\(\int (g+h x)^4 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\) [293]

Optimal result

Integrand size = 31, antiderivative size = 365 \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {B (b c-a d) h \left (a^3 d^3 h^3-a^2 b d^2 h^2 (5 d g-c h)+a b^2 d h \left (10 d^2 g^2-5 c d g h+c^2 h^2\right )-b^3 \left (10 d^3 g^3-10 c d^2 g^2 h+5 c^2 d g h^2-c^3 h^3\right )\right ) n x}{5 b^4 d^4}-\frac {B (b c-a d) h^2 \left (a^2 d^2 h^2-a b d h (5 d g-c h)+b^2 \left (10 d^2 g^2-5 c d g h+c^2 h^2\right )\right ) n x^2}{10 b^3 d^3}-\frac {B (b c-a d) h^3 (5 b d g-b c h-a d h) n x^3}{15 b^2 d^2}-\frac {B (b c-a d) h^4 n x^4}{20 b d}-\frac {B (b g-a h)^5 n \log (a+b x)}{5 b^5 h}+\frac {B (d g-c h)^5 n \log (c+d x)}{5 d^5 h}+\frac {(g+h x)^5 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{5 h} \]

[Out]

1/5*B*(-a*d+b*c)*h*(a^3*d^3*h^3-a^2*b*d^2*h^2*(-c*h+5*d*g)+a*b^2*d*h*(c^2*h^2-5*c*d*g*h+10*d^2*g^2)-b^3*(-c^3*
h^3+5*c^2*d*g*h^2-10*c*d^2*g^2*h+10*d^3*g^3))*n*x/b^4/d^4-1/10*B*(-a*d+b*c)*h^2*(a^2*d^2*h^2-a*b*d*h*(-c*h+5*d
*g)+b^2*(c^2*h^2-5*c*d*g*h+10*d^2*g^2))*n*x^2/b^3/d^3-1/15*B*(-a*d+b*c)*h^3*(-a*d*h-b*c*h+5*b*d*g)*n*x^3/b^2/d
^2-1/20*B*(-a*d+b*c)*h^4*n*x^4/b/d-1/5*B*(-a*h+b*g)^5*n*ln(b*x+a)/b^5/h+1/5*B*(-c*h+d*g)^5*n*ln(d*x+c)/d^5/h+1
/5*(h*x+g)^5*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/h

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 84} \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=-\frac {B h^2 n x^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (5 d g-c h)+b^2 \left (c^2 h^2-5 c d g h+10 d^2 g^2\right )\right )}{10 b^3 d^3}+\frac {B h n x (b c-a d) \left (a^3 d^3 h^3-a^2 b d^2 h^2 (5 d g-c h)+a b^2 d h \left (c^2 h^2-5 c d g h+10 d^2 g^2\right )-\left (b^3 \left (-c^3 h^3+5 c^2 d g h^2-10 c d^2 g^2 h+10 d^3 g^3\right )\right )\right )}{5 b^4 d^4}+\frac {(g+h x)^5 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{5 h}-\frac {B n (b g-a h)^5 \log (a+b x)}{5 b^5 h}-\frac {B h^3 n x^3 (b c-a d) (-a d h-b c h+5 b d g)}{15 b^2 d^2}-\frac {B h^4 n x^4 (b c-a d)}{20 b d}+\frac {B n (d g-c h)^5 \log (c+d x)}{5 d^5 h} \]

[In]

Int[(g + h*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(B*(b*c - a*d)*h*(a^3*d^3*h^3 - a^2*b*d^2*h^2*(5*d*g - c*h) + a*b^2*d*h*(10*d^2*g^2 - 5*c*d*g*h + c^2*h^2) - b
^3*(10*d^3*g^3 - 10*c*d^2*g^2*h + 5*c^2*d*g*h^2 - c^3*h^3))*n*x)/(5*b^4*d^4) - (B*(b*c - a*d)*h^2*(a^2*d^2*h^2
- a*b*d*h*(5*d*g - c*h) + b^2*(10*d^2*g^2 - 5*c*d*g*h + c^2*h^2))*n*x^2)/(10*b^3*d^3) - (B*(b*c - a*d)*h^3*(5
*b*d*g - b*c*h - a*d*h)*n*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*h^4*n*x^4)/(20*b*d) - (B*(b*g - a*h)^5*n*Log[a +
b*x])/(5*b^5*h) + (B*(d*g - c*h)^5*n*Log[c + d*x])/(5*d^5*h) + ((g + h*x)^5*(A + B*Log[(e*(a + b*x)^n)/(c + d*
x)^n]))/(5*h)

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 {(g+h x)^5 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{5 h}-\frac {(B (b c-a d) n) \int \frac {(g+h x)^5}{(a+b x) (c+d x)} \, dx}{5 h} \\ & = \frac {(g+h x)^5 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{5 h}-\frac {(B (b c-a d) n) \int \left (\frac {h^2 \left (-a^3 d^3 h^3+a^2 b d^2 h^2 (5 d g-c h)-a b^2 d h \left (10 d^2 g^2-5 c d g h+c^2 h^2\right )+b^3 \left (10 d^3 g^3-10 c d^2 g^2 h+5 c^2 d g h^2-c^3 h^3\right )\right )}{b^4 d^4}+\frac {h^3 \left (a^2 d^2 h^2-a b d h (5 d g-c h)+b^2 \left (10 d^2 g^2-5 c d g h+c^2 h^2\right )\right ) x}{b^3 d^3}+\frac {h^4 (5 b d g-b c h-a d h) x^2}{b^2 d^2}+\frac {h^5 x^3}{b d}+\frac {(b g-a h)^5}{b^4 (b c-a d) (a+b x)}+\frac {(d g-c h)^5}{d^4 (-b c+a d) (c+d x)}\right ) \, dx}{5 h} \\ & = \frac {B (b c-a d) h \left (a^3 d^3 h^3-a^2 b d^2 h^2 (5 d g-c h)+a b^2 d h \left (10 d^2 g^2-5 c d g h+c^2 h^2\right )-b^3 \left (10 d^3 g^3-10 c d^2 g^2 h+5 c^2 d g h^2-c^3 h^3\right )\right ) n x}{5 b^4 d^4}-\frac {B (b c-a d) h^2 \left (a^2 d^2 h^2-a b d h (5 d g-c h)+b^2 \left (10 d^2 g^2-5 c d g h+c^2 h^2\right )\right ) n x^2}{10 b^3 d^3}-\frac {B (b c-a d) h^3 (5 b d g-b c h-a d h) n x^3}{15 b^2 d^2}-\frac {B (b c-a d) h^4 n x^4}{20 b d}-\frac {B (b g-a h)^5 n \log (a+b x)}{5 b^5 h}+\frac {B (d g-c h)^5 n \log (c+d x)}{5 d^5 h}+\frac {(g+h x)^5 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{5 h} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.27 \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {b d x \left (12 A b^4 d^4 \left (5 g^4+10 g^3 h x+10 g^2 h^2 x^2+5 g h^3 x^3+h^4 x^4\right )+B (b c-a d) h n \left (12 a^3 d^3 h^3-6 a^2 b d^2 h^2 (10 d g-2 c h+d h x)+2 a b^2 d h \left (6 c^2 h^2-3 c d h (10 g+h x)+d^2 \left (60 g^2+15 g h x+2 h^2 x^2\right )\right )-b^3 \left (-12 c^3 h^3+6 c^2 d h^2 (10 g+h x)-2 c d^2 h \left (60 g^2+15 g h x+2 h^2 x^2\right )+d^3 \left (120 g^3+60 g^2 h x+20 g h^2 x^2+3 h^3 x^3\right )\right )\right )\right )+12 a^2 B d^5 h \left (-10 b^3 g^3+10 a b^2 g^2 h-5 a^2 b g h^2+a^3 h^3\right ) n \log (a+b x)-12 b^4 B \left (-5 a d^5 g^4+b c \left (5 d^4 g^4-10 c d^3 g^3 h+10 c^2 d^2 g^2 h^2-5 c^3 d g h^3+c^4 h^4\right )\right ) n \log (c+d x)+12 b^4 B d^5 \left (5 a g^4+b x \left (5 g^4+10 g^3 h x+10 g^2 h^2 x^2+5 g h^3 x^3+h^4 x^4\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{60 b^5 d^5} \]

[In]

Integrate[(g + h*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(b*d*x*(12*A*b^4*d^4*(5*g^4 + 10*g^3*h*x + 10*g^2*h^2*x^2 + 5*g*h^3*x^3 + h^4*x^4) + B*(b*c - a*d)*h*n*(12*a^3
*d^3*h^3 - 6*a^2*b*d^2*h^2*(10*d*g - 2*c*h + d*h*x) + 2*a*b^2*d*h*(6*c^2*h^2 - 3*c*d*h*(10*g + h*x) + d^2*(60*
g^2 + 15*g*h*x + 2*h^2*x^2)) - b^3*(-12*c^3*h^3 + 6*c^2*d*h^2*(10*g + h*x) - 2*c*d^2*h*(60*g^2 + 15*g*h*x + 2*
h^2*x^2) + d^3*(120*g^3 + 60*g^2*h*x + 20*g*h^2*x^2 + 3*h^3*x^3)))) + 12*a^2*B*d^5*h*(-10*b^3*g^3 + 10*a*b^2*g
^2*h - 5*a^2*b*g*h^2 + a^3*h^3)*n*Log[a + b*x] - 12*b^4*B*(-5*a*d^5*g^4 + b*c*(5*d^4*g^4 - 10*c*d^3*g^3*h + 10
*c^2*d^2*g^2*h^2 - 5*c^3*d*g*h^3 + c^4*h^4))*n*Log[c + d*x] + 12*b^4*B*d^5*(5*a*g^4 + b*x*(5*g^4 + 10*g^3*h*x
+ 10*g^2*h^2*x^2 + 5*g*h^3*x^3 + h^4*x^4))*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(60*b^5*d^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1169\) vs. \(2(351)=702\).

Time = 85.50 (sec) , antiderivative size = 1170, normalized size of antiderivative = 3.21

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1170\)
risch	\(\text {Expression too large to display}\)	\(2612\)

[In]

int((h*x+g)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x,method=_RETURNVERBOSE)

[Out]

1/60*(12*B*ln(b*x+a)*a^6*c*d^5*h^4*n^2-12*B*ln(b*x+a)*a*b^5*c^6*h^4*n^2+12*A*x^5*a*b^5*c*d^5*h^4*n+12*B*ln(e*(
b*x+a)^n/((d*x+c)^n))*a*b^5*c^6*h^4*n+12*B*x^5*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^5*c*d^5*h^4*n+60*A*x^4*a*b^5*c*
d^5*g*h^3*n+20*B*x^3*a^2*b^4*c*d^5*g*h^3*n^2-20*B*x^3*a*b^5*c^2*d^4*g*h^3*n^2+120*A*x^3*a*b^5*c*d^5*g^2*h^2*n-
30*B*x^2*a^3*b^3*c*d^5*g*h^3*n^2+60*B*x^2*a^2*b^4*c*d^5*g^2*h^2*n^2+30*B*x^2*a*b^5*c^3*d^3*g*h^3*n^2-60*B*x^2*
a*b^5*c^2*d^4*g^2*h^2*n^2+120*A*x^2*a*b^5*c*d^5*g^3*h*n+60*B*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^5*c*d^5*g^4*n+6
0*B*x*a^4*b^2*c*d^5*g*h^3*n^2-120*B*x*a^3*b^3*c*d^5*g^2*h^2*n^2+120*B*x*a^2*b^4*c*d^5*g^3*h*n^2-60*B*x*a*b^5*c
^4*d^2*g*h^3*n^2+120*B*x*a*b^5*c^3*d^3*g^2*h^2*n^2-120*B*x*a*b^5*c^2*d^4*g^3*h*n^2-60*B*ln(e*(b*x+a)^n/((d*x+c
)^n))*a*b^5*c^5*d*g*h^3*n+120*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^5*c^4*d^2*g^2*h^2*n-120*B*ln(e*(b*x+a)^n/((d*x
+c)^n))*a*b^5*c^3*d^3*g^3*h*n-60*B*ln(b*x+a)*a^5*b*c*d^5*g*h^3*n^2+120*B*ln(b*x+a)*a^4*b^2*c*d^5*g^2*h^2*n^2-1
20*B*ln(b*x+a)*a^3*b^3*c*d^5*g^3*h*n^2+60*B*ln(b*x+a)*a*b^5*c^5*d*g*h^3*n^2-120*B*ln(b*x+a)*a*b^5*c^4*d^2*g^2*
h^2*n^2+120*B*ln(b*x+a)*a*b^5*c^3*d^3*g^3*h*n^2+3*B*x^4*a^2*b^4*c*d^5*h^4*n^2-3*B*x^4*a*b^5*c^2*d^4*h^4*n^2-4*
B*x^3*a^3*b^3*c*d^5*h^4*n^2+4*B*x^3*a*b^5*c^3*d^3*h^4*n^2+6*B*x^2*a^4*b^2*c*d^5*h^4*n^2-6*B*x^2*a*b^5*c^4*d^2*
h^4*n^2-12*B*x*a^5*b*c*d^5*h^4*n^2+12*B*x*a*b^5*c^5*d*h^4*n^2+60*A*x*a*b^5*c*d^5*g^4*n+60*B*ln(e*(b*x+a)^n/((d
*x+c)^n))*a*b^5*c^2*d^4*g^4*n+60*B*ln(b*x+a)*a^2*b^4*c*d^5*g^4*n^2-60*B*ln(b*x+a)*a*b^5*c^2*d^4*g^4*n^2+60*B*x
^4*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^5*c*d^5*g*h^3*n+120*B*x^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^5*c*d^5*g^2*h^2*n
+120*B*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^5*c*d^5*g^3*h*n)/a/c/d^5/n/b^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (351) = 702\).

Time = 0.34 (sec) , antiderivative size = 805, normalized size of antiderivative = 2.21 \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {12 \, A b^{5} d^{5} h^{4} x^{5} + 3 \, {\left (20 \, A b^{5} d^{5} g h^{3} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} h^{4} n\right )} x^{4} + 4 \, {\left (30 \, A b^{5} d^{5} g^{2} h^{2} - {\left (5 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g h^{3} - {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} h^{4}\right )} n\right )} x^{3} + 6 \, {\left (20 \, A b^{5} d^{5} g^{3} h - {\left (10 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{2} h^{2} - 5 \, {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} g h^{3} + {\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} h^{4}\right )} n\right )} x^{2} + 12 \, {\left (5 \, A b^{5} d^{5} g^{4} - {\left (10 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{3} h - 10 \, {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} g^{2} h^{2} + 5 \, {\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} g h^{3} - {\left (B b^{5} c^{4} d - B a^{4} b d^{5}\right )} h^{4}\right )} n\right )} x + 12 \, {\left (B b^{5} d^{5} h^{4} n x^{5} + 5 \, B b^{5} d^{5} g h^{3} n x^{4} + 10 \, B b^{5} d^{5} g^{2} h^{2} n x^{3} + 10 \, B b^{5} d^{5} g^{3} h n x^{2} + 5 \, B b^{5} d^{5} g^{4} n x + {\left (5 \, B a b^{4} d^{5} g^{4} - 10 \, B a^{2} b^{3} d^{5} g^{3} h + 10 \, B a^{3} b^{2} d^{5} g^{2} h^{2} - 5 \, B a^{4} b d^{5} g h^{3} + B a^{5} d^{5} h^{4}\right )} n\right )} \log \left (b x + a\right ) - 12 \, {\left (B b^{5} d^{5} h^{4} n x^{5} + 5 \, B b^{5} d^{5} g h^{3} n x^{4} + 10 \, B b^{5} d^{5} g^{2} h^{2} n x^{3} + 10 \, B b^{5} d^{5} g^{3} h n x^{2} + 5 \, B b^{5} d^{5} g^{4} n x + {\left (5 \, B b^{5} c d^{4} g^{4} - 10 \, B b^{5} c^{2} d^{3} g^{3} h + 10 \, B b^{5} c^{3} d^{2} g^{2} h^{2} - 5 \, B b^{5} c^{4} d g h^{3} + B b^{5} c^{5} h^{4}\right )} n\right )} \log \left (d x + c\right ) + 12 \, {\left (B b^{5} d^{5} h^{4} x^{5} + 5 \, B b^{5} d^{5} g h^{3} x^{4} + 10 \, B b^{5} d^{5} g^{2} h^{2} x^{3} + 10 \, B b^{5} d^{5} g^{3} h x^{2} + 5 \, B b^{5} d^{5} g^{4} x\right )} \log \left (e\right )}{60 \, b^{5} d^{5}} \]

[In]

integrate((h*x+g)^4*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

1/60*(12*A*b^5*d^5*h^4*x^5 + 3*(20*A*b^5*d^5*g*h^3 - (B*b^5*c*d^4 - B*a*b^4*d^5)*h^4*n)*x^4 + 4*(30*A*b^5*d^5*
g^2*h^2 - (5*(B*b^5*c*d^4 - B*a*b^4*d^5)*g*h^3 - (B*b^5*c^2*d^3 - B*a^2*b^3*d^5)*h^4)*n)*x^3 + 6*(20*A*b^5*d^5
*g^3*h - (10*(B*b^5*c*d^4 - B*a*b^4*d^5)*g^2*h^2 - 5*(B*b^5*c^2*d^3 - B*a^2*b^3*d^5)*g*h^3 + (B*b^5*c^3*d^2 -
B*a^3*b^2*d^5)*h^4)*n)*x^2 + 12*(5*A*b^5*d^5*g^4 - (10*(B*b^5*c*d^4 - B*a*b^4*d^5)*g^3*h - 10*(B*b^5*c^2*d^3 -
B*a^2*b^3*d^5)*g^2*h^2 + 5*(B*b^5*c^3*d^2 - B*a^3*b^2*d^5)*g*h^3 - (B*b^5*c^4*d - B*a^4*b*d^5)*h^4)*n)*x + 12
*(B*b^5*d^5*h^4*n*x^5 + 5*B*b^5*d^5*g*h^3*n*x^4 + 10*B*b^5*d^5*g^2*h^2*n*x^3 + 10*B*b^5*d^5*g^3*h*n*x^2 + 5*B*
b^5*d^5*g^4*n*x + (5*B*a*b^4*d^5*g^4 - 10*B*a^2*b^3*d^5*g^3*h + 10*B*a^3*b^2*d^5*g^2*h^2 - 5*B*a^4*b*d^5*g*h^3
+ B*a^5*d^5*h^4)*n)*log(b*x + a) - 12*(B*b^5*d^5*h^4*n*x^5 + 5*B*b^5*d^5*g*h^3*n*x^4 + 10*B*b^5*d^5*g^2*h^2*n
*x^3 + 10*B*b^5*d^5*g^3*h*n*x^2 + 5*B*b^5*d^5*g^4*n*x + (5*B*b^5*c*d^4*g^4 - 10*B*b^5*c^2*d^3*g^3*h + 10*B*b^5
*c^3*d^2*g^2*h^2 - 5*B*b^5*c^4*d*g*h^3 + B*b^5*c^5*h^4)*n)*log(d*x + c) + 12*(B*b^5*d^5*h^4*x^5 + 5*B*b^5*d^5*
g*h^3*x^4 + 10*B*b^5*d^5*g^2*h^2*x^3 + 10*B*b^5*d^5*g^3*h*x^2 + 5*B*b^5*d^5*g^4*x)*log(e))/(b^5*d^5)

Sympy [F(-2)]

Exception generated. \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((h*x+g)**4*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n))),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.84 \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {1}{5} \, B h^{4} x^{5} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{5} \, A h^{4} x^{5} + B g h^{3} x^{4} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A g h^{3} x^{4} + 2 \, B g^{2} h^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + 2 \, A g^{2} h^{2} x^{3} + 2 \, B g^{3} h x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + 2 \, A g^{3} h x^{2} + B g^{4} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A g^{4} x + \frac {{\left (\frac {a e n \log \left (b x + a\right )}{b} - \frac {c e n \log \left (d x + c\right )}{d}\right )} B g^{4}}{e} - \frac {2 \, {\left (\frac {a^{2} e n \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} e n \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c e n - a d e n\right )} x}{b d}\right )} B g^{3} h}{e} + \frac {{\left (\frac {2 \, a^{3} e n \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} e n \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d e n - a b d^{2} e n\right )} x^{2} - 2 \, {\left (b^{2} c^{2} e n - a^{2} d^{2} e n\right )} x}{b^{2} d^{2}}\right )} B g^{2} h^{2}}{e} - \frac {{\left (\frac {6 \, a^{4} e n \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} e n \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} e n - a b^{2} d^{3} e n\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d e n - a^{2} b d^{3} e n\right )} x^{2} + 6 \, {\left (b^{3} c^{3} e n - a^{3} d^{3} e n\right )} x}{b^{3} d^{3}}\right )} B g h^{3}}{6 \, e} + \frac {{\left (\frac {12 \, a^{5} e n \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} e n \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} e n - a b^{3} d^{4} e n\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} e n - a^{2} b^{2} d^{4} e n\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d e n - a^{3} b d^{4} e n\right )} x^{2} - 12 \, {\left (b^{4} c^{4} e n - a^{4} d^{4} e n\right )} x}{b^{4} d^{4}}\right )} B h^{4}}{60 \, e} \]

[In]

integrate((h*x+g)^4*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

1/5*B*h^4*x^5*log((b*x + a)^n*e/(d*x + c)^n) + 1/5*A*h^4*x^5 + B*g*h^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + A*
g*h^3*x^4 + 2*B*g^2*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 2*A*g^2*h^2*x^3 + 2*B*g^3*h*x^2*log((b*x + a)^n*e
/(d*x + c)^n) + 2*A*g^3*h*x^2 + B*g^4*x*log((b*x + a)^n*e/(d*x + c)^n) + A*g^4*x + (a*e*n*log(b*x + a)/b - c*e
*n*log(d*x + c)/d)*B*g^4/e - 2*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b
*d))*B*g^3*h/e + (2*a^3*e*n*log(b*x + a)/b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 -
2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*B*g^2*h^2/e - 1/6*(6*a^4*e*n*log(b*x + a)/b^4 - 6*c^4*e*n*log(d*x
+ c)/d^4 + (2*(b^3*c*d^2*e*n - a*b^2*d^3*e*n)*x^3 - 3*(b^3*c^2*d*e*n - a^2*b*d^3*e*n)*x^2 + 6*(b^3*c^3*e*n -
a^3*d^3*e*n)*x)/(b^3*d^3))*B*g*h^3/e + 1/60*(12*a^5*e*n*log(b*x + a)/b^5 - 12*c^5*e*n*log(d*x + c)/d^5 - (3*(b
^4*c*d^3*e*n - a*b^3*d^4*e*n)*x^4 - 4*(b^4*c^2*d^2*e*n - a^2*b^2*d^4*e*n)*x^3 + 6*(b^4*c^3*d*e*n - a^3*b*d^4*e
*n)*x^2 - 12*(b^4*c^4*e*n - a^4*d^4*e*n)*x)/(b^4*d^4))*B*h^4/e

Giac [F(-1)]

Timed out. \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\text {Timed out} \]

[In]

integrate((h*x+g)^4*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 1.80 (sec) , antiderivative size = 1434, normalized size of antiderivative = 3.93 \[ \int (g+h x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\text {Too large to display} \]

[In]

int((g + h*x)^4*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n)),x)

[Out]

x*((5*A*b*d*g^4 + 20*A*a*d*g^3*h + 20*A*b*c*g^3*h + 30*A*a*c*g^2*h^2 + 10*B*a*d*g^3*h*n - 10*B*b*c*g^3*h*n)/(5
*b*d) - ((5*a*d + 5*b*c)*((20*A*a*c*g*h^3 + 20*A*b*d*g^3*h + 30*A*a*d*g^2*h^2 + 30*A*b*c*g^2*h^2 + 10*B*a*d*g^
2*h^2*n - 10*B*b*c*g^2*h^2*n)/(5*b*d) + ((5*a*d + 5*b*c)*((((5*A*a*d*h^4 + 5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*
d*h^4*n - B*b*c*h^4*n)/(5*b*d) - (A*h^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(5*b*d) - (5*A*a*c*h^4 + 20
*A*a*d*g*h^3 + 20*A*b*c*g*h^3 + 30*A*b*d*g^2*h^2 + 5*B*a*d*g*h^3*n - 5*B*b*c*g*h^3*n)/(5*b*d) + (A*a*c*h^4)/(b
*d)))/(5*b*d) - (a*c*((5*A*a*d*h^4 + 5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*d*h^4*n - B*b*c*h^4*n)/(5*b*d) - (A*h^
4*(5*a*d + 5*b*c))/(5*b*d)))/(b*d)))/(5*b*d) + (a*c*((((5*A*a*d*h^4 + 5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*d*h^4
*n - B*b*c*h^4*n)/(5*b*d) - (A*h^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(5*b*d) - (5*A*a*c*h^4 + 20*A*a*
d*g*h^3 + 20*A*b*c*g*h^3 + 30*A*b*d*g^2*h^2 + 5*B*a*d*g*h^3*n - 5*B*b*c*g*h^3*n)/(5*b*d) + (A*a*c*h^4)/(b*d)))
/(b*d)) + log((e*(a + b*x)^n)/(c + d*x)^n)*((B*h^4*x^5)/5 + B*g^4*x + 2*B*g^2*h^2*x^3 + 2*B*g^3*h*x^2 + B*g*h^
3*x^4) + x^4*((5*A*a*d*h^4 + 5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*d*h^4*n - B*b*c*h^4*n)/(20*b*d) - (A*h^4*(5*a*
d + 5*b*c))/(20*b*d)) - x^3*((((5*A*a*d*h^4 + 5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*d*h^4*n - B*b*c*h^4*n)/(5*b*d
) - (A*h^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(15*b*d) - (5*A*a*c*h^4 + 20*A*a*d*g*h^3 + 20*A*b*c*g*h^
3 + 30*A*b*d*g^2*h^2 + 5*B*a*d*g*h^3*n - 5*B*b*c*g*h^3*n)/(15*b*d) + (A*a*c*h^4)/(3*b*d)) + x^2*((20*A*a*c*g*h
^3 + 20*A*b*d*g^3*h + 30*A*a*d*g^2*h^2 + 30*A*b*c*g^2*h^2 + 10*B*a*d*g^2*h^2*n - 10*B*b*c*g^2*h^2*n)/(10*b*d)
+ ((5*a*d + 5*b*c)*((((5*A*a*d*h^4 + 5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*d*h^4*n - B*b*c*h^4*n)/(5*b*d) - (A*h^
4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(5*b*d) - (5*A*a*c*h^4 + 20*A*a*d*g*h^3 + 20*A*b*c*g*h^3 + 30*A*b
*d*g^2*h^2 + 5*B*a*d*g*h^3*n - 5*B*b*c*g*h^3*n)/(5*b*d) + (A*a*c*h^4)/(b*d)))/(10*b*d) - (a*c*((5*A*a*d*h^4 +
5*A*b*c*h^4 + 20*A*b*d*g*h^3 + B*a*d*h^4*n - B*b*c*h^4*n)/(5*b*d) - (A*h^4*(5*a*d + 5*b*c))/(5*b*d)))/(2*b*d))
+ (A*h^4*x^5)/5 + (log(a + b*x)*((B*a^5*h^4*n)/5 + B*a*b^4*g^4*n + 2*B*a^3*b^2*g^2*h^2*n - B*a^4*b*g*h^3*n -
2*B*a^2*b^3*g^3*h*n))/b^5 - (log(c + d*x)*(B*c^5*h^4*n + 5*B*c*d^4*g^4*n + 10*B*c^3*d^2*g^2*h^2*n - 5*B*c^4*d*
g*h^3*n - 10*B*c^2*d^3*g^3*h*n))/(5*d^5)