\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(f+g x)^3} \, dx\) [237]

Optimal result

Integrand size = 27, antiderivative size = 183 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=-\frac {B (b c-a d)}{2 (b f-a g) (d f-c g) (f+g x)}+\frac {b^2 B \log (a+b x)}{2 g (b f-a g)^2}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 g (f+g x)^2}-\frac {B d^2 \log (c+d x)}{2 g (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \log (f+g x)}{2 (b f-a g)^2 (d f-c g)^2} \]

[Out]

-1/2*B*(-a*d+b*c)/(-a*g+b*f)/(-c*g+d*f)/(g*x+f)+1/2*b^2*B*ln(b*x+a)/g/(-a*g+b*f)^2+1/2*(-A-B*ln(e*(b*x+a)/(d*x
+c)))/g/(g*x+f)^2-1/2*B*d^2*ln(d*x+c)/g/(-c*g+d*f)^2+1/2*B*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*ln(g*x+f)/(-a*g+b
*f)^2/(-c*g+d*f)^2

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2548, 84} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 g (f+g x)^2}+\frac {b^2 B \log (a+b x)}{2 g (b f-a g)^2}-\frac {B (b c-a d)}{2 (f+g x) (b f-a g) (d f-c g)}+\frac {B (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f)}{2 (b f-a g)^2 (d f-c g)^2}-\frac {B d^2 \log (c+d x)}{2 g (d f-c g)^2} \]

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^3,x]

[Out]

-1/2*(B*(b*c - a*d))/((b*f - a*g)*(d*f - c*g)*(f + g*x)) + (b^2*B*Log[a + b*x])/(2*g*(b*f - a*g)^2) - (A + B*L
og[(e*(a + b*x))/(c + d*x)])/(2*g*(f + g*x)^2) - (B*d^2*Log[c + d*x])/(2*g*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*
b*d*f - b*c*g - a*d*g)*Log[f + g*x])/(2*(b*f - a*g)^2*(d*f - c*g)^2)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = -\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 g (f+g x)^2}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x) (f+g x)^2} \, dx}{2 g} \\ & = -\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 g (f+g x)^2}+\frac {(B (b c-a d)) \int \left (\frac {b^3}{(b c-a d) (b f-a g)^2 (a+b x)}-\frac {d^3}{(b c-a d) (-d f+c g)^2 (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^2}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}\right ) \, dx}{2 g} \\ & = -\frac {B (b c-a d)}{2 (b f-a g) (d f-c g) (f+g x)}+\frac {b^2 B \log (a+b x)}{2 g (b f-a g)^2}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 g (f+g x)^2}-\frac {B d^2 \log (c+d x)}{2 g (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \log (f+g x)}{2 (b f-a g)^2 (d f-c g)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.92 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^2}+B (b c-a d) \left (\frac {b^2 \log (a+b x)}{(b c-a d) (b f-a g)^2}+\frac {\frac {g (-d f+c g)}{(b f-a g) (f+g x)}+\frac {d^2 \log (c+d x)}{-b c+a d}-\frac {g (-2 b d f+b c g+a d g) \log (f+g x)}{(b f-a g)^2}}{(d f-c g)^2}\right )}{2 g} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(693\) vs. \(2(176)=352\).

Time = 1.89 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.79

method	result	size
parts	\(-\frac {A}{2 \left (g x +f \right )^{2} g}-\frac {B \left (a d -c b \right ) e \left (-\frac {g \,d^{2} \left (a d -c b \right ) e \left (-\frac {\frac {\ln \left (c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f \right )}{c g -d f}+\frac {e \left (a g -b f \right )}{\left (c g -d f \right ) \left (c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f \right )}}{2 \left (a g -b f \right )^{2} e^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-2 a e g +2 b e f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f \right )^{2} \left (a g -b f \right )^{2} e^{2}}\right )}{c g -d f}-\frac {d^{3} \left (\frac {\ln \left (\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f \right )}{e \left (a g -b f \right ) \left (c g -d f \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \left (a g -b f \right ) \left (c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f \right )}\right )}{c g -d f}\right )}{d^{2}}\)	\(694\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-A \,d^{2} \left (-\frac {d}{\left (c g -d f \right ) \left (-c g +d f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}+\frac {e g \left (a d -c b \right )}{2 \left (c g -d f \right ) \left (-c g +d f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )^{2}}\right )-B \,d^{2} \left (\frac {d \left (-\frac {\ln \left (\left (-c g +d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+a e g -b e f \right )}{e \left (a g -b f \right ) \left (-c g +d f \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \left (a g -b f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}\right )}{c g -d f}+\frac {e g \left (a d -c b \right ) \left (-\frac {\frac {e \left (a g -b f \right )}{\left (-c g +d f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}-\frac {\ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}{-c g +d f}}{2 \left (a g -b f \right )^{2} e^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 a e g -2 b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )^{2} \left (a g -b f \right )^{2} e^{2}}\right )}{c g -d f}\right )\right )}{d^{2}}\)	\(888\)
default	\(-\frac {e \left (a d -c b \right ) \left (-A \,d^{2} \left (-\frac {d}{\left (c g -d f \right ) \left (-c g +d f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}+\frac {e g \left (a d -c b \right )}{2 \left (c g -d f \right ) \left (-c g +d f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )^{2}}\right )-B \,d^{2} \left (\frac {d \left (-\frac {\ln \left (\left (-c g +d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+a e g -b e f \right )}{e \left (a g -b f \right ) \left (-c g +d f \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \left (a g -b f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}\right )}{c g -d f}+\frac {e g \left (a d -c b \right ) \left (-\frac {\frac {e \left (a g -b f \right )}{\left (-c g +d f \right ) \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}-\frac {\ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}{-c g +d f}}{2 \left (a g -b f \right )^{2} e^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 a e g -2 b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )^{2} \left (a g -b f \right )^{2} e^{2}}\right )}{c g -d f}\right )\right )}{d^{2}}\)	\(888\)
risch	\(\text {Expression too large to display}\)	\(1023\)
parallelrisch	\(\text {Expression too large to display}\)	\(1339\)

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(g*x+f)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (173) = 346\).

Time = 45.30 (sec) , antiderivative size = 1017, normalized size of antiderivative = 5.56 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=-\frac {A b^{2} d^{2} f^{4} + A a^{2} c^{2} g^{4} - {\left ({\left (2 \, A - B\right )} b^{2} c d + {\left (2 \, A + B\right )} a b d^{2}\right )} f^{3} g + {\left ({\left (A - B\right )} b^{2} c^{2} + 4 \, A a b c d + {\left (A + B\right )} a^{2} d^{2}\right )} f^{2} g^{2} - {\left ({\left (2 \, A - B\right )} a b c^{2} + {\left (2 \, A + B\right )} a^{2} c d\right )} f g^{3} + {\left ({\left (B b^{2} c d - B a b d^{2}\right )} f^{2} g^{2} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} f g^{3} + {\left (B a b c^{2} - B a^{2} c d\right )} g^{4}\right )} x - {\left (B b^{2} d^{2} f^{4} - 2 \, B b^{2} c d f^{3} g + B b^{2} c^{2} f^{2} g^{2} + {\left (B b^{2} d^{2} f^{2} g^{2} - 2 \, B b^{2} c d f g^{3} + B b^{2} c^{2} g^{4}\right )} x^{2} + 2 \, {\left (B b^{2} d^{2} f^{3} g - 2 \, B b^{2} c d f^{2} g^{2} + B b^{2} c^{2} f g^{3}\right )} x\right )} \log \left (b x + a\right ) + {\left (B b^{2} d^{2} f^{4} - 2 \, B a b d^{2} f^{3} g + B a^{2} d^{2} f^{2} g^{2} + {\left (B b^{2} d^{2} f^{2} g^{2} - 2 \, B a b d^{2} f g^{3} + B a^{2} d^{2} g^{4}\right )} x^{2} + 2 \, {\left (B b^{2} d^{2} f^{3} g - 2 \, B a b d^{2} f^{2} g^{2} + B a^{2} d^{2} f g^{3}\right )} x\right )} \log \left (d x + c\right ) - {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} f^{3} g - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} f^{2} g^{2} + {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} f g^{3} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} g^{4}\right )} x^{2} + 2 \, {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} f^{2} g^{2} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} f g^{3}\right )} x\right )} \log \left (g x + f\right ) + {\left (B b^{2} d^{2} f^{4} + B a^{2} c^{2} g^{4} - 2 \, {\left (B b^{2} c d + B a b d^{2}\right )} f^{3} g + {\left (B b^{2} c^{2} + 4 \, B a b c d + B a^{2} d^{2}\right )} f^{2} g^{2} - 2 \, {\left (B a b c^{2} + B a^{2} c d\right )} f g^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, {\left (b^{2} d^{2} f^{6} g + a^{2} c^{2} f^{2} g^{5} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{5} g^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{4} g^{3} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3} g^{4} + {\left (b^{2} d^{2} f^{4} g^{3} + a^{2} c^{2} g^{7} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g^{4} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{5} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{6}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} f^{5} g^{2} + a^{2} c^{2} f g^{6} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{4} g^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3} g^{4} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} g^{5}\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(g*x+f)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (173) = 346\).

Time = 0.21 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.92 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=\frac {1}{2} \, {\left (\frac {b^{2} \log \left (b x + a\right )}{b^{2} f^{2} g - 2 \, a b f g^{2} + a^{2} g^{3}} - \frac {d^{2} \log \left (d x + c\right )}{d^{2} f^{2} g - 2 \, c d f g^{2} + c^{2} g^{3}} + \frac {{\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} f - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} g\right )} \log \left (g x + f\right )}{b^{2} d^{2} f^{4} + a^{2} c^{2} g^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{3}} - \frac {b c - a d}{b d f^{3} + a c f g^{2} - {\left (b c + a d\right )} f^{2} g + {\left (b d f^{2} g + a c g^{3} - {\left (b c + a d\right )} f g^{2}\right )} x} - \frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g}\right )} B - \frac {A}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \]

[In]

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

[Out]

1/2*(b^2*log(b*x + a)/(b^2*f^2*g - 2*a*b*f*g^2 + a^2*g^3) - d^2*log(d*x + c)/(d^2*f^2*g - 2*c*d*f*g^2 + c^2*g^
3) + (2*(b^2*c*d - a*b*d^2)*f - (b^2*c^2 - a^2*d^2)*g)*log(g*x + f)/(b^2*d^2*f^4 + a^2*c^2*g^4 - 2*(b^2*c*d +
a*b*d^2)*f^3*g + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2*g^2 - 2*(a*b*c^2 + a^2*c*d)*f*g^3) - (b*c - a*d)/(b*d*f^3
+ a*c*f*g^2 - (b*c + a*d)*f^2*g + (b*d*f^2*g + a*c*g^3 - (b*c + a*d)*f*g^2)*x) - log(b*e*x/(d*x + c) + a*e/(d
*x + c))/(g^3*x^2 + 2*f*g^2*x + f^2*g))*B - 1/2*A/(g^3*x^2 + 2*f*g^2*x + f^2*g)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2969 vs. \(2 (173) = 346\).

Time = 0.52 (sec) , antiderivative size = 2969, normalized size of antiderivative = 16.22 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((2*B*b^3*c^2*d*e*f - 4*B*a*b^2*c*d^2*e*f + 2*B*a^2*b*d^3*e*f - B*b^3*c^3*e*g + B*a*b^2*c^2*d*e*g + B*a^2*
b*c*d^2*e*g - B*a^3*d^3*e*g)*log(-b*e*f + a*e*g + (b*e*x + a*e)*d*f/(d*x + c) - (b*e*x + a*e)*c*g/(d*x + c))/(
b^2*d^2*f^4 - 2*b^2*c*d*f^3*g - 2*a*b*d^2*f^3*g + b^2*c^2*f^2*g^2 + 4*a*b*c*d*f^2*g^2 + a^2*d^2*f^2*g^2 - 2*a*
b*c^2*f*g^3 - 2*a^2*c*d*f*g^3 + a^2*c^2*g^4) + (2*B*b^3*c^2*d*e^3*f - 4*B*a*b^2*c*d^2*e^3*f + 2*B*a^2*b*d^3*e^
3*f - B*b^3*c^3*e^3*g + B*a*b^2*c^2*d*e^3*g + B*a^2*b*c*d^2*e^3*g - B*a^3*d^3*e^3*g - 2*(b*e*x + a*e)*B*b^2*c^
2*d^2*e^2*f/(d*x + c) + 4*(b*e*x + a*e)*B*a*b*c*d^3*e^2*f/(d*x + c) - 2*(b*e*x + a*e)*B*a^2*d^4*e^2*f/(d*x + c
) + 2*(b*e*x + a*e)*B*b^2*c^3*d*e^2*g/(d*x + c) - 4*(b*e*x + a*e)*B*a*b*c^2*d^2*e^2*g/(d*x + c) + 2*(b*e*x + a
*e)*B*a^2*c*d^3*e^2*g/(d*x + c))*log((b*e*x + a*e)/(d*x + c))/(b^2*d^2*e^2*f^4 - 2*b^2*c*d*e^2*f^3*g - 2*a*b*d
^2*e^2*f^3*g + b^2*c^2*e^2*f^2*g^2 + 4*a*b*c*d*e^2*f^2*g^2 + a^2*d^2*e^2*f^2*g^2 - 2*a*b*c^2*e^2*f*g^3 - 2*a^2
*c*d*e^2*f*g^3 + a^2*c^2*e^2*g^4 - 2*(b*e*x + a*e)*b*d^3*e*f^4/(d*x + c) + 6*(b*e*x + a*e)*b*c*d^2*e*f^3*g/(d*
x + c) + 2*(b*e*x + a*e)*a*d^3*e*f^3*g/(d*x + c) - 6*(b*e*x + a*e)*b*c^2*d*e*f^2*g^2/(d*x + c) - 6*(b*e*x + a*
e)*a*c*d^2*e*f^2*g^2/(d*x + c) + 2*(b*e*x + a*e)*b*c^3*e*f*g^3/(d*x + c) + 6*(b*e*x + a*e)*a*c^2*d*e*f*g^3/(d*
x + c) - 2*(b*e*x + a*e)*a*c^3*e*g^4/(d*x + c) + (b*e*x + a*e)^2*d^4*f^4/(d*x + c)^2 - 4*(b*e*x + a*e)^2*c*d^3
*f^3*g/(d*x + c)^2 + 6*(b*e*x + a*e)^2*c^2*d^2*f^2*g^2/(d*x + c)^2 - 4*(b*e*x + a*e)^2*c^3*d*f*g^3/(d*x + c)^2
+ (b*e*x + a*e)^2*c^4*g^4/(d*x + c)^2) - (2*B*b^3*c^2*d*e*f - 4*B*a*b^2*c*d^2*e*f + 2*B*a^2*b*d^3*e*f - B*b^3
*c^3*e*g + B*a*b^2*c^2*d*e*g + B*a^2*b*c*d^2*e*g - B*a^3*d^3*e*g)*log((b*e*x + a*e)/(d*x + c))/(b^2*d^2*f^4 -
2*b^2*c*d*f^3*g - 2*a*b*d^2*f^3*g + b^2*c^2*f^2*g^2 + 4*a*b*c*d*f^2*g^2 + a^2*d^2*f^2*g^2 - 2*a*b*c^2*f*g^3 -
2*a^2*c*d*f*g^3 + a^2*c^2*g^4) + (2*A*b^4*c^2*d*e^3*f^2 - 4*A*a*b^3*c*d^2*e^3*f^2 + 2*A*a^2*b^2*d^3*e^3*f^2 -
A*b^4*c^3*e^3*f*g + B*b^4*c^3*e^3*f*g - A*a*b^3*c^2*d*e^3*f*g - 3*B*a*b^3*c^2*d*e^3*f*g + 5*A*a^2*b^2*c*d^2*e^
3*f*g + 3*B*a^2*b^2*c*d^2*e^3*f*g - 3*A*a^3*b*d^3*e^3*f*g - B*a^3*b*d^3*e^3*f*g + A*a*b^3*c^3*e^3*g^2 - B*a*b^
3*c^3*e^3*g^2 - A*a^2*b^2*c^2*d*e^3*g^2 + 3*B*a^2*b^2*c^2*d*e^3*g^2 - A*a^3*b*c*d^2*e^3*g^2 - 3*B*a^3*b*c*d^2*
e^3*g^2 + A*a^4*d^3*e^3*g^2 + B*a^4*d^3*e^3*g^2 - 2*(b*e*x + a*e)*A*b^3*c^2*d^2*e^2*f^2/(d*x + c) + 4*(b*e*x +
a*e)*A*a*b^2*c*d^3*e^2*f^2/(d*x + c) - 2*(b*e*x + a*e)*A*a^2*b*d^4*e^2*f^2/(d*x + c) + 2*(b*e*x + a*e)*A*b^3*
c^3*d*e^2*f*g/(d*x + c) - (b*e*x + a*e)*B*b^3*c^3*d*e^2*f*g/(d*x + c) - 2*(b*e*x + a*e)*A*a*b^2*c^2*d^2*e^2*f*
g/(d*x + c) + 3*(b*e*x + a*e)*B*a*b^2*c^2*d^2*e^2*f*g/(d*x + c) - 2*(b*e*x + a*e)*A*a^2*b*c*d^3*e^2*f*g/(d*x +
c) - 3*(b*e*x + a*e)*B*a^2*b*c*d^3*e^2*f*g/(d*x + c) + 2*(b*e*x + a*e)*A*a^3*d^4*e^2*f*g/(d*x + c) + (b*e*x +
a*e)*B*a^3*d^4*e^2*f*g/(d*x + c) + (b*e*x + a*e)*B*b^3*c^4*e^2*g^2/(d*x + c) - 2*(b*e*x + a*e)*A*a*b^2*c^3*d*
e^2*g^2/(d*x + c) - 3*(b*e*x + a*e)*B*a*b^2*c^3*d*e^2*g^2/(d*x + c) + 4*(b*e*x + a*e)*A*a^2*b*c^2*d^2*e^2*g^2/
(d*x + c) + 3*(b*e*x + a*e)*B*a^2*b*c^2*d^2*e^2*g^2/(d*x + c) - 2*(b*e*x + a*e)*A*a^3*c*d^3*e^2*g^2/(d*x + c)
- (b*e*x + a*e)*B*a^3*c*d^3*e^2*g^2/(d*x + c))/(b^3*d^2*e^2*f^5 - 2*b^3*c*d*e^2*f^4*g - 3*a*b^2*d^2*e^2*f^4*g
+ b^3*c^2*e^2*f^3*g^2 + 6*a*b^2*c*d*e^2*f^3*g^2 + 3*a^2*b*d^2*e^2*f^3*g^2 - 3*a*b^2*c^2*e^2*f^2*g^3 - 6*a^2*b*
c*d*e^2*f^2*g^3 - a^3*d^2*e^2*f^2*g^3 + 3*a^2*b*c^2*e^2*f*g^4 + 2*a^3*c*d*e^2*f*g^4 - a^3*c^2*e^2*g^5 - 2*(b*e
*x + a*e)*b^2*d^3*e*f^5/(d*x + c) + 6*(b*e*x + a*e)*b^2*c*d^2*e*f^4*g/(d*x + c) + 4*(b*e*x + a*e)*a*b*d^3*e*f^
4*g/(d*x + c) - 6*(b*e*x + a*e)*b^2*c^2*d*e*f^3*g^2/(d*x + c) - 12*(b*e*x + a*e)*a*b*c*d^2*e*f^3*g^2/(d*x + c)
- 2*(b*e*x + a*e)*a^2*d^3*e*f^3*g^2/(d*x + c) + 2*(b*e*x + a*e)*b^2*c^3*e*f^2*g^3/(d*x + c) + 12*(b*e*x + a*e
)*a*b*c^2*d*e*f^2*g^3/(d*x + c) + 6*(b*e*x + a*e)*a^2*c*d^2*e*f^2*g^3/(d*x + c) - 4*(b*e*x + a*e)*a*b*c^3*e*f*
g^4/(d*x + c) - 6*(b*e*x + a*e)*a^2*c^2*d*e*f*g^4/(d*x + c) + 2*(b*e*x + a*e)*a^2*c^3*e*g^5/(d*x + c) + (b*e*x
+ a*e)^2*b*d^4*f^5/(d*x + c)^2 - 4*(b*e*x + a*e)^2*b*c*d^3*f^4*g/(d*x + c)^2 - (b*e*x + a*e)^2*a*d^4*f^4*g/(d
*x + c)^2 + 6*(b*e*x + a*e)^2*b*c^2*d^2*f^3*g^2/(d*x + c)^2 + 4*(b*e*x + a*e)^2*a*c*d^3*f^3*g^2/(d*x + c)^2 -
4*(b*e*x + a*e)^2*b*c^3*d*f^2*g^3/(d*x + c)^2 - 6*(b*e*x + a*e)^2*a*c^2*d^2*f^2*g^3/(d*x + c)^2 + (b*e*x + a*e
)^2*b*c^4*f*g^4/(d*x + c)^2 + 4*(b*e*x + a*e)^2*a*c^3*d*f*g^4/(d*x + c)^2 - (b*e*x + a*e)^2*a*c^4*g^5/(d*x + c
)^2))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))

Mupad [B] (verification not implemented)

Time = 3.69 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.28 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3} \, dx=\frac {\ln \left (f+g\,x\right )\,\left (g\,\left (B\,a^2\,d^2-B\,b^2\,c^2\right )-2\,B\,a\,b\,d^2\,f+2\,B\,b^2\,c\,d\,f\right )}{2\,a^2\,c^2\,g^4-4\,a^2\,c\,d\,f\,g^3+2\,a^2\,d^2\,f^2\,g^2-4\,a\,b\,c^2\,f\,g^3+8\,a\,b\,c\,d\,f^2\,g^2-4\,a\,b\,d^2\,f^3\,g+2\,b^2\,c^2\,f^2\,g^2-4\,b^2\,c\,d\,f^3\,g+2\,b^2\,d^2\,f^4}-\frac {\frac {A\,a\,c\,g^2+A\,b\,d\,f^2-A\,a\,d\,f\,g-A\,b\,c\,f\,g-B\,a\,d\,f\,g+B\,b\,c\,f\,g}{a\,c\,g^2+b\,d\,f^2-a\,d\,f\,g-b\,c\,f\,g}-\frac {x\,\left (B\,a\,d\,g^2-B\,b\,c\,g^2\right )}{a\,c\,g^2+b\,d\,f^2-a\,d\,f\,g-b\,c\,f\,g}}{2\,f^2\,g+4\,f\,g^2\,x+2\,g^3\,x^2}+\frac {B\,b^2\,\ln \left (a+b\,x\right )}{2\,a^2\,g^3-4\,a\,b\,f\,g^2+2\,b^2\,f^2\,g}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g\,\left (f^2+2\,f\,g\,x+g^2\,x^2\right )}-\frac {B\,d^2\,\ln \left (c+d\,x\right )}{2\,c^2\,g^3-4\,c\,d\,f\,g^2+2\,d^2\,f^2\,g} \]

[In]

int((A + B*log((e*(a + b*x))/(c + d*x)))/(f + g*x)^3,x)

[Out]

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