3.12.52 \(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^2} \, dx\) [1152]

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

[Out]

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

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Rubi [A]
time = 0.22, 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 = {785} \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 (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} \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 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*} \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 = 0.22, size = 201, normalized size = 1.00 \begin {gather*} -\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 (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} \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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Maple [A]
time = 0.69, size = 203, normalized size = 1.01

method	result	size
default	\(\frac {c^{2} \left (4 A b c e -2 A \,c^{2} d -3 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b e -c d \right )^{3}}-\frac {\left (A c -B b \right ) c^{2}}{b^{2} \left (b e -c d \right )^{2} \left (c x +b \right )}-\frac {A}{x \,b^{2} d^{2}}+\frac {\left (-2 A b e -2 A c d +B b d \right ) \ln \left (x \right )}{b^{3} d^{3}}+\frac {e^{2} \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right ) \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}-\frac {\left (A e -B d \right ) e^{2}}{\left (b e -c d \right )^{2} d^{2} \left (e x +d \right )}\)	\(203\)
norman	\(\frac {\frac {\left (2 A \,b^{4} e^{4}-A \,b^{3} c d \,e^{3}-A b \,c^{3} d^{3} e +2 A \,c^{4} d^{4}-B \,b^{4} d \,e^{3}-B b \,c^{3} d^{4}\right ) x^{2}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (2 A \,b^{3} e^{3}-A \,b^{2} c d \,e^{2}-A b \,c^{2} d^{2} e +2 A \,c^{3} d^{3}-B \,b^{3} d \,e^{2}-B b \,c^{2} d^{3}\right ) c e \,x^{3}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {A}{d b}}{x \left (e x +d \right ) \left (c x +b \right )}+\frac {c^{2} \left (4 A b c e -2 A \,c^{2} d -3 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b^{3}}+\frac {e^{2} \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {\left (2 A b e +2 A c d -B b d \right ) \ln \left (x \right )}{b^{3} d^{3}}\)	\(401\)
risch	\(\frac {-\frac {c e \left (2 A \,b^{2} e^{2}-2 A b c d e +2 A \,c^{2} d^{2}-B \,b^{2} d e -B b c \,d^{2}\right ) x^{2}}{b^{2} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (2 A \,b^{3} e^{3}-A \,b^{2} c d \,e^{2}-A b \,c^{2} d^{2} e +2 A \,c^{3} d^{3}-B \,b^{3} d \,e^{2}-B b \,c^{2} d^{3}\right ) x}{b^{2} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {A}{d b}}{x \left (e x +d \right ) \left (c x +b \right )}-\frac {2 \ln \left (-x \right ) A e}{b^{2} d^{3}}-\frac {2 \ln \left (-x \right ) A c}{b^{3} d^{2}}+\frac {\ln \left (-x \right ) B}{b^{2} d^{2}}+\frac {4 c^{3} \ln \left (c x +b \right ) A e}{\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b^{2}}-\frac {2 c^{4} \ln \left (c x +b \right ) A d}{\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b^{3}}-\frac {3 c^{2} \ln \left (c x +b \right ) B e}{\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b}+\frac {c^{3} \ln \left (c x +b \right ) B d}{\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b^{2}}+\frac {2 e^{4} \ln \left (-e x -d \right ) A b}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {4 e^{3} \ln \left (-e x -d \right ) A c}{d^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {e^{3} \ln \left (-e x -d \right ) B b}{d^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {3 e^{2} \ln \left (-e x -d \right ) B c}{d \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\)	\(684\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (212) = 424\).
time = 0.29, size = 468, normalized size = 2.33 \begin {gather*} \frac {{\left (3 \, B b^{2} c^{2} e - 4 \, A b c^{3} e - {\left (B b c^{3} - 2 \, A c^{4}\right )} d\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 e^{3} + 4 \, A c e^{3}\right )} d\right )} \log \left (x e + 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} e - 2 \, A c^{3} e\right )} d^{2} - {\left (B b^{2} c e^{2} + 2 \, A b c^{2} e^{2}\right )} d\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} e^{2} + A b^{2} c e^{2}\right )} d\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 \left (x\right )}{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]

(3*B*b^2*c^2*e - 4*A*b*c^3*e - (B*b*c^3 - 2*A*c^4)*d)*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*e^3 + 4*A*c*e^3)*d)*log(x*e + 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*e -
2*A*c^3*e)*d^2 - (B*b^2*c*e^2 + 2*A*b*c^2*e^2)*d)*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*e^2 + A*b^2*c*e^2)*d)*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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (212) = 424\).
time = 242.45, size = 1044, normalized size = 5.19 \begin {gather*} -\frac {A b^{2} c^{3} d^{5} - {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} x - 2 \, {\left (A b^{4} c d x^{2} + A b^{5} d x\right )} e^{4} - {\left (A b^{5} d^{2} - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d^{2} x^{2} - {\left (B b^{5} + 3 \, A b^{4} c\right )} d^{2} x\right )} e^{3} - {\left (4 \, A b^{2} c^{3} d^{3} x^{2} + B b^{4} c d^{3} x - 3 \, A b^{4} c d^{3}\right )} e^{2} - {\left (3 \, A b^{3} c^{2} d^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{4} x^{2} - {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{4} x\right )} e + {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{5} x^{2} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} x - {\left ({\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{3} x^{3} + {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{3} x^{2}\right )} e^{2} + {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} x^{3} - 2 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{4} x^{2} - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{4} x\right )} e\right )} \log \left (c x + b\right ) + {\left (2 \, {\left (A b^{4} c x^{3} + A b^{5} x^{2}\right )} e^{5} + {\left (2 \, A b^{5} d x - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d x^{3} - {\left (B b^{5} + 2 \, A b^{4} c\right )} d x^{2}\right )} e^{4} + {\left (3 \, B b^{3} c^{2} d^{2} x^{3} + 2 \, {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2} x^{2} - {\left (B b^{5} + 4 \, A b^{4} c\right )} d^{2} x\right )} e^{3} + 3 \, {\left (B b^{3} c^{2} d^{3} x^{2} + B b^{4} c d^{3} x\right )} e^{2}\right )} \log \left (x e + d\right ) - {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{5} x^{2} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} x + 2 \, {\left (A b^{4} c x^{3} + A b^{5} x^{2}\right )} e^{5} + {\left (2 \, A b^{5} d x - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d x^{3} - {\left (B b^{5} + 2 \, A b^{4} c\right )} d x^{2}\right )} e^{4} + {\left (3 \, B b^{3} c^{2} d^{2} x^{3} + 2 \, {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2} x^{2} - {\left (B b^{5} + 4 \, A b^{4} c\right )} d^{2} x\right )} e^{3} + {\left (4 \, A b^{2} c^{3} d^{3} x^{2} + 3 \, B b^{4} c d^{3} x - {\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{3} x^{3}\right )} e^{2} + {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} x^{3} - 2 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{4} x^{2} - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{4} x\right )} e\right )} \log \left (x\right )}{b^{3} c^{4} d^{7} x^{2} + b^{4} c^{3} d^{7} x - {\left (b^{6} c d^{3} x^{3} + b^{7} d^{3} x^{2}\right )} e^{4} + {\left (3 \, b^{5} c^{2} d^{4} x^{3} + 2 \, b^{6} c d^{4} x^{2} - b^{7} d^{4} x\right )} e^{3} - 3 \, {\left (b^{4} c^{3} d^{5} x^{3} - b^{6} c d^{5} x\right )} e^{2} + {\left (b^{3} c^{4} d^{6} x^{3} - 2 \, b^{4} c^{3} d^{6} x^{2} - 3 \, b^{5} c^{2} d^{6} x\right )} e} \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 - (B*b^2*c^3 - 2*A*b*c^4)*d^5*x - 2*(A*b^4*c*d*x^2 + A*b^5*d*x)*e^4 - (A*b^5*d^2 - (B*b^4*c +
4*A*b^3*c^2)*d^2*x^2 - (B*b^5 + 3*A*b^4*c)*d^2*x)*e^3 - (4*A*b^2*c^3*d^3*x^2 + B*b^4*c*d^3*x - 3*A*b^4*c*d^3)*
e^2 - (3*A*b^3*c^2*d^4 + (B*b^2*c^3 - 2*A*b*c^4)*d^4*x^2 - (B*b^3*c^2 - 3*A*b^2*c^3)*d^4*x)*e + ((B*b*c^4 - 2*
A*c^5)*d^5*x^2 + (B*b^2*c^3 - 2*A*b*c^4)*d^5*x - ((3*B*b^2*c^3 - 4*A*b*c^4)*d^3*x^3 + (3*B*b^3*c^2 - 4*A*b^2*c
^3)*d^3*x^2)*e^2 + ((B*b*c^4 - 2*A*c^5)*d^4*x^3 - 2*(B*b^2*c^3 - A*b*c^4)*d^4*x^2 - (3*B*b^3*c^2 - 4*A*b^2*c^3
)*d^4*x)*e)*log(c*x + b) + (2*(A*b^4*c*x^3 + A*b^5*x^2)*e^5 + (2*A*b^5*d*x - (B*b^4*c + 4*A*b^3*c^2)*d*x^3 - (
B*b^5 + 2*A*b^4*c)*d*x^2)*e^4 + (3*B*b^3*c^2*d^2*x^3 + 2*(B*b^4*c - 2*A*b^3*c^2)*d^2*x^2 - (B*b^5 + 4*A*b^4*c)
*d^2*x)*e^3 + 3*(B*b^3*c^2*d^3*x^2 + B*b^4*c*d^3*x)*e^2)*log(x*e + d) - ((B*b*c^4 - 2*A*c^5)*d^5*x^2 + (B*b^2*
c^3 - 2*A*b*c^4)*d^5*x + 2*(A*b^4*c*x^3 + A*b^5*x^2)*e^5 + (2*A*b^5*d*x - (B*b^4*c + 4*A*b^3*c^2)*d*x^3 - (B*b
^5 + 2*A*b^4*c)*d*x^2)*e^4 + (3*B*b^3*c^2*d^2*x^3 + 2*(B*b^4*c - 2*A*b^3*c^2)*d^2*x^2 - (B*b^5 + 4*A*b^4*c)*d^
2*x)*e^3 + (4*A*b^2*c^3*d^3*x^2 + 3*B*b^4*c*d^3*x - (3*B*b^2*c^3 - 4*A*b*c^4)*d^3*x^3)*e^2 + ((B*b*c^4 - 2*A*c
^5)*d^4*x^3 - 2*(B*b^2*c^3 - A*b*c^4)*d^4*x^2 - (3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*x)*e)*log(x))/(b^3*c^4*d^7*x^2
+ b^4*c^3*d^7*x - (b^6*c*d^3*x^3 + b^7*d^3*x^2)*e^4 + (3*b^5*c^2*d^4*x^3 + 2*b^6*c*d^4*x^2 - b^7*d^4*x)*e^3 -
3*(b^4*c^3*d^5*x^3 - b^6*c*d^5*x)*e^2 + (b^3*c^4*d^6*x^3 - 2*b^4*c^3*d^6*x^2 - 3*b^5*c^2*d^6*x)*e)

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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (212) = 424\).
time = 0.99, 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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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 \left (x\right )\,\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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