e^ 
2))/(2*a*(c*d^2 + a*e^2))
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 657

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

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, 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 = 1.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {\left (A e -B d \right ) e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {\frac {\frac {\left (A a c d \,e^{2}+A \,c^{2} d^{3}+B \,e^{3} a^{2}+B a c \,d^{2} e \right ) x}{2 a}+\frac {A a \,e^{3}}{2}+\frac {A c \,d^{2} e}{2}-\frac {B a d \,e^{2}}{2}-\frac {B c \,d^{3}}{2}}{c \,x^{2}+a}+\frac {\frac {\left (-2 A a c \,e^{3}+2 B a c d \,e^{2}\right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A a c d \,e^{2}+A \,c^{2} d^{3}+B \,e^{3} a^{2}-B a c \,d^{2} e \right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(215\)
risch	\(\frac {\frac {\left (A c d +B a e \right ) x}{2 a \left (a \,e^{2}+c \,d^{2}\right )}+\frac {A e -B d}{2 a \,e^{2}+2 c \,d^{2}}}{c \,x^{2}+a}+\frac {e^{3} \ln \left (e x +d \right ) A}{a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}-\frac {e^{2} \ln \left (e x +d \right ) B d}{a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{5} c \,e^{4}+2 a^{4} c^{2} d^{2} e^{2}+a^{3} c^{3} d^{4}\right ) \textit {\_Z}^{2}+\left (4 A \,a^{3} c \,e^{3}-4 B \,a^{3} c d \,e^{2}\right ) \textit {\_Z} +4 A^{2} a c \,e^{2}+A^{2} d^{2} c^{2}-2 A B a c d e +B^{2} a^{2} e^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{5} c \,e^{6}+5 a^{4} c^{2} d^{2} e^{4}+a^{3} c^{3} d^{4} e^{2}-a^{2} c^{4} d^{6}\right ) \textit {\_R}^{2}+\left (6 A \,a^{3} c \,e^{5}+8 A \,a^{2} c^{2} d^{2} e^{3}+2 A a \,c^{3} d^{4} e -4 B \,a^{3} c d \,e^{4}-4 B \,a^{2} c^{2} d^{3} e^{2}\right ) \textit {\_R} +2 A^{2} c^{2} d^{2} e^{2}+4 A B a c d \,e^{3}+2 B^{2} a^{2} e^{4}\right ) x +\left (4 a^{5} c d \,e^{5}+8 a^{4} c^{2} d^{3} e^{3}+4 a^{3} c^{3} d^{5} e \right ) \textit {\_R}^{2}+\left (A \,a^{3} c d \,e^{4}+2 A \,a^{2} c^{2} d^{3} e^{2}+A \,d^{5} c^{3} a -B \,e^{5} a^{4}-2 B \,a^{3} c \,d^{2} e^{3}-B \,a^{2} c^{2} d^{4} e \right ) \textit {\_R} +4 A^{2} a c d \,e^{3}+2 A^{2} c^{2} d^{3} e +4 A B \,a^{2} e^{4}-2 B^{2} a^{2} d \,e^{3}\right )\right )}{4}\)	\(565\)

Input:

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

Output:

(A*e-B*d)*e^2/(a*e^2+c*d^2)^2*ln(e*x+d)+1/(a*e^2+c*d^2)^2*((1/2*(A*a*c*d*e 
^2+A*c^2*d^3+B*a^2*e^3+B*a*c*d^2*e)/a*x+1/2*A*a*e^3+1/2*A*c*d^2*e-1/2*B*a* 
d*e^2-1/2*B*c*d^3)/(c*x^2+a)+1/2/a*(1/2*(-2*A*a*c*e^3+2*B*a*c*d*e^2)/c*ln( 
c*x^2+a)+(3*A*a*c*d*e^2+A*c^2*d^3+B*a^2*e^3-B*a*c*d^2*e)/(a*c)^(1/2)*arcta 
n(c*x/(a*c)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (181) = 362\).

Time = 9.01 (sec) , antiderivative size = 795, normalized size of antiderivative = 4.08 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx=\left [-\frac {2 \, B a^{2} c^{2} d^{3} - 2 \, A a^{2} c^{2} d^{2} e + 2 \, B a^{3} c d e^{2} - 2 \, A a^{3} c e^{3} + {\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} + {\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x - 2 \, {\left (B a^{3} c d e^{2} - A a^{3} c e^{3} + {\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \, {\left (B a^{3} c d e^{2} - A a^{3} c e^{3} + {\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{4 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}, -\frac {B a^{2} c^{2} d^{3} - A a^{2} c^{2} d^{2} e + B a^{3} c d e^{2} - A a^{3} c e^{3} - {\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} + {\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x - {\left (B a^{3} c d e^{2} - A a^{3} c e^{3} + {\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B a^{3} c d e^{2} - A a^{3} c e^{3} + {\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(2*B*a^2*c^2*d^3 - 2*A*a^2*c^2*d^2*e + 2*B*a^3*c*d*e^2 - 2*A*a^3*c*e 
^3 + (A*a*c^2*d^3 - B*a^2*c*d^2*e + 3*A*a^2*c*d*e^2 + B*a^3*e^3 + (A*c^3*d 
^3 - B*a*c^2*d^2*e + 3*A*a*c^2*d*e^2 + B*a^2*c*e^3)*x^2)*sqrt(-a*c)*log((c 
*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*(A*a*c^3*d^3 + B*a^2*c^2*d^2*e 
 + A*a^2*c^2*d*e^2 + B*a^3*c*e^3)*x - 2*(B*a^3*c*d*e^2 - A*a^3*c*e^3 + (B* 
a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(c*x^2 + a) + 4*(B*a^3*c*d*e^2 - A* 
a^3*c*e^3 + (B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(e*x + d))/(a^3*c^3* 
d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a 
^4*c^2*e^4)*x^2), -1/2*(B*a^2*c^2*d^3 - A*a^2*c^2*d^2*e + B*a^3*c*d*e^2 - 
A*a^3*c*e^3 - (A*a*c^2*d^3 - B*a^2*c*d^2*e + 3*A*a^2*c*d*e^2 + B*a^3*e^3 + 
 (A*c^3*d^3 - B*a*c^2*d^2*e + 3*A*a*c^2*d*e^2 + B*a^2*c*e^3)*x^2)*sqrt(a*c 
)*arctan(sqrt(a*c)*x/a) - (A*a*c^3*d^3 + B*a^2*c^2*d^2*e + A*a^2*c^2*d*e^2 
 + B*a^3*c*e^3)*x - (B*a^3*c*d*e^2 - A*a^3*c*e^3 + (B*a^2*c^2*d*e^2 - A*a^ 
2*c^2*e^3)*x^2)*log(c*x^2 + a) + 2*(B*a^3*c*d*e^2 - A*a^3*c*e^3 + (B*a^2*c 
^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(e*x + d))/(a^3*c^3*d^4 + 2*a^4*c^2*d^2* 
e^2 + a^5*c*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^2)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx=\frac {{\left (B d e^{2} - A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (A c^{2} d^{3} - B a c d^{2} e + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {B a d - A a e - {\left (A c d + B a e\right )} x}{2 \, {\left (a^{2} c d^{2} + a^{3} e^{2} + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{2}\right )}} \] Input:

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

Output:

1/2*(B*d*e^2 - A*e^3)*log(c*x^2 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - 
 (B*d*e^2 - A*e^3)*log(e*x + d)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/2* 
(A*c^2*d^3 - B*a*c*d^2*e + 3*A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/sqrt(a*c) 
)/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*sqrt(a*c)) - 1/2*(B*a*d - A*a*e 
 - (A*c*d + B*a*e)*x)/(a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx=\frac {{\left (B d e^{2} - A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {{\left (B d e^{3} - A e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac {{\left (A c^{2} d^{3} - B a c d^{2} e + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {B a c d^{3} - A a c d^{2} e + B a^{2} d e^{2} - A a^{2} e^{3} - {\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (c x^{2} + a\right )} a} \] Input:

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

Output:

1/2*(B*d*e^2 - A*e^3)*log(c*x^2 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - 
 (B*d*e^3 - A*e^4)*log(abs(e*x + d))/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) 
 + 1/2*(A*c^2*d^3 - B*a*c*d^2*e + 3*A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/sq 
rt(a*c))/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*sqrt(a*c)) - 1/2*(B*a*c* 
d^3 - A*a*c*d^2*e + B*a^2*d*e^2 - A*a^2*e^3 - (A*c^2*d^3 + B*a*c*d^2*e + A 
*a*c*d*e^2 + B*a^2*e^3)*x)/((c*d^2 + a*e^2)^2*(c*x^2 + a)*a)
 

Mupad [B] (verification not implemented)

Time = 7.28 (sec) , antiderivative size = 1086, normalized size of antiderivative = 5.57 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((A*e - B*d)/(2*(a*e^2 + c*d^2)) + (x*(A*c*d + B*a*e))/(2*a*(a*e^2 + c*d^2 
)))/(a + c*x^2) - (log(A*c^3*d^5*(-a^3*c)^(1/2) - B*a^3*e^5*(-a^3*c)^(1/2) 
 - 6*A*a^4*c*e^5 + B*a^4*c*e^5*x + 2*A*a^2*c^3*d^4*e + 12*A*a^3*c^2*d^2*e^ 
3 - 8*B*a^3*c^2*d^3*e^2 + 8*B*a^4*c*d*e^4 - A*a*c^4*d^5*x - 2*A*a^2*c^3*d^ 
3*e^2*x - 14*B*a^3*c^2*d^2*e^3*x + 2*A*a*c^2*d^3*e^2*(-a^3*c)^(1/2) + 14*B 
*a^2*c*d^2*e^3*(-a^3*c)^(1/2) + 15*A*a^3*c^2*d*e^4*x + B*a^2*c^3*d^4*e*x - 
 15*A*a^2*c*d*e^4*(-a^3*c)^(1/2) - B*a*c^2*d^4*e*(-a^3*c)^(1/2) - 6*A*a^2* 
c*e^5*x*(-a^3*c)^(1/2) + 2*A*c^3*d^4*e*x*(-a^3*c)^(1/2) + 8*B*a^2*c*d*e^4* 
x*(-a^3*c)^(1/2) + 12*A*a*c^2*d^2*e^3*x*(-a^3*c)^(1/2) - 8*B*a*c^2*d^3*e^2 
*x*(-a^3*c)^(1/2))*(c*(a*((3*A*d*e^2*(-a^3*c)^(1/2))/4 - (B*d^2*e*(-a^3*c) 
^(1/2))/4) + a^3*((A*e^3)/2 - (B*d*e^2)/2)) + (A*c^2*d^3*(-a^3*c)^(1/2))/4 
 + (B*a^2*e^3*(-a^3*c)^(1/2))/4))/(a^5*c*e^4 + a^3*c^3*d^4 + 2*a^4*c^2*d^2 
*e^2) + (log(A*c^3*d^5*(-a^3*c)^(1/2) - B*a^3*e^5*(-a^3*c)^(1/2) + 6*A*a^4 
*c*e^5 - B*a^4*c*e^5*x - 2*A*a^2*c^3*d^4*e - 12*A*a^3*c^2*d^2*e^3 + 8*B*a^ 
3*c^2*d^3*e^2 - 8*B*a^4*c*d*e^4 + A*a*c^4*d^5*x + 2*A*a^2*c^3*d^3*e^2*x + 
14*B*a^3*c^2*d^2*e^3*x + 2*A*a*c^2*d^3*e^2*(-a^3*c)^(1/2) + 14*B*a^2*c*d^2 
*e^3*(-a^3*c)^(1/2) - 15*A*a^3*c^2*d*e^4*x - B*a^2*c^3*d^4*e*x - 15*A*a^2* 
c*d*e^4*(-a^3*c)^(1/2) - B*a*c^2*d^4*e*(-a^3*c)^(1/2) - 6*A*a^2*c*e^5*x*(- 
a^3*c)^(1/2) + 2*A*c^3*d^4*e*x*(-a^3*c)^(1/2) + 8*B*a^2*c*d*e^4*x*(-a^3*c) 
^(1/2) + 12*A*a*c^2*d^2*e^3*x*(-a^3*c)^(1/2) - 8*B*a*c^2*d^3*e^2*x*(-a^...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.70 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2} b \,e^{3}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2} c d \,e^{2}-\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a b c \,d^{2} e +\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a b c \,e^{3} x^{2}+\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a \,c^{2} d^{3}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a \,c^{2} d \,e^{2} x^{2}-\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) b \,c^{2} d^{2} e \,x^{2}+\sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) c^{3} d^{3} x^{2}-\mathrm {log}\left (c \,x^{2}+a \right ) a^{3} c \,e^{3}+\mathrm {log}\left (c \,x^{2}+a \right ) a^{2} b c d \,e^{2}-\mathrm {log}\left (c \,x^{2}+a \right ) a^{2} c^{2} e^{3} x^{2}+\mathrm {log}\left (c \,x^{2}+a \right ) a b \,c^{2} d \,e^{2} x^{2}+2 \,\mathrm {log}\left (e x +d \right ) a^{3} c \,e^{3}-2 \,\mathrm {log}\left (e x +d \right ) a^{2} b c d \,e^{2}+2 \,\mathrm {log}\left (e x +d \right ) a^{2} c^{2} e^{3} x^{2}-2 \,\mathrm {log}\left (e x +d \right ) a b \,c^{2} d \,e^{2} x^{2}+a^{2} b c \,e^{3} x +a^{2} c^{2} d \,e^{2} x -a^{2} c^{2} e^{3} x^{2}+a b \,c^{2} d^{2} e x +a b \,c^{2} d \,e^{2} x^{2}+a \,c^{3} d^{3} x -a \,c^{3} d^{2} e \,x^{2}+b \,c^{3} d^{3} x^{2}}{2 a c \left (a^{2} c \,e^{4} x^{2}+2 a \,c^{2} d^{2} e^{2} x^{2}+c^{3} d^{4} x^{2}+a^{3} e^{4}+2 a^{2} c \,d^{2} e^{2}+a \,c^{2} d^{4}\right )} \] Input:

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

Output:

(sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a**2*b*e**3 + 3*sqrt(c)*sqr 
t(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a**2*c*d*e**2 - sqrt(c)*sqrt(a)*atan((c 
*x)/(sqrt(c)*sqrt(a)))*a*b*c*d**2*e + sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)* 
sqrt(a)))*a*b*c*e**3*x**2 + sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))* 
a*c**2*d**3 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a*c**2*d*e** 
2*x**2 - sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*b*c**2*d**2*e*x**2 
+ sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*c**3*d**3*x**2 - log(a + c 
*x**2)*a**3*c*e**3 + log(a + c*x**2)*a**2*b*c*d*e**2 - log(a + c*x**2)*a** 
2*c**2*e**3*x**2 + log(a + c*x**2)*a*b*c**2*d*e**2*x**2 + 2*log(d + e*x)*a 
**3*c*e**3 - 2*log(d + e*x)*a**2*b*c*d*e**2 + 2*log(d + e*x)*a**2*c**2*e** 
3*x**2 - 2*log(d + e*x)*a*b*c**2*d*e**2*x**2 + a**2*b*c*e**3*x + a**2*c**2 
*d*e**2*x - a**2*c**2*e**3*x**2 + a*b*c**2*d**2*e*x + a*b*c**2*d*e**2*x**2 
 + a*c**3*d**3*x - a*c**3*d**2*e*x**2 + b*c**3*d**3*x**2)/(2*a*c*(a**3*e** 
4 + 2*a**2*c*d**2*e**2 + a**2*c*e**4*x**2 + a*c**2*d**4 + 2*a*c**2*d**2*e* 
*2*x**2 + c**3*d**4*x**2))