Integrand size = 20, antiderivative size = 160 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=-24 a b^3 p^3 q^3 x+24 b^4 p^4 q^4 x-\frac {24 b^4 p^3 q^3 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+\frac {12 b^2 p^2 q^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-\frac {4 b p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{f} \]
-24*a*b^3*p^3*q^3*x+24*b^4*p^4*q^4*x-24*b^4*p^3*q^3*(f*x+e)*ln(c*(d*(f*x+e )^p)^q)/f+12*b^2*p^2*q^2*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f-4*b*p*q*( f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f+(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q)) ^4/f
Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4-4 b p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-3 b p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{f} \]
((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^4 - 4*b*p*q*((e + f*x)*(a + b* Log[c*(d*(e + f*x)^p)^q])^3 - 3*b*p*q*((e + f*x)*(a + b*Log[c*(d*(e + f*x) ^p)^q])^2 - 2*b*p*q*(f*(a - b*p*q)*x + b*(e + f*x)*Log[c*(d*(e + f*x)^p)^q ]))))/f
Time = 0.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2895, 2836, 2733, 2733, 2733, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4d(e+f x)}{f}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4-4 b p q \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3d(e+f x)}{f}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4-4 b p q \left ((e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3-3 b p q \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2d(e+f x)\right )}{f}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4-4 b p q \left ((e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3-3 b p q \left ((e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2-2 b p q \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )d(e+f x)\right )\right )}{f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4-4 b p q \left ((e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3-3 b p q \left ((e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2-2 b p q \left (a (e+f x)+b (e+f x) \log \left (c d^q (e+f x)^{p q}\right )-b p q (e+f x)\right )\right )\right )}{f}\) |
((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)])^4 - 4*b*p*q*((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)])^3 - 3*b*p*q*((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)])^2 - 2*b*p*q*(a*(e + f*x) - b*p*q*(e + f*x) + b*(e + f*x)*Log [c*d^q*(e + f*x)^(p*q)]))))/f
3.5.41.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(160)=320\).
Time = 5.17 (sec) , antiderivative size = 642, normalized size of antiderivative = 4.01
method | result | size |
parallelrisch | \(\frac {24 x \,b^{4} e f \,p^{4} q^{4}-12 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} a \,b^{3} e^{2} p q +24 \ln \left (f x +e \right ) a \,b^{3} e^{2} p^{3} q^{3}-24 b^{4} e^{2} p^{4} q^{4}+12 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{4} e^{2} p^{2} q^{2}-4 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{3} b^{4} e^{2} p q -24 \ln \left (f x +e \right ) b^{4} e^{2} p^{4} q^{4}+x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{4} b^{4} e f +24 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a \,b^{3} e f \,p^{2} q^{2}+{\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{4} b^{4} e^{2}-12 x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} a \,b^{3} e f p q -12 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a^{2} b^{2} e f p q -a^{4} e^{2}+4 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{3} a \,b^{3} e^{2}+6 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} a^{2} b^{2} e^{2}+x \,a^{4} e f +24 a \,b^{3} e^{2} p^{3} q^{3}-12 a^{2} b^{2} e^{2} p^{2} q^{2}+4 a^{3} b \,e^{2} p q -24 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{4} e f \,p^{3} q^{3}+12 x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{4} e f \,p^{2} q^{2}-24 x a \,b^{3} e f \,p^{3} q^{3}-4 x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{3} b^{4} e f p q +12 x \,a^{2} b^{2} e f \,p^{2} q^{2}-4 x \,a^{3} b e f p q -12 \ln \left (f x +e \right ) a^{2} b^{2} e^{2} p^{2} q^{2}+4 \ln \left (f x +e \right ) a^{3} b \,e^{2} p q +4 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a^{3} b e f +4 x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{3} a \,b^{3} e f +6 x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} a^{2} b^{2} e f}{e f}\) | \(642\) |
(24*x*b^4*e*f*p^4*q^4-12*ln(c*(d*(f*x+e)^p)^q)^2*a*b^3*e^2*p*q+24*ln(f*x+e )*a*b^3*e^2*p^3*q^3-24*b^4*e^2*p^4*q^4+12*ln(c*(d*(f*x+e)^p)^q)^2*b^4*e^2* p^2*q^2-4*ln(c*(d*(f*x+e)^p)^q)^3*b^4*e^2*p*q-24*ln(f*x+e)*b^4*e^2*p^4*q^4 +x*ln(c*(d*(f*x+e)^p)^q)^4*b^4*e*f+24*x*ln(c*(d*(f*x+e)^p)^q)*a*b^3*e*f*p^ 2*q^2+ln(c*(d*(f*x+e)^p)^q)^4*b^4*e^2-12*x*ln(c*(d*(f*x+e)^p)^q)^2*a*b^3*e *f*p*q-12*x*ln(c*(d*(f*x+e)^p)^q)*a^2*b^2*e*f*p*q-a^4*e^2+4*ln(c*(d*(f*x+e )^p)^q)^3*a*b^3*e^2+6*ln(c*(d*(f*x+e)^p)^q)^2*a^2*b^2*e^2+x*a^4*e*f+24*a*b ^3*e^2*p^3*q^3-12*a^2*b^2*e^2*p^2*q^2+4*a^3*b*e^2*p*q-24*x*ln(c*(d*(f*x+e) ^p)^q)*b^4*e*f*p^3*q^3+12*x*ln(c*(d*(f*x+e)^p)^q)^2*b^4*e*f*p^2*q^2-24*x*a *b^3*e*f*p^3*q^3-4*x*ln(c*(d*(f*x+e)^p)^q)^3*b^4*e*f*p*q+12*x*a^2*b^2*e*f* p^2*q^2-4*x*a^3*b*e*f*p*q-12*ln(f*x+e)*a^2*b^2*e^2*p^2*q^2+4*ln(f*x+e)*a^3 *b*e^2*p*q+4*x*ln(c*(d*(f*x+e)^p)^q)*a^3*b*e*f+4*x*ln(c*(d*(f*x+e)^p)^q)^3 *a*b^3*e*f+6*x*ln(c*(d*(f*x+e)^p)^q)^2*a^2*b^2*e*f)/e/f
Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (160) = 320\).
Time = 0.31 (sec) , antiderivative size = 1409, normalized size of antiderivative = 8.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\text {Too large to display} \]
(b^4*f*q^4*x*log(d)^4 + b^4*f*x*log(c)^4 + (b^4*f*p^4*q^4*x + b^4*e*p^4*q^ 4)*log(f*x + e)^4 - 4*(b^4*f*p*q - a*b^3*f)*x*log(c)^3 - 4*(b^4*e*p^4*q^4 - a*b^3*e*p^3*q^3 + (b^4*f*p^4*q^4 - a*b^3*f*p^3*q^3)*x - (b^4*f*p^3*q^3*x + b^4*e*p^3*q^3)*log(c) - (b^4*f*p^3*q^4*x + b^4*e*p^3*q^4)*log(d))*log(f *x + e)^3 + 6*(2*b^4*f*p^2*q^2 - 2*a*b^3*f*p*q + a^2*b^2*f)*x*log(c)^2 + 4 *(b^4*f*q^3*x*log(c) - (b^4*f*p*q^4 - a*b^3*f*q^3)*x)*log(d)^3 + 6*(2*b^4* e*p^4*q^4 - 2*a*b^3*e*p^3*q^3 + a^2*b^2*e*p^2*q^2 + (b^4*f*p^2*q^2*x + b^4 *e*p^2*q^2)*log(c)^2 + (b^4*f*p^2*q^4*x + b^4*e*p^2*q^4)*log(d)^2 + (2*b^4 *f*p^4*q^4 - 2*a*b^3*f*p^3*q^3 + a^2*b^2*f*p^2*q^2)*x - 2*(b^4*e*p^3*q^3 - a*b^3*e*p^2*q^2 + (b^4*f*p^3*q^3 - a*b^3*f*p^2*q^2)*x)*log(c) - 2*(b^4*e* p^3*q^4 - a*b^3*e*p^2*q^3 + (b^4*f*p^3*q^4 - a*b^3*f*p^2*q^3)*x - (b^4*f*p ^2*q^3*x + b^4*e*p^2*q^3)*log(c))*log(d))*log(f*x + e)^2 - 4*(6*b^4*f*p^3* q^3 - 6*a*b^3*f*p^2*q^2 + 3*a^2*b^2*f*p*q - a^3*b*f)*x*log(c) + 6*(b^4*f*q ^2*x*log(c)^2 - 2*(b^4*f*p*q^3 - a*b^3*f*q^2)*x*log(c) + (2*b^4*f*p^2*q^4 - 2*a*b^3*f*p*q^3 + a^2*b^2*f*q^2)*x)*log(d)^2 + (24*b^4*f*p^4*q^4 - 24*a* b^3*f*p^3*q^3 + 12*a^2*b^2*f*p^2*q^2 - 4*a^3*b*f*p*q + a^4*f)*x - 4*(6*b^4 *e*p^4*q^4 - 6*a*b^3*e*p^3*q^3 + 3*a^2*b^2*e*p^2*q^2 - a^3*b*e*p*q - (b^4* f*p*q*x + b^4*e*p*q)*log(c)^3 - (b^4*f*p*q^4*x + b^4*e*p*q^4)*log(d)^3 + 3 *(b^4*e*p^2*q^2 - a*b^3*e*p*q + (b^4*f*p^2*q^2 - a*b^3*f*p*q)*x)*log(c)^2 + 3*(b^4*e*p^2*q^4 - a*b^3*e*p*q^3 + (b^4*f*p^2*q^4 - a*b^3*f*p*q^3)*x ...
Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (155) = 310\).
Time = 2.07 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - 4 a^{3} b p q x + 4 a^{3} b x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {12 a^{2} b^{2} e p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {6 a^{2} b^{2} e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} + 12 a^{2} b^{2} p^{2} q^{2} x - 12 a^{2} b^{2} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + 6 a^{2} b^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {24 a b^{3} e p^{2} q^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - \frac {12 a b^{3} e p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} + \frac {4 a b^{3} e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3}}{f} - 24 a b^{3} p^{3} q^{3} x + 24 a b^{3} p^{2} q^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - 12 a b^{3} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + 4 a b^{3} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3} - \frac {24 b^{4} e p^{3} q^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {12 b^{4} e p^{2} q^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} - \frac {4 b^{4} e p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3}}{f} + \frac {b^{4} e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{4}}{f} + 24 b^{4} p^{4} q^{4} x - 24 b^{4} p^{3} q^{3} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + 12 b^{4} p^{2} q^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} - 4 b^{4} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3} + b^{4} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{4} & \text {for}\: f \neq 0 \\x \left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*x + 4*a**3*b*e*log(c*(d*(e + f*x)**p)**q)/f - 4*a**3*b*p*q *x + 4*a**3*b*x*log(c*(d*(e + f*x)**p)**q) - 12*a**2*b**2*e*p*q*log(c*(d*( e + f*x)**p)**q)/f + 6*a**2*b**2*e*log(c*(d*(e + f*x)**p)**q)**2/f + 12*a* *2*b**2*p**2*q**2*x - 12*a**2*b**2*p*q*x*log(c*(d*(e + f*x)**p)**q) + 6*a* *2*b**2*x*log(c*(d*(e + f*x)**p)**q)**2 + 24*a*b**3*e*p**2*q**2*log(c*(d*( e + f*x)**p)**q)/f - 12*a*b**3*e*p*q*log(c*(d*(e + f*x)**p)**q)**2/f + 4*a *b**3*e*log(c*(d*(e + f*x)**p)**q)**3/f - 24*a*b**3*p**3*q**3*x + 24*a*b** 3*p**2*q**2*x*log(c*(d*(e + f*x)**p)**q) - 12*a*b**3*p*q*x*log(c*(d*(e + f *x)**p)**q)**2 + 4*a*b**3*x*log(c*(d*(e + f*x)**p)**q)**3 - 24*b**4*e*p**3 *q**3*log(c*(d*(e + f*x)**p)**q)/f + 12*b**4*e*p**2*q**2*log(c*(d*(e + f*x )**p)**q)**2/f - 4*b**4*e*p*q*log(c*(d*(e + f*x)**p)**q)**3/f + b**4*e*log (c*(d*(e + f*x)**p)**q)**4/f + 24*b**4*p**4*q**4*x - 24*b**4*p**3*q**3*x*l og(c*(d*(e + f*x)**p)**q) + 12*b**4*p**2*q**2*x*log(c*(d*(e + f*x)**p)**q) **2 - 4*b**4*p*q*x*log(c*(d*(e + f*x)**p)**q)**3 + b**4*x*log(c*(d*(e + f* x)**p)**q)**4, Ne(f, 0)), (x*(a + b*log(c*(d*e**p)**q))**4, True))
Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (160) = 320\).
Time = 0.23 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.49 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=b^{4} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{4} - 4 \, a^{3} b f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + 4 \, a b^{3} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 6 \, a^{2} b^{2} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 4 \, a^{3} b x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - 6 \, {\left (2 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f}\right )} a^{2} b^{2} - 4 \, {\left (3 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} - {\left (\frac {{\left (e \log \left (f x + e\right )^{3} + 3 \, e \log \left (f x + e\right )^{2} - 6 \, f x + 6 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{2}} - \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p q \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{f^{2}}\right )} f p q\right )} a b^{3} - {\left (4 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + {\left ({\left (\frac {{\left (e \log \left (f x + e\right )^{4} + 4 \, e \log \left (f x + e\right )^{3} + 12 \, e \log \left (f x + e\right )^{2} - 24 \, f x + 24 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{3}} - \frac {4 \, {\left (e \log \left (f x + e\right )^{3} + 3 \, e \log \left (f x + e\right )^{2} - 6 \, f x + 6 \, e \log \left (f x + e\right )\right )} p q \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{f^{3}}\right )} f p q + \frac {6 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p q \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2}}{f^{2}}\right )} f p q\right )} b^{4} + a^{4} x \]
b^4*x*log(((f*x + e)^p*d)^q*c)^4 - 4*a^3*b*f*p*q*(x/f - e*log(f*x + e)/f^2 ) + 4*a*b^3*x*log(((f*x + e)^p*d)^q*c)^3 + 6*a^2*b^2*x*log(((f*x + e)^p*d) ^q*c)^2 + 4*a^3*b*x*log(((f*x + e)^p*d)^q*c) - 6*(2*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*c) + (e*log(f*x + e)^2 - 2*f*x + 2*e*log( f*x + e))*p^2*q^2/f)*a^2*b^2 - 4*(3*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(( (f*x + e)^p*d)^q*c)^2 - ((e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e))*p^2*q^2/f^2 - 3*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*p*q*log(((f*x + e)^p*d)^q*c)/f^2)*f*p*q)*a*b^3 - (4*f*p*q*(x/f - e*l og(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*c)^3 + (((e*log(f*x + e)^4 + 4*e*lo g(f*x + e)^3 + 12*e*log(f*x + e)^2 - 24*f*x + 24*e*log(f*x + e))*p^2*q^2/f ^3 - 4*(e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e))* p*q*log(((f*x + e)^p*d)^q*c)/f^3)*f*p*q + 6*(e*log(f*x + e)^2 - 2*f*x + 2* e*log(f*x + e))*p*q*log(((f*x + e)^p*d)^q*c)^2/f^2)*f*p*q)*b^4 + a^4*x
Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (160) = 320\).
Time = 0.36 (sec) , antiderivative size = 1697, normalized size of antiderivative = 10.61 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\text {Too large to display} \]
(f*x + e)*b^4*p^4*q^4*log(f*x + e)^4/f - 4*(f*x + e)*b^4*p^4*q^4*log(f*x + e)^3/f + 4*(f*x + e)*b^4*p^3*q^4*log(f*x + e)^3*log(d)/f + 12*(f*x + e)*b ^4*p^4*q^4*log(f*x + e)^2/f + 4*(f*x + e)*b^4*p^3*q^3*log(f*x + e)^3*log(c )/f - 12*(f*x + e)*b^4*p^3*q^4*log(f*x + e)^2*log(d)/f + 6*(f*x + e)*b^4*p ^2*q^4*log(f*x + e)^2*log(d)^2/f - 24*(f*x + e)*b^4*p^4*q^4*log(f*x + e)/f + 4*(f*x + e)*a*b^3*p^3*q^3*log(f*x + e)^3/f - 12*(f*x + e)*b^4*p^3*q^3*l og(f*x + e)^2*log(c)/f + 24*(f*x + e)*b^4*p^3*q^4*log(f*x + e)*log(d)/f + 12*(f*x + e)*b^4*p^2*q^3*log(f*x + e)^2*log(c)*log(d)/f - 12*(f*x + e)*b^4 *p^2*q^4*log(f*x + e)*log(d)^2/f + 4*(f*x + e)*b^4*p*q^4*log(f*x + e)*log( d)^3/f + 24*(f*x + e)*b^4*p^4*q^4/f - 12*(f*x + e)*a*b^3*p^3*q^3*log(f*x + e)^2/f + 24*(f*x + e)*b^4*p^3*q^3*log(f*x + e)*log(c)/f + 6*(f*x + e)*b^4 *p^2*q^2*log(f*x + e)^2*log(c)^2/f - 24*(f*x + e)*b^4*p^3*q^4*log(d)/f + 1 2*(f*x + e)*a*b^3*p^2*q^3*log(f*x + e)^2*log(d)/f - 24*(f*x + e)*b^4*p^2*q ^3*log(f*x + e)*log(c)*log(d)/f + 12*(f*x + e)*b^4*p^2*q^4*log(d)^2/f + 12 *(f*x + e)*b^4*p*q^3*log(f*x + e)*log(c)*log(d)^2/f - 4*(f*x + e)*b^4*p*q^ 4*log(d)^3/f + (f*x + e)*b^4*q^4*log(d)^4/f + 24*(f*x + e)*a*b^3*p^3*q^3*l og(f*x + e)/f - 24*(f*x + e)*b^4*p^3*q^3*log(c)/f + 12*(f*x + e)*a*b^3*p^2 *q^2*log(f*x + e)^2*log(c)/f - 12*(f*x + e)*b^4*p^2*q^2*log(f*x + e)*log(c )^2/f - 24*(f*x + e)*a*b^3*p^2*q^3*log(f*x + e)*log(d)/f + 24*(f*x + e)*b^ 4*p^2*q^3*log(c)*log(d)/f + 12*(f*x + e)*b^4*p*q^2*log(f*x + e)*log(c)^...
Time = 1.60 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.38 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx={\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^3\,\left (\frac {4\,\left (a\,b^3\,e-b^4\,e\,p\,q\right )}{f}+4\,b^3\,x\,\left (a-b\,p\,q\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^4\,\left (b^4\,x+\frac {b^4\,e}{f}\right )+x\,\left (a^4-4\,a^3\,b\,p\,q+12\,a^2\,b^2\,p^2\,q^2-24\,a\,b^3\,p^3\,q^3+24\,b^4\,p^4\,q^4\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (\frac {6\,\left (e\,a^2\,b^2-2\,e\,a\,b^3\,p\,q+2\,e\,b^4\,p^2\,q^2\right )}{f}+6\,b^2\,x\,\left (a^2-2\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )\right )+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (4\,b\,f\,\left (a^3-3\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-6\,b^3\,p^3\,q^3\right )\,x^2+4\,b\,e\,\left (a^3-3\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-6\,b^3\,p^3\,q^3\right )\,x\right )}{e+f\,x}-\frac {\ln \left (e+f\,x\right )\,\left (-4\,e\,a^3\,b\,p\,q+12\,e\,a^2\,b^2\,p^2\,q^2-24\,e\,a\,b^3\,p^3\,q^3+24\,e\,b^4\,p^4\,q^4\right )}{f} \]
log(c*(d*(e + f*x)^p)^q)^3*((4*(a*b^3*e - b^4*e*p*q))/f + 4*b^3*x*(a - b*p *q)) + log(c*(d*(e + f*x)^p)^q)^4*(b^4*x + (b^4*e)/f) + x*(a^4 + 24*b^4*p^ 4*q^4 - 24*a*b^3*p^3*q^3 - 4*a^3*b*p*q + 12*a^2*b^2*p^2*q^2) + log(c*(d*(e + f*x)^p)^q)^2*((6*(a^2*b^2*e + 2*b^4*e*p^2*q^2 - 2*a*b^3*e*p*q))/f + 6*b ^2*x*(a^2 + 2*b^2*p^2*q^2 - 2*a*b*p*q)) + (log(c*(d*(e + f*x)^p)^q)*(4*b*e *x*(a^3 - 6*b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 3*a^2*b*p*q) + 4*b*f*x^2*(a^3 - 6*b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 3*a^2*b*p*q)))/(e + f*x) - (log(e + f* x)*(24*b^4*e*p^4*q^4 - 24*a*b^3*e*p^3*q^3 - 4*a^3*b*e*p*q + 12*a^2*b^2*e*p ^2*q^2))/f