\(\int \frac {A+B x}{(d+e x)^3 (b x+c x^2)^2} \, dx\) [1153]

Optimal result

Integrand size = 24, antiderivative size = 283 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d^3 x}+\frac {c^3 (b B-A c)}{b^2 (c d-b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}-\frac {e^2 \left (B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \]

[Out]

-A/b^2/d^3/x+c^3*(-A*c+B*b)/b^2/(-b*e+c*d)^3/(c*x+b)+1/2*e^2*(-A*e+B*d)/d^2/(-b*e+c*d)^2/(e*x+d)^2-e^2*(2*A*e*
(-b*e+2*c*d)-B*d*(-b*e+3*c*d))/d^3/(-b*e+c*d)^3/(e*x+d)+(-3*A*b*e-2*A*c*d+B*b*d)*ln(x)/b^3/d^4+c^3*(2*A*c^2*d+
4*b^2*B*e-b*c*(5*A*e+B*d))*ln(c*x+b)/b^3/(-b*e+c*d)^4-e^2*(B*d*(b^2*e^2-4*b*c*d*e+6*c^2*d^2)-A*e*(3*b^2*e^2-10
*b*c*d*e+10*c^2*d^2))*ln(e*x+d)/d^4/(-b*e+c*d)^4

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\frac {\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}+\frac {c^3 (b B-A c)}{b^2 (b+c x) (c d-b e)^3}-\frac {e^2 \log (d+e x) \left (B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )-A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}-\frac {A}{b^2 d^3 x}+\frac {c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac {e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \]

[In]

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

[Out]

-(A/(b^2*d^3*x)) + (c^3*(b*B - A*c))/(b^2*(c*d - b*e)^3*(b + c*x)) + (e^2*(B*d - A*e))/(2*d^2*(c*d - b*e)^2*(d
+ e*x)^2) - (e^2*(2*A*e*(2*c*d - b*e) - B*d*(3*c*d - b*e)))/(d^3*(c*d - b*e)^3*(d + e*x)) + ((b*B*d - 2*A*c*d
- 3*A*b*e)*Log[x])/(b^3*d^4) + (c^3*(2*A*c^2*d + 4*b^2*B*e - b*c*(B*d + 5*A*e))*Log[b + c*x])/(b^3*(c*d - b*e
)^4) - (e^2*(B*d*(6*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2))*Log[d + e*x])/
(d^4*(c*d - b*e)^4)

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{b^2 d^3 x^2}+\frac {b B d-2 A c d-3 A b e}{b^3 d^4 x}+\frac {c^4 (b B-A c)}{b^2 (-c d+b e)^3 (b+c x)^2}+\frac {c^4 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right )}{b^3 (c d-b e)^4 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)^3}+\frac {e^3 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)^2}+\frac {e^3 \left (-B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right )}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx \\ & = -\frac {A}{b^2 d^3 x}+\frac {c^3 (b B-A c)}{b^2 (c d-b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}-\frac {e^2 \left (B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d^3 x}+\frac {c^3 (-b B+A c)}{b^2 (-c d+b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}+\frac {e^2 (B d (3 c d-b e)+2 A e (-2 c d+b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}+\frac {e^2 \left (-B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \]

[In]

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

[Out]

-(A/(b^2*d^3*x)) + (c^3*(-(b*B) + A*c))/(b^2*(-(c*d) + b*e)^3*(b + c*x)) + (e^2*(B*d - A*e))/(2*d^2*(c*d - b*e
)^2*(d + e*x)^2) + (e^2*(B*d*(3*c*d - b*e) + 2*A*e*(-2*c*d + b*e)))/(d^3*(c*d - b*e)^3*(d + e*x)) + ((b*B*d -
2*A*c*d - 3*A*b*e)*Log[x])/(b^3*d^4) + (c^3*(2*A*c^2*d + 4*b^2*B*e - b*c*(B*d + 5*A*e))*Log[b + c*x])/(b^3*(c*
d - b*e)^4) + (e^2*(-(B*d*(6*c^2*d^2 - 4*b*c*d*e + b^2*e^2)) + A*e*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2))*Log[
d + e*x])/(d^4*(c*d - b*e)^4)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {A}{b^{2} d^{3} x}+\frac {\left (-3 A b e -2 A c d +B b d \right ) \ln \left (x \right )}{b^{3} d^{4}}-\frac {c^{3} \left (5 A b c e -2 A \,c^{2} d -4 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b e -c d \right )^{4}}+\frac {\left (A c -B b \right ) c^{3}}{b^{2} \left (b e -c d \right )^{3} \left (c x +b \right )}-\frac {e^{2} \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right )}{d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )}+\frac {e^{2} \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{d^{4} \left (b e -c d \right )^{4}}-\frac {\left (A e -B d \right ) e^{2}}{2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{2}}\)	\(285\)
norman	\(\frac {\frac {\left (6 A \,b^{5} e^{5}-12 A \,b^{4} c d \,e^{4}+4 A \,b^{3} c^{2} d^{2} e^{3}+A b \,c^{4} d^{4} e -2 A \,c^{5} d^{5}-2 B \,b^{5} d \,e^{4}+4 B \,b^{4} c \,d^{2} e^{3}+B b \,c^{4} d^{5}\right ) x^{2}}{d^{3} b^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {A}{d b}+\frac {e \left (9 A \,b^{5} e^{5}-7 A \,b^{4} c d \,e^{4}-18 A \,b^{3} c^{2} d^{2} e^{3}+8 A \,b^{2} c^{3} d^{3} e^{2}+4 A b \,c^{4} d^{4} e -8 A \,c^{5} d^{5}-3 B \,b^{5} d \,e^{4}+3 B \,b^{4} c \,d^{2} e^{3}+8 B \,b^{3} c^{2} d^{3} e^{2}+4 B b \,c^{4} d^{5}\right ) x^{3}}{2 d^{4} b^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {e^{2} c \left (9 A \,b^{4} e^{4}-19 A \,b^{3} c d \,e^{3}+6 A \,b^{2} c^{2} d^{2} e^{2}+2 A b \,c^{3} d^{3} e -4 A \,c^{4} d^{4}-3 B \,b^{4} d \,e^{3}+7 B \,b^{3} c \,d^{2} e^{2}+2 B b \,c^{3} d^{4}\right ) x^{4}}{2 d^{4} b^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}}{\left (e x +d \right )^{2} x \left (c x +b \right )}+\frac {e^{2} \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{d^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {\left (3 A b e +2 A c d -B b d \right ) \ln \left (x \right )}{b^{3} d^{4}}-\frac {c^{3} \left (5 A b c e -2 A \,c^{2} d -4 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) b^{3}}\)	\(714\)
risch	\(\text {Expression too large to display}\)	\(1180\)
parallelrisch	\(\text {Expression too large to display}\)	\(2554\)

[In]

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

[Out]

-A/b^2/d^3/x+(-3*A*b*e-2*A*c*d+B*b*d)*ln(x)/b^3/d^4-c^3*(5*A*b*c*e-2*A*c^2*d-4*B*b^2*e+B*b*c*d)/b^3/(b*e-c*d)^
4*ln(c*x+b)+(A*c-B*b)*c^3/b^2/(b*e-c*d)^3/(c*x+b)-e^2*(2*A*b*e^2-4*A*c*d*e-B*b*d*e+3*B*c*d^2)/d^3/(b*e-c*d)^3/
(e*x+d)+e^2*(3*A*b^2*e^3-10*A*b*c*d*e^2+10*A*c^2*d^2*e-B*b^2*d*e^2+4*B*b*c*d^2*e-6*B*c^2*d^3)/d^4/(b*e-c*d)^4*
ln(e*x+d)-1/2*(A*e-B*d)*e^2/d^2/(b*e-c*d)^2/(e*x+d)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1924 vs. \(2 (279) = 558\).

Time = 264.40 (sec) , antiderivative size = 1924, normalized size of antiderivative = 6.80 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(2*A*b^2*c^4*d^7 - 8*A*b^3*c^3*d^6*e + 12*A*b^4*c^2*d^5*e^2 - 8*A*b^5*c*d^4*e^3 + 2*A*b^6*d^3*e^4 + 2*(3*
A*b^5*c*d*e^6 - (B*b^2*c^4 - 2*A*b*c^5)*d^5*e^2 - (2*B*b^3*c^3 + 5*A*b^2*c^4)*d^4*e^3 + 2*(2*B*b^4*c^2 + 5*A*b
^3*c^3)*d^3*e^4 - (B*b^5*c + 10*A*b^4*c^2)*d^2*e^5)*x^3 + (6*A*b^6*d*e^6 - 4*(B*b^2*c^4 - 2*A*b*c^5)*d^6*e - 3
*(B*b^3*c^3 + 6*A*b^2*c^4)*d^5*e^2 + (4*B*b^4*c^2 + 25*A*b^3*c^3)*d^4*e^3 + 5*(B*b^5*c - 2*A*b^4*c^2)*d^3*e^4
- (2*B*b^6 + 11*A*b^5*c)*d^2*e^5)*x^2 + (9*A*b^6*d^2*e^5 - 2*(B*b^2*c^4 - 2*A*b*c^5)*d^7 + 2*(B*b^3*c^3 - 3*A*
b^2*c^4)*d^6*e - (7*B*b^4*c^2 + 4*A*b^3*c^3)*d^5*e^2 + 5*(2*B*b^5*c + 5*A*b^4*c^2)*d^4*e^3 - (3*B*b^6 + 28*A*b
^5*c)*d^3*e^4)*x + 2*(((B*b*c^5 - 2*A*c^6)*d^5*e^2 - (4*B*b^2*c^4 - 5*A*b*c^5)*d^4*e^3)*x^4 + (2*(B*b*c^5 - 2*
A*c^6)*d^6*e - (7*B*b^2*c^4 - 8*A*b*c^5)*d^5*e^2 - (4*B*b^3*c^3 - 5*A*b^2*c^4)*d^4*e^3)*x^3 + ((B*b*c^5 - 2*A*
c^6)*d^7 - (2*B*b^2*c^4 - A*b*c^5)*d^6*e - 2*(4*B*b^3*c^3 - 5*A*b^2*c^4)*d^5*e^2)*x^2 + ((B*b^2*c^4 - 2*A*b*c^
5)*d^7 - (4*B*b^3*c^3 - 5*A*b^2*c^4)*d^6*e)*x)*log(c*x + b) + 2*((6*B*b^3*c^3*d^3*e^4 - 3*A*b^5*c*e^7 - 2*(2*B
*b^4*c^2 + 5*A*b^3*c^3)*d^2*e^5 + (B*b^5*c + 10*A*b^4*c^2)*d*e^6)*x^4 + (12*B*b^3*c^3*d^4*e^3 - 3*A*b^6*e^7 -
2*(B*b^4*c^2 + 10*A*b^3*c^3)*d^3*e^4 - 2*(B*b^5*c - 5*A*b^4*c^2)*d^2*e^5 + (B*b^6 + 4*A*b^5*c)*d*e^6)*x^3 + (6
*B*b^3*c^3*d^5*e^2 - 6*A*b^6*d*e^6 + 2*(4*B*b^4*c^2 - 5*A*b^3*c^3)*d^4*e^3 - (7*B*b^5*c + 10*A*b^4*c^2)*d^3*e^
4 + (2*B*b^6 + 17*A*b^5*c)*d^2*e^5)*x^2 + (6*B*b^4*c^2*d^5*e^2 - 3*A*b^6*d^2*e^5 - 2*(2*B*b^5*c + 5*A*b^4*c^2)
*d^4*e^3 + (B*b^6 + 10*A*b^5*c)*d^3*e^4)*x)*log(e*x + d) - 2*((6*B*b^3*c^3*d^3*e^4 - 3*A*b^5*c*e^7 + (B*b*c^5
- 2*A*c^6)*d^5*e^2 - (4*B*b^2*c^4 - 5*A*b*c^5)*d^4*e^3 - 2*(2*B*b^4*c^2 + 5*A*b^3*c^3)*d^2*e^5 + (B*b^5*c + 10
*A*b^4*c^2)*d*e^6)*x^4 - (3*A*b^6*e^7 - 2*(B*b*c^5 - 2*A*c^6)*d^6*e + (7*B*b^2*c^4 - 8*A*b*c^5)*d^5*e^2 - (8*B
*b^3*c^3 + 5*A*b^2*c^4)*d^4*e^3 + 2*(B*b^4*c^2 + 10*A*b^3*c^3)*d^3*e^4 + 2*(B*b^5*c - 5*A*b^4*c^2)*d^2*e^5 - (
B*b^6 + 4*A*b^5*c)*d*e^6)*x^3 - (6*A*b^6*d*e^6 - (B*b*c^5 - 2*A*c^6)*d^7 + (2*B*b^2*c^4 - A*b*c^5)*d^6*e + 2*(
B*b^3*c^3 - 5*A*b^2*c^4)*d^5*e^2 - 2*(4*B*b^4*c^2 - 5*A*b^3*c^3)*d^4*e^3 + (7*B*b^5*c + 10*A*b^4*c^2)*d^3*e^4
- (2*B*b^6 + 17*A*b^5*c)*d^2*e^5)*x^2 + (6*B*b^4*c^2*d^5*e^2 - 3*A*b^6*d^2*e^5 + (B*b^2*c^4 - 2*A*b*c^5)*d^7 -
(4*B*b^3*c^3 - 5*A*b^2*c^4)*d^6*e - 2*(2*B*b^5*c + 5*A*b^4*c^2)*d^4*e^3 + (B*b^6 + 10*A*b^5*c)*d^3*e^4)*x)*lo
g(x))/((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^4 - 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*
b^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^2 + 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^5 + b^8*d^4*e^6)*x^3
+ (b^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7*c*d^6*e^4 + 2*b^8*d^5*e^5)*
x^2 + (b^4*c^4*d^10 - 4*b^5*c^3*d^9*e + 6*b^6*c^2*d^8*e^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)/(e*x+d)**3/(c*x**2+b*x)**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (279) = 558\).

Time = 0.23 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.87 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {{\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d - {\left (4 \, B b^{2} c^{3} - 5 \, A b c^{4}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}} - \frac {{\left (6 \, B c^{2} d^{3} e^{2} - 3 \, A b^{2} e^{5} - 2 \, {\left (2 \, B b c + 5 \, A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 10 \, A b c\right )} d e^{4}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac {2 \, A b c^{3} d^{5} - 6 \, A b^{2} c^{2} d^{4} e + 6 \, A b^{3} c d^{3} e^{2} - 2 \, A b^{4} d^{2} e^{3} - 2 \, {\left (3 \, A b^{3} c e^{5} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} c^{2} + A b c^{3}\right )} d^{2} e^{3} - {\left (B b^{3} c + 7 \, A b^{2} c^{2}\right )} d e^{4}\right )} x^{3} - {\left (6 \, A b^{4} e^{5} + 4 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{4} e + {\left (7 \, B b^{2} c^{2} + 10 \, A b c^{3}\right )} d^{3} e^{2} + 3 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} d^{2} e^{3} - {\left (2 \, B b^{4} + 5 \, A b^{3} c\right )} d e^{4}\right )} x^{2} - {\left (2 \, A b c^{3} d^{4} e + 9 \, A b^{4} d e^{4} + 2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{5} + {\left (7 \, B b^{3} c + 6 \, A b^{2} c^{2}\right )} d^{3} e^{2} - {\left (3 \, B b^{4} + 19 \, A b^{3} c\right )} d^{2} e^{3}\right )} x}{2 \, {\left ({\left (b^{2} c^{4} d^{6} e^{2} - 3 \, b^{3} c^{3} d^{5} e^{3} + 3 \, b^{4} c^{2} d^{4} e^{4} - b^{5} c d^{3} e^{5}\right )} x^{4} + {\left (2 \, b^{2} c^{4} d^{7} e - 5 \, b^{3} c^{3} d^{6} e^{2} + 3 \, b^{4} c^{2} d^{5} e^{3} + b^{5} c d^{4} e^{4} - b^{6} d^{3} e^{5}\right )} x^{3} + {\left (b^{2} c^{4} d^{8} - b^{3} c^{3} d^{7} e - 3 \, b^{4} c^{2} d^{6} e^{2} + 5 \, b^{5} c d^{5} e^{3} - 2 \, b^{6} d^{4} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{8} - 3 \, b^{4} c^{2} d^{7} e + 3 \, b^{5} c d^{6} e^{2} - b^{6} d^{5} e^{3}\right )} x\right )}} - \frac {{\left (3 \, A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3} d^{4}} \]

[In]

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

[Out]

-((B*b*c^4 - 2*A*c^5)*d - (4*B*b^2*c^3 - 5*A*b*c^4)*e)*log(c*x + b)/(b^3*c^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2
*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4) - (6*B*c^2*d^3*e^2 - 3*A*b^2*e^5 - 2*(2*B*b*c + 5*A*c^2)*d^2*e^3 + (B*b^2
+ 10*A*b*c)*d*e^4)*log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)
- 1/2*(2*A*b*c^3*d^5 - 6*A*b^2*c^2*d^4*e + 6*A*b^3*c*d^3*e^2 - 2*A*b^4*d^2*e^3 - 2*(3*A*b^3*c*e^5 + (B*b*c^3 -
2*A*c^4)*d^3*e^2 + 3*(B*b^2*c^2 + A*b*c^3)*d^2*e^3 - (B*b^3*c + 7*A*b^2*c^2)*d*e^4)*x^3 - (6*A*b^4*e^5 + 4*(B
*b*c^3 - 2*A*c^4)*d^4*e + (7*B*b^2*c^2 + 10*A*b*c^3)*d^3*e^2 + 3*(B*b^3*c - 5*A*b^2*c^2)*d^2*e^3 - (2*B*b^4 +
5*A*b^3*c)*d*e^4)*x^2 - (2*A*b*c^3*d^4*e + 9*A*b^4*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^5 + (7*B*b^3*c + 6*A*b^2*c^
2)*d^3*e^2 - (3*B*b^4 + 19*A*b^3*c)*d^2*e^3)*x)/((b^2*c^4*d^6*e^2 - 3*b^3*c^3*d^5*e^3 + 3*b^4*c^2*d^4*e^4 - b^
5*c*d^3*e^5)*x^4 + (2*b^2*c^4*d^7*e - 5*b^3*c^3*d^6*e^2 + 3*b^4*c^2*d^5*e^3 + b^5*c*d^4*e^4 - b^6*d^3*e^5)*x^3
+ (b^2*c^4*d^8 - b^3*c^3*d^7*e - 3*b^4*c^2*d^6*e^2 + 5*b^5*c*d^5*e^3 - 2*b^6*d^4*e^4)*x^2 + (b^3*c^3*d^8 - 3*
b^4*c^2*d^7*e + 3*b^5*c*d^6*e^2 - b^6*d^5*e^3)*x) - (3*A*b*e - (B*b - 2*A*c)*d)*log(x)/(b^3*d^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (279) = 558\).

Time = 0.28 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (B b c^{5} d - 2 \, A c^{6} d - 4 \, B b^{2} c^{4} e + 5 \, A b c^{5} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{5} d^{4} - 4 \, b^{4} c^{4} d^{3} e + 6 \, b^{5} c^{3} d^{2} e^{2} - 4 \, b^{6} c^{2} d e^{3} + b^{7} c e^{4}} - \frac {{\left (6 \, B c^{2} d^{3} e^{3} - 4 \, B b c d^{2} e^{4} - 10 \, A c^{2} d^{2} e^{4} + B b^{2} d e^{5} + 10 \, A b c d e^{5} - 3 \, A b^{2} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, b c^{3} d^{7} e^{2} + 6 \, b^{2} c^{2} d^{6} e^{3} - 4 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5}} + \frac {{\left (B b d - 2 \, A c d - 3 \, A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{4}} - \frac {2 \, A b c^{4} d^{7} - 8 \, A b^{2} c^{3} d^{6} e + 12 \, A b^{3} c^{2} d^{5} e^{2} - 8 \, A b^{4} c d^{4} e^{3} + 2 \, A b^{5} d^{3} e^{4} - 2 \, {\left (B b c^{4} d^{5} e^{2} - 2 \, A c^{5} d^{5} e^{2} + 2 \, B b^{2} c^{3} d^{4} e^{3} + 5 \, A b c^{4} d^{4} e^{3} - 4 \, B b^{3} c^{2} d^{3} e^{4} - 10 \, A b^{2} c^{3} d^{3} e^{4} + B b^{4} c d^{2} e^{5} + 10 \, A b^{3} c^{2} d^{2} e^{5} - 3 \, A b^{4} c d e^{6}\right )} x^{3} - {\left (4 \, B b c^{4} d^{6} e - 8 \, A c^{5} d^{6} e + 3 \, B b^{2} c^{3} d^{5} e^{2} + 18 \, A b c^{4} d^{5} e^{2} - 4 \, B b^{3} c^{2} d^{4} e^{3} - 25 \, A b^{2} c^{3} d^{4} e^{3} - 5 \, B b^{4} c d^{3} e^{4} + 10 \, A b^{3} c^{2} d^{3} e^{4} + 2 \, B b^{5} d^{2} e^{5} + 11 \, A b^{4} c d^{2} e^{5} - 6 \, A b^{5} d e^{6}\right )} x^{2} - {\left (2 \, B b c^{4} d^{7} - 4 \, A c^{5} d^{7} - 2 \, B b^{2} c^{3} d^{6} e + 6 \, A b c^{4} d^{6} e + 7 \, B b^{3} c^{2} d^{5} e^{2} + 4 \, A b^{2} c^{3} d^{5} e^{2} - 10 \, B b^{4} c d^{4} e^{3} - 25 \, A b^{3} c^{2} d^{4} e^{3} + 3 \, B b^{5} d^{3} e^{4} + 28 \, A b^{4} c d^{3} e^{4} - 9 \, A b^{5} d^{2} e^{5}\right )} x}{2 \, {\left (c d - b e\right )}^{4} {\left (c x + b\right )} {\left (e x + d\right )}^{2} b^{2} d^{4} x} \]

[In]

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

[Out]

-(B*b*c^5*d - 2*A*c^6*d - 4*B*b^2*c^4*e + 5*A*b*c^5*e)*log(abs(c*x + b))/(b^3*c^5*d^4 - 4*b^4*c^4*d^3*e + 6*b^
5*c^3*d^2*e^2 - 4*b^6*c^2*d*e^3 + b^7*c*e^4) - (6*B*c^2*d^3*e^3 - 4*B*b*c*d^2*e^4 - 10*A*c^2*d^2*e^4 + B*b^2*d
*e^5 + 10*A*b*c*d*e^5 - 3*A*b^2*e^6)*log(abs(e*x + d))/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 - 4*b^
3*c*d^5*e^4 + b^4*d^4*e^5) + (B*b*d - 2*A*c*d - 3*A*b*e)*log(abs(x))/(b^3*d^4) - 1/2*(2*A*b*c^4*d^7 - 8*A*b^2*
c^3*d^6*e + 12*A*b^3*c^2*d^5*e^2 - 8*A*b^4*c*d^4*e^3 + 2*A*b^5*d^3*e^4 - 2*(B*b*c^4*d^5*e^2 - 2*A*c^5*d^5*e^2
+ 2*B*b^2*c^3*d^4*e^3 + 5*A*b*c^4*d^4*e^3 - 4*B*b^3*c^2*d^3*e^4 - 10*A*b^2*c^3*d^3*e^4 + B*b^4*c*d^2*e^5 + 10*
A*b^3*c^2*d^2*e^5 - 3*A*b^4*c*d*e^6)*x^3 - (4*B*b*c^4*d^6*e - 8*A*c^5*d^6*e + 3*B*b^2*c^3*d^5*e^2 + 18*A*b*c^4
*d^5*e^2 - 4*B*b^3*c^2*d^4*e^3 - 25*A*b^2*c^3*d^4*e^3 - 5*B*b^4*c*d^3*e^4 + 10*A*b^3*c^2*d^3*e^4 + 2*B*b^5*d^2
*e^5 + 11*A*b^4*c*d^2*e^5 - 6*A*b^5*d*e^6)*x^2 - (2*B*b*c^4*d^7 - 4*A*c^5*d^7 - 2*B*b^2*c^3*d^6*e + 6*A*b*c^4*
d^6*e + 7*B*b^3*c^2*d^5*e^2 + 4*A*b^2*c^3*d^5*e^2 - 10*B*b^4*c*d^4*e^3 - 25*A*b^3*c^2*d^4*e^3 + 3*B*b^5*d^3*e^
4 + 28*A*b^4*c*d^3*e^4 - 9*A*b^5*d^2*e^5)*x)/((c*d - b*e)^4*(c*x + b)*(e*x + d)^2*b^2*d^4*x)

Mupad [B] (verification not implemented)

Time = 12.03 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.57 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (\left (3\,A\,e^5-B\,d\,e^4\right )\,b^2+\left (4\,B\,d^2\,e^3-10\,A\,d\,e^4\right )\,b\,c+\left (10\,A\,d^2\,e^3-6\,B\,d^3\,e^2\right )\,c^2\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\frac {A}{b\,d}+\frac {x^2\,\left (-2\,B\,b^4\,d\,e^4+6\,A\,b^4\,e^5+3\,B\,b^3\,c\,d^2\,e^3-5\,A\,b^3\,c\,d\,e^4+7\,B\,b^2\,c^2\,d^3\,e^2-15\,A\,b^2\,c^2\,d^2\,e^3+4\,B\,b\,c^3\,d^4\,e+10\,A\,b\,c^3\,d^3\,e^2-8\,A\,c^4\,d^4\,e\right )}{2\,b^2\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x\,\left (-3\,B\,b^4\,d\,e^3+9\,A\,b^4\,e^4+7\,B\,b^3\,c\,d^2\,e^2-19\,A\,b^3\,c\,d\,e^3+6\,A\,b^2\,c^2\,d^2\,e^2+2\,B\,b\,c^3\,d^4+2\,A\,b\,c^3\,d^3\,e-4\,A\,c^4\,d^4\right )}{2\,b^2\,d^2\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {c\,e^2\,x^3\,\left (-B\,b^3\,d\,e^2+3\,A\,b^3\,e^3+3\,B\,b^2\,c\,d^2\,e-7\,A\,b^2\,c\,d\,e^2+B\,b\,c^2\,d^3+3\,A\,b\,c^2\,d^2\,e-2\,A\,c^3\,d^3\right )}{b^2\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^2\,\left (c\,d^2+2\,b\,e\,d\right )+x^3\,\left (b\,e^2+2\,c\,d\,e\right )+c\,e^2\,x^4+b\,d^2\,x}+\frac {\ln \left (b+c\,x\right )\,\left (e\,\left (4\,B\,b^2\,c^3-5\,A\,b\,c^4\right )+d\,\left (2\,A\,c^5-B\,b\,c^4\right )\right )}{b^7\,e^4-4\,b^6\,c\,d\,e^3+6\,b^5\,c^2\,d^2\,e^2-4\,b^4\,c^3\,d^3\,e+b^3\,c^4\,d^4}-\frac {\ln \left (x\right )\,\left (d\,\left (2\,A\,c-B\,b\right )+3\,A\,b\,e\right )}{b^3\,d^4} \]

[In]

int((A + B*x)/((b*x + c*x^2)^2*(d + e*x)^3),x)

[Out]

(log(d + e*x)*(b^2*(3*A*e^5 - B*d*e^4) + c^2*(10*A*d^2*e^3 - 6*B*d^3*e^2) + b*c*(4*B*d^2*e^3 - 10*A*d*e^4)))/(
c^4*d^8 + b^4*d^4*e^4 - 4*b^3*c*d^5*e^3 + 6*b^2*c^2*d^6*e^2 - 4*b*c^3*d^7*e) - (A/(b*d) + (x^2*(6*A*b^4*e^5 -
8*A*c^4*d^4*e - 2*B*b^4*d*e^4 + 10*A*b*c^3*d^3*e^2 + 3*B*b^3*c*d^2*e^3 - 15*A*b^2*c^2*d^2*e^3 + 7*B*b^2*c^2*d^
3*e^2 - 5*A*b^3*c*d*e^4 + 4*B*b*c^3*d^4*e))/(2*b^2*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)) +
(x*(9*A*b^4*e^4 - 4*A*c^4*d^4 + 2*B*b*c^3*d^4 - 3*B*b^4*d*e^3 + 7*B*b^3*c*d^2*e^2 + 6*A*b^2*c^2*d^2*e^2 + 2*A*
b*c^3*d^3*e - 19*A*b^3*c*d*e^3))/(2*b^2*d^2*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)) + (c*e^2*x^3*
(3*A*b^3*e^3 - 2*A*c^3*d^3 + B*b*c^2*d^3 - B*b^3*d*e^2 + 3*A*b*c^2*d^2*e - 7*A*b^2*c*d*e^2 + 3*B*b^2*c*d^2*e))
/(b^2*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)))/(x^2*(c*d^2 + 2*b*d*e) + x^3*(b*e^2 + 2*c*d*e)
+ c*e^2*x^4 + b*d^2*x) + (log(b + c*x)*(e*(4*B*b^2*c^3 - 5*A*b*c^4) + d*(2*A*c^5 - B*b*c^4)))/(b^7*e^4 + b^3*
c^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2*d^2*e^2 - 4*b^6*c*d*e^3) - (log(x)*(d*(2*A*c - B*b) + 3*A*b*e))/(b^3*d^4
)