\(\int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx\) [1129]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=-\frac {A b-a B}{(b d-a e)^2 (a+b x)}+\frac {B d-A e}{(b d-a e)^2 (d+e x)}+\frac {(b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^3}-\frac {(b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=-\frac {A b-a B}{(a+b x) (b d-a e)^2}+\frac {B d-A e}{(d+e x) (b d-a e)^2}+\frac {\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac {\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]

[In]

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

[Out]

-((A*b - a*B)/((b*d - a*e)^2*(a + b*x))) + (B*d - A*e)/((b*d - a*e)^2*(d + e*x)) + ((b*B*d - 2*A*b*e + a*B*e)*
Log[a + b*x])/(b*d - a*e)^3 - ((b*B*d - 2*A*b*e + a*B*e)*Log[d + e*x])/(b*d - a*e)^3

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b (A b-a B)}{(b d-a e)^2 (a+b x)^2}+\frac {b (b B d-2 A b e+a B e)}{(b d-a e)^3 (a+b x)}+\frac {e (-B d+A e)}{(b d-a e)^2 (d+e x)^2}+\frac {e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)}\right ) \, dx \\ & = -\frac {A b-a B}{(b d-a e)^2 (a+b x)}+\frac {B d-A e}{(b d-a e)^2 (d+e x)}+\frac {(b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^3}-\frac {(b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=\frac {\frac {(-A b+a B) (b d-a e)}{a+b x}+\frac {(b d-a e) (B d-A e)}{d+e x}+(b B d-2 A b e+a B e) \log (a+b x)-(b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^3} \]

[In]

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

[Out]

(((-(A*b) + a*B)*(b*d - a*e))/(a + b*x) + ((b*d - a*e)*(B*d - A*e))/(d + e*x) + (b*B*d - 2*A*b*e + a*B*e)*Log[
a + b*x] - (b*B*d - 2*A*b*e + a*B*e)*Log[d + e*x])/(b*d - a*e)^3

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {\left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}-\frac {A b -B a}{\left (a e -b d \right )^{2} \left (b x +a \right )}-\frac {A e -B d}{\left (a e -b d \right )^{2} \left (e x +d \right )}-\frac {\left (2 A b e -B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}\)	\(123\)
norman	\(\frac {-\frac {A a b \,e^{2}+A \,b^{2} d e -2 B a b d e}{e b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {\left (2 A \,b^{2} e^{2}-B a b \,e^{2}-b^{2} B d e \right ) x}{e b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}}{\left (b x +a \right ) \left (e x +d \right )}+\frac {\left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {\left (2 A b e -B a e -B b d \right ) \ln \left (e x +d \right )}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}\)	\(252\)
risch	\(\frac {-\frac {\left (2 A b e -B a e -B b d \right ) x}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {A a e +A b d -2 B a d}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}}{\left (b x +a \right ) \left (e x +d \right )}-\frac {2 \ln \left (e x +d \right ) A b e}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\ln \left (e x +d \right ) B a e}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\ln \left (e x +d \right ) B b d}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {2 \ln \left (-b x -a \right ) A b e}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {\ln \left (-b x -a \right ) B a e}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {\ln \left (-b x -a \right ) B b d}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}\)	\(400\)
parallelrisch	\(\frac {-B \ln \left (b x +a \right ) x^{2} a \,b^{2} e^{3}-B \ln \left (b x +a \right ) x^{2} b^{3} d \,e^{2}+B \ln \left (e x +d \right ) x^{2} a \,b^{2} e^{3}+B \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{2}+2 A \ln \left (b x +a \right ) x a \,b^{2} e^{3}+2 A \ln \left (b x +a \right ) x \,b^{3} d \,e^{2}-2 A \ln \left (e x +d \right ) x a \,b^{2} e^{3}-2 A \ln \left (e x +d \right ) x \,b^{3} d \,e^{2}-B \ln \left (b x +a \right ) x \,a^{2} b \,e^{3}-B \ln \left (b x +a \right ) x \,b^{3} d^{2} e +B \ln \left (e x +d \right ) x \,a^{2} b \,e^{3}+B \ln \left (e x +d \right ) x \,b^{3} d^{2} e +2 A \ln \left (b x +a \right ) a \,b^{2} d \,e^{2}-2 A \ln \left (e x +d \right ) a \,b^{2} d \,e^{2}-B \ln \left (b x +a \right ) a^{2} b d \,e^{2}-B \ln \left (b x +a \right ) a \,b^{2} d^{2} e +B \ln \left (e x +d \right ) a^{2} b d \,e^{2}+B \ln \left (e x +d \right ) a \,b^{2} d^{2} e +2 A \ln \left (b x +a \right ) x^{2} b^{3} e^{3}-2 A \ln \left (e x +d \right ) x^{2} b^{3} e^{3}-2 A x a \,b^{2} e^{3}+2 A x \,b^{3} d \,e^{2}+B x \,a^{2} b \,e^{3}-B x \,b^{3} d^{2} e -2 B \ln \left (b x +a \right ) x a \,b^{2} d \,e^{2}+2 B \ln \left (e x +d \right ) x a \,b^{2} d \,e^{2}+2 B \,a^{2} b d \,e^{2}-2 B a \,b^{2} d^{2} e -A \,a^{2} b \,e^{3}+A \,b^{3} d^{2} e}{\left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right ) \left (b x +a \right ) b e}\)	\(525\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (116) = 232\).

Time = 0.23 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.38 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=-\frac {2 \, B a^{2} d e - A a^{2} e^{2} - {\left (2 \, B a b - A b^{2}\right )} d^{2} - {\left (B b^{2} d^{2} - 2 \, A b^{2} d e - {\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x - {\left (B a b d^{2} + {\left (B a^{2} - 2 \, A a b\right )} d e + {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (B b^{2} d^{2} + 2 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (b x + a\right ) + {\left (B a b d^{2} + {\left (B a^{2} - 2 \, A a b\right )} d e + {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (B b^{2} d^{2} + 2 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} + {\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x} \]

[In]

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

[Out]

-(2*B*a^2*d*e - A*a^2*e^2 - (2*B*a*b - A*b^2)*d^2 - (B*b^2*d^2 - 2*A*b^2*d*e - (B*a^2 - 2*A*a*b)*e^2)*x - (B*a
*b*d^2 + (B*a^2 - 2*A*a*b)*d*e + (B*b^2*d*e + (B*a*b - 2*A*b^2)*e^2)*x^2 + (B*b^2*d^2 + 2*(B*a*b - A*b^2)*d*e
+ (B*a^2 - 2*A*a*b)*e^2)*x)*log(b*x + a) + (B*a*b*d^2 + (B*a^2 - 2*A*a*b)*d*e + (B*b^2*d*e + (B*a*b - 2*A*b^2)
*e^2)*x^2 + (B*b^2*d^2 + 2*(B*a*b - A*b^2)*d*e + (B*a^2 - 2*A*a*b)*e^2)*x)*log(e*x + d))/(a*b^3*d^4 - 3*a^2*b^
2*d^3*e + 3*a^3*b*d^2*e^2 - a^4*d*e^3 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x^2 + (b^4
*d^4 - 2*a*b^3*d^3*e + 2*a^3*b*d*e^3 - a^4*e^4)*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (104) = 208\).

Time = 1.36 (sec) , antiderivative size = 706, normalized size of antiderivative = 6.03 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=\frac {- A a e - A b d + 2 B a d + x \left (- 2 A b e + B a e + B b d\right )}{a^{3} d e^{2} - 2 a^{2} b d^{2} e + a b^{2} d^{3} + x^{2} \left (a^{2} b e^{3} - 2 a b^{2} d e^{2} + b^{3} d^{2} e\right ) + x \left (a^{3} e^{3} - a^{2} b d e^{2} - a b^{2} d^{2} e + b^{3} d^{3}\right )} + \frac {\left (- 2 A b e + B a e + B b d\right ) \log {\left (x + \frac {- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} - \frac {a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} - \frac {\left (- 2 A b e + B a e + B b d\right ) \log {\left (x + \frac {- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} + \frac {a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} \]

[In]

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

[Out]

(-A*a*e - A*b*d + 2*B*a*d + x*(-2*A*b*e + B*a*e + B*b*d))/(a**3*d*e**2 - 2*a**2*b*d**2*e + a*b**2*d**3 + x**2*
(a**2*b*e**3 - 2*a*b**2*d*e**2 + b**3*d**2*e) + x*(a**3*e**3 - a**2*b*d*e**2 - a*b**2*d**2*e + b**3*d**3)) + (
-2*A*b*e + B*a*e + B*b*d)*log(x + (-2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**2 + 2*B*a*b*d*e + B*b**2*d**2 - a*
*4*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 + 4*a**3*b*d*e**3*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3
- 6*a**2*b**2*d**2*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 + 4*a*b**3*d**3*e*(-2*A*b*e + B*a*e + B*b*d)
/(a*e - b*d)**3 - b**4*d**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3)/(-4*A*b**2*e**2 + 2*B*a*b*e**2 + 2*B*b*
*2*d*e))/(a*e - b*d)**3 - (-2*A*b*e + B*a*e + B*b*d)*log(x + (-2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**2 + 2*B
*a*b*d*e + B*b**2*d**2 + a**4*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 - 4*a**3*b*d*e**3*(-2*A*b*e + B*a
*e + B*b*d)/(a*e - b*d)**3 + 6*a**2*b**2*d**2*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 - 4*a*b**3*d**3*e
*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 + b**4*d**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3)/(-4*A*b**2*e
**2 + 2*B*a*b*e**2 + 2*B*b**2*d*e))/(a*e - b*d)**3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (116) = 232\).

Time = 0.21 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.19 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=\frac {{\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {{\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {A a e - {\left (2 \, B a - A b\right )} d - {\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} x}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + {\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \]

[In]

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

[Out]

(B*b*d + (B*a - 2*A*b)*e)*log(b*x + a)/(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3) - (B*b*d + (B*a - 2
*A*b)*e)*log(e*x + d)/(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3) - (A*a*e - (2*B*a - A*b)*d - (B*b*d
+ (B*a - 2*A*b)*e)*x)/(a*b^2*d^3 - 2*a^2*b*d^2*e + a^3*d*e^2 + (b^3*d^2*e - 2*a*b^2*d*e^2 + a^2*b*e^3)*x^2 + (
b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.66 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=-\frac {{\left (B b^{2} d + B a b e - 2 \, A b^{2} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac {\frac {B a b^{2}}{b x + a} - \frac {A b^{3}}{b x + a}}{b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}} - \frac {B b d e - A b e^{2}}{{\left (b d - a e\right )}^{3} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}} \]

[In]

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

[Out]

-(B*b^2*d + B*a*b*e - 2*A*b^2*e)*log(abs(b*d/(b*x + a) - a*e/(b*x + a) + e))/(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*
b^2*d*e^2 - a^3*b*e^3) + (B*a*b^2/(b*x + a) - A*b^3/(b*x + a))/(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2) - (B*b*d*
e - A*b*e^2)/((b*d - a*e)^3*(b*d/(b*x + a) - a*e/(b*x + a) + e))

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.25 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx=-\frac {\frac {A\,a\,e+A\,b\,d-2\,B\,a\,d}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}-\frac {x\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}}{b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d}-\frac {2\,\mathrm {atanh}\left (\frac {\left (\frac {a^3\,e^3-a^2\,b\,d\,e^2-a\,b^2\,d^2\,e+b^3\,d^3}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}+2\,b\,e\,x\right )\,\left (e\,\left (2\,A\,b-B\,a\right )-B\,b\,d\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}\right )\,\left (e\,\left (2\,A\,b-B\,a\right )-B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^3} \]

[In]

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

[Out]

- ((A*a*e + A*b*d - 2*B*a*d)/(a^2*e^2 + b^2*d^2 - 2*a*b*d*e) - (x*(B*a*e - 2*A*b*e + B*b*d))/(a^2*e^2 + b^2*d^
2 - 2*a*b*d*e))/(a*d + x*(a*e + b*d) + b*e*x^2) - (2*atanh((((a^3*e^3 + b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2)/(
a^2*e^2 + b^2*d^2 - 2*a*b*d*e) + 2*b*e*x)*(e*(2*A*b - B*a) - B*b*d)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e))/((a*e - b
*d)^3*(B*a*e - 2*A*b*e + B*b*d)))*(e*(2*A*b - B*a) - B*b*d))/(a*e - b*d)^3