3.11.7 \(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=201 \[ \frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}+\frac {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac {A}{b^2 d^2 x}+\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (c d-b e)^3}+\frac {e^2 \log (d+e x) (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2} \]

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Rubi [A]  time = 0.33, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}+\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (c d-b e)^3}+\frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}-\frac {A}{b^2 d^2 x}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2}+\frac {e^2 \log (d+e x) (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

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*} \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {A}{b^2 d^2 x^2}+\frac {b B d-2 A c d-2 A b e}{b^3 d^3 x}-\frac {c^3 (b B-A c)}{b^2 (-c d+b e)^2 (b+c x)^2}+\frac {c^3 \left (2 A c^2 d+3 b^2 B e-b c (B d+4 A e)\right )}{b^3 (c d-b e)^3 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)^2}+\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)}\right ) \, dx\\ &=-\frac {A}{b^2 d^2 x}+\frac {c^2 (b B-A c)}{b^2 (c d-b e)^2 (b+c x)}+\frac {e^2 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {(b B d-2 A c d-2 A b e) \log (x)}{b^3 d^3}+\frac {c^2 \left (2 A c^2 d+3 b^2 B e-b c (B d+4 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e)) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 201, normalized size = 1.00 \begin {gather*} \frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}+\frac {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac {A}{b^2 d^2 x}-\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (b e-c d)^3}-\frac {e^2 \log (d+e x) (2 A e (b e-2 c d)+B d (3 c d-b e))}{d^3 (c d-b e)^3}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 134.90, size = 1034, normalized size = 5.14 \begin {gather*} -\frac {A b^{2} c^{3} d^{5} - 3 \, A b^{3} c^{2} d^{4} e + 3 \, A b^{4} c d^{3} e^{2} - A b^{5} d^{2} e^{3} - {\left (4 \, A b^{2} c^{3} d^{3} e^{2} + 2 \, A b^{4} c d e^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{4} e - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d^{2} e^{3}\right )} x^{2} - {\left (B b^{4} c d^{3} e^{2} + 2 \, A b^{5} d e^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} - {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{4} e - {\left (B b^{5} + 3 \, A b^{4} c\right )} d^{2} e^{3}\right )} x + {\left ({\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} e - {\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{3} e^{2}\right )} x^{3} + {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{5} - 2 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{4} e - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{3} e^{2}\right )} x^{2} + {\left ({\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{4} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (3 \, B b^{3} c^{2} d^{2} e^{3} + 2 \, A b^{4} c e^{5} - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d e^{4}\right )} x^{3} + {\left (3 \, B b^{3} c^{2} d^{3} e^{2} + 2 \, A b^{5} e^{5} + 2 \, {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2} e^{3} - {\left (B b^{5} + 2 \, A b^{4} c\right )} d e^{4}\right )} x^{2} + {\left (3 \, B b^{4} c d^{3} e^{2} + 2 \, A b^{5} d e^{4} - {\left (B b^{5} + 4 \, A b^{4} c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right ) - {\left ({\left (3 \, B b^{3} c^{2} d^{2} e^{3} + 2 \, A b^{4} c e^{5} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} e - {\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{3} e^{2} - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d e^{4}\right )} x^{3} + {\left (4 \, A b^{2} c^{3} d^{3} e^{2} + 2 \, A b^{5} e^{5} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{5} - 2 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{4} e + 2 \, {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2} e^{3} - {\left (B b^{5} + 2 \, A b^{4} c\right )} d e^{4}\right )} x^{2} + {\left (3 \, B b^{4} c d^{3} e^{2} + 2 \, A b^{5} d e^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{4} e - {\left (B b^{5} + 4 \, A b^{4} c\right )} d^{2} e^{3}\right )} x\right )} \log \relax (x)}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} + {\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} + {\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.28, size = 670, normalized size = 3.33 \begin {gather*} \frac {{\left (2 \, B b c^{3} d^{4} e^{2} - 4 \, A c^{4} d^{4} e^{2} - 6 \, B b^{2} c^{2} d^{3} e^{3} + 8 \, A b c^{3} d^{3} e^{3} + 3 \, B b^{3} c d^{2} e^{4} - B b^{4} d e^{5} - 4 \, A b^{3} c d e^{5} + 2 \, A b^{4} e^{6}\right )} e^{\left (-2\right )} \log \left (\frac {{\left | 2 \, c d e - \frac {2 \, c d^{2} e}{x e + d} - b e^{2} + \frac {2 \, b d e^{2}}{x e + d} - {\left | b \right |} e^{2} \right |}}{{\left | 2 \, c d e - \frac {2 \, c d^{2} e}{x e + d} - b e^{2} + \frac {2 \, b d e^{2}}{x e + d} + {\left | b \right |} e^{2} \right |}}\right )}{2 \, {\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left | b \right |}} + \frac {{\left (3 \, B c d^{2} e^{2} - B b d e^{3} - 4 \, A c d e^{3} + 2 \, A b e^{4}\right )} \log \left ({\left | c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}} + \frac {\frac {B d e^{6}}{x e + d} - \frac {A e^{7}}{x e + d}}{c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}} + \frac {\frac {B b c^{3} d^{3} e - 2 \, A c^{4} d^{3} e + 3 \, A b c^{3} d^{2} e^{2} - 3 \, A b^{2} c^{2} d e^{3} + A b^{3} c e^{4}}{c d^{2} - b d e} - \frac {{\left (B b c^{3} d^{4} e^{2} - 2 \, A c^{4} d^{4} e^{2} + 4 \, A b c^{3} d^{3} e^{3} - 6 \, A b^{2} c^{2} d^{2} e^{4} + 4 \, A b^{3} c d e^{5} - A b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )} {\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 357, normalized size = 1.78 \begin {gather*} \frac {2 A b \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}+\frac {4 A \,c^{3} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}-\frac {2 A \,c^{4} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {4 A c \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {B b \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {3 B \,c^{2} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b}+\frac {B \,c^{3} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}+\frac {3 B c \,e^{2} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d}-\frac {A \,c^{3}}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{2}}-\frac {A \,e^{3}}{\left (b e -c d \right )^{2} \left (e x +d \right ) d^{2}}+\frac {B \,c^{2}}{\left (b e -c d \right )^{2} \left (c x +b \right ) b}+\frac {B \,e^{2}}{\left (b e -c d \right )^{2} \left (e x +d \right ) d}-\frac {2 A e \ln \relax (x )}{b^{2} d^{3}}-\frac {2 A c \ln \relax (x )}{b^{3} d^{2}}+\frac {B \ln \relax (x )}{b^{2} d^{2}}-\frac {A}{b^{2} d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.71, size = 467, normalized size = 2.32 \begin {gather*} -\frac {{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d - {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} - \frac {{\left (3 \, B c d^{2} e^{2} + 2 \, A b e^{4} - {\left (B b + 4 \, A c\right )} d e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} + {\left (2 \, A b^{2} c e^{3} - {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} e - {\left (B b^{2} c + 2 \, A b c^{2}\right )} d e^{2}\right )} x^{2} - {\left (A b c^{2} d^{2} e - 2 \, A b^{3} e^{3} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{3} + {\left (B b^{3} + A b^{2} c\right )} d e^{2}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} + {\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} + {\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac {{\left (2 \, A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \relax (x)}{b^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.41, size = 410, normalized size = 2.04 \begin {gather*} \frac {\frac {x\,\left (B\,b^3\,d\,e^2-2\,A\,b^3\,e^3+A\,b^2\,c\,d\,e^2+B\,b\,c^2\,d^3+A\,b\,c^2\,d^2\,e-2\,A\,c^3\,d^3\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}-\frac {A}{b\,d}+\frac {x^2\,\left (B\,b^2\,c\,d\,e^2-2\,A\,b^2\,c\,e^3+B\,b\,c^2\,d^2\,e+2\,A\,b\,c^2\,d\,e^2-2\,A\,c^3\,d^2\,e\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+b\,d\,x}-\frac {\ln \left (b+c\,x\right )\,\left (e\,\left (3\,B\,b^2\,c^2-4\,A\,b\,c^3\right )+d\,\left (2\,A\,c^4-B\,b\,c^3\right )\right )}{b^6\,e^3-3\,b^5\,c\,d\,e^2+3\,b^4\,c^2\,d^2\,e-b^3\,c^3\,d^3}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (3\,B\,d^2\,e^2-4\,A\,d\,e^3\right )+b\,\left (2\,A\,e^4-B\,d\,e^3\right )\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {\ln \relax (x)\,\left (d\,\left (2\,A\,c-B\,b\right )+2\,A\,b\,e\right )}{b^3\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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