3.441 \(\int (a+b \log (c (d (e+f x)^p)^q))^4 \, dx\)

Optimal. Leaf size=160 \[ -24 a b^3 p^3 q^3 x+\frac {12 b^2 p^2 q^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-\frac {4 b p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{f}-\frac {24 b^4 p^3 q^3 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+24 b^4 p^4 q^4 x \]

[Out]

-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

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Rubi [A]  time = 0.20, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2389, 2296, 2295, 2445} \[ \frac {12 b^2 p^2 q^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f}-24 a b^3 p^3 q^3 x-\frac {4 b p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{f}-\frac {24 b^4 p^3 q^3 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}+24 b^4 p^4 q^4 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^4,x]

[Out]

-24*a*b^3*p^3*q^3*x + 24*b^4*p^4*q^4*x - (24*b^4*p^3*q^3*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f + (12*b^2*p^2*q
^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f - (4*b*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f
+ ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^4)/f

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[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]

Rule 2445

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]]

Rubi steps

\begin {align*} \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx &=\operatorname {Subst}\left (\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^4 \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{f}-\operatorname {Subst}\left (\frac {(4 b p q) \operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\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}+\operatorname {Subst}\left (\frac {\left (12 b^2 p^2 q^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\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}-\operatorname {Subst}\left (\frac {\left (24 b^3 p^3 q^3\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-24 a b^3 p^3 q^3 x+\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}-\operatorname {Subst}\left (\frac {\left (24 b^4 p^3 q^3\right ) \operatorname {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-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}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 132, normalized size = 0.82 \[ \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4-4 b p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-3 b p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f x (a-b p q)+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^4,x]

[Out]

((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

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fricas [B]  time = 0.48, size = 1409, normalized size = 8.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4,x, algorithm="fricas")

[Out]

(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 - (b^4*f*p*q^3*x + b^4*e*p*q^3)*log(c))*log(d)^2 + (6*b^4*f*p^4
*q^4 - 6*a*b^3*f*p^3*q^3 + 3*a^2*b^2*f*p^2*q^2 - a^3*b*f*p*q)*x - 3*(2*b^4*e*p^3*q^3 - 2*a*b^3*e*p^2*q^2 + a^2
*b^2*e*p*q + (2*b^4*f*p^3*q^3 - 2*a*b^3*f*p^2*q^2 + a^2*b^2*f*p*q)*x)*log(c) - 3*(2*b^4*e*p^3*q^4 - 2*a*b^3*e*
p^2*q^3 + a^2*b^2*e*p*q^2 + (b^4*f*p*q^2*x + b^4*e*p*q^2)*log(c)^2 + (2*b^4*f*p^3*q^4 - 2*a*b^3*f*p^2*q^3 + a^
2*b^2*f*p*q^2)*x - 2*(b^4*e*p^2*q^3 - a*b^3*e*p*q^2 + (b^4*f*p^2*q^3 - a*b^3*f*p*q^2)*x)*log(c))*log(d))*log(f
*x + e) + 4*(b^4*f*q*x*log(c)^3 - 3*(b^4*f*p*q^2 - a*b^3*f*q)*x*log(c)^2 + 3*(2*b^4*f*p^2*q^3 - 2*a*b^3*f*p*q^
2 + a^2*b^2*f*q)*x*log(c) - (6*b^4*f*p^3*q^4 - 6*a*b^3*f*p^2*q^3 + 3*a^2*b^2*f*p*q^2 - a^3*b*f*q)*x)*log(d))/f

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giac [B]  time = 0.30, size = 1802, normalized size = 11.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4,x, algorithm="giac")

[Out]

(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*lo
g(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*log(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 + 12*(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*log(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)^2*log(d)/f + 12*(f*x + e)*a*b^3*p*q^3*log(f*x + e)*log(d)^2/f -
12*(f*x + e)*b^4*p*q^3*log(c)*log(d)^2/f + 4*(f*x + e)*b^4*q^3*log(c)*log(d)^3/f - 24*(f*x + e)*a*b^3*p^3*q^3
/f + 6*(f*x + e)*a^2*b^2*p^2*q^2*log(f*x + e)^2/f - 24*(f*x + e)*a*b^3*p^2*q^2*log(f*x + e)*log(c)/f + 12*(f*x
+ e)*b^4*p^2*q^2*log(c)^2/f + 4*(f*x + e)*b^4*p*q*log(f*x + e)*log(c)^3/f + 24*(f*x + e)*a*b^3*p^2*q^3*log(d)
/f + 24*(f*x + e)*a*b^3*p*q^2*log(f*x + e)*log(c)*log(d)/f - 12*(f*x + e)*b^4*p*q^2*log(c)^2*log(d)/f - 12*(f*
x + e)*a*b^3*p*q^3*log(d)^2/f + 6*(f*x + e)*b^4*q^2*log(c)^2*log(d)^2/f + 4*(f*x + e)*a*b^3*q^3*log(d)^3/f - 1
2*(f*x + e)*a^2*b^2*p^2*q^2*log(f*x + e)/f + 24*(f*x + e)*a*b^3*p^2*q^2*log(c)/f + 12*(f*x + e)*a*b^3*p*q*log(
f*x + e)*log(c)^2/f - 4*(f*x + e)*b^4*p*q*log(c)^3/f + 12*(f*x + e)*a^2*b^2*p*q^2*log(f*x + e)*log(d)/f - 24*(
f*x + e)*a*b^3*p*q^2*log(c)*log(d)/f + 4*(f*x + e)*b^4*q*log(c)^3*log(d)/f + 12*(f*x + e)*a*b^3*q^2*log(c)*log
(d)^2/f + 12*(f*x + e)*a^2*b^2*p^2*q^2/f + 12*(f*x + e)*a^2*b^2*p*q*log(f*x + e)*log(c)/f - 12*(f*x + e)*a*b^3
*p*q*log(c)^2/f + (f*x + e)*b^4*log(c)^4/f - 12*(f*x + e)*a^2*b^2*p*q^2*log(d)/f + 12*(f*x + e)*a*b^3*q*log(c)
^2*log(d)/f + 6*(f*x + e)*a^2*b^2*q^2*log(d)^2/f + 4*(f*x + e)*a^3*b*p*q*log(f*x + e)/f - 12*(f*x + e)*a^2*b^2
*p*q*log(c)/f + 4*(f*x + e)*a*b^3*log(c)^3/f + 12*(f*x + e)*a^2*b^2*q*log(c)*log(d)/f - 4*(f*x + e)*a^3*b*p*q/
f + 6*(f*x + e)*a^2*b^2*log(c)^2/f + 4*(f*x + e)*a^3*b*q*log(d)/f + 4*(f*x + e)*a^3*b*log(c)/f + (f*x + e)*a^4
/f

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(d*(f*x+e)^p)^q)+a)^4,x)

[Out]

int((b*ln(c*(d*(f*x+e)^p)^q)+a)^4,x)

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maxima [B]  time = 0.88, size = 559, normalized size = 3.49 \[ 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 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4,x, algorithm="maxima")

[Out]

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*log(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*c)^3 + (((e*log(f*x + e)^4 +
4*e*log(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

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mupad [B]  time = 0.56, size = 380, normalized size = 2.38 \[ {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^3\,\left (\frac {4\,\left (a\,b^3\,e-b^4\,e\,p\,q\right )}{f}+4\,b^3\,x\,\left (a-b\,p\,q\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^4\,\left (b^4\,x+\frac {b^4\,e}{f}\right )+x\,\left (a^4-4\,a^3\,b\,p\,q+12\,a^2\,b^2\,p^2\,q^2-24\,a\,b^3\,p^3\,q^3+24\,b^4\,p^4\,q^4\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (\frac {6\,\left (e\,a^2\,b^2-2\,e\,a\,b^3\,p\,q+2\,e\,b^4\,p^2\,q^2\right )}{f}+6\,b^2\,x\,\left (a^2-2\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )\right )+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (4\,b\,f\,\left (a^3-3\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-6\,b^3\,p^3\,q^3\right )\,x^2+4\,b\,e\,\left (a^3-3\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-6\,b^3\,p^3\,q^3\right )\,x\right )}{e+f\,x}-\frac {\ln \left (e+f\,x\right )\,\left (-4\,e\,a^3\,b\,p\,q+12\,e\,a^2\,b^2\,p^2\,q^2-24\,e\,a\,b^3\,p^3\,q^3+24\,e\,b^4\,p^4\,q^4\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d*(e + f*x)^p)^q))^4,x)

[Out]

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

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sympy [A]  time = 22.87, size = 2390, normalized size = 14.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*e*p*q*log(e + f*x)/f + 4*a**3*b*p*q*x*log(e + f*x) - 4*a**3*b*p*q*x + 4*a**3*b*q*
x*log(d) + 4*a**3*b*x*log(c) + 6*a**2*b**2*e*p**2*q**2*log(e + f*x)**2/f - 12*a**2*b**2*e*p**2*q**2*log(e + f*
x)/f + 12*a**2*b**2*e*p*q**2*log(d)*log(e + f*x)/f + 12*a**2*b**2*e*p*q*log(c)*log(e + f*x)/f + 6*a**2*b**2*p*
*2*q**2*x*log(e + f*x)**2 - 12*a**2*b**2*p**2*q**2*x*log(e + f*x) + 12*a**2*b**2*p**2*q**2*x + 12*a**2*b**2*p*
q**2*x*log(d)*log(e + f*x) - 12*a**2*b**2*p*q**2*x*log(d) + 12*a**2*b**2*p*q*x*log(c)*log(e + f*x) - 12*a**2*b
**2*p*q*x*log(c) + 6*a**2*b**2*q**2*x*log(d)**2 + 12*a**2*b**2*q*x*log(c)*log(d) + 6*a**2*b**2*x*log(c)**2 + 4
*a*b**3*e*p**3*q**3*log(e + f*x)**3/f - 12*a*b**3*e*p**3*q**3*log(e + f*x)**2/f + 24*a*b**3*e*p**3*q**3*log(e
+ f*x)/f + 12*a*b**3*e*p**2*q**3*log(d)*log(e + f*x)**2/f - 24*a*b**3*e*p**2*q**3*log(d)*log(e + f*x)/f + 12*a
*b**3*e*p**2*q**2*log(c)*log(e + f*x)**2/f - 24*a*b**3*e*p**2*q**2*log(c)*log(e + f*x)/f + 12*a*b**3*e*p*q**3*
log(d)**2*log(e + f*x)/f + 24*a*b**3*e*p*q**2*log(c)*log(d)*log(e + f*x)/f + 12*a*b**3*e*p*q*log(c)**2*log(e +
f*x)/f + 4*a*b**3*p**3*q**3*x*log(e + f*x)**3 - 12*a*b**3*p**3*q**3*x*log(e + f*x)**2 + 24*a*b**3*p**3*q**3*x
*log(e + f*x) - 24*a*b**3*p**3*q**3*x + 12*a*b**3*p**2*q**3*x*log(d)*log(e + f*x)**2 - 24*a*b**3*p**2*q**3*x*l
og(d)*log(e + f*x) + 24*a*b**3*p**2*q**3*x*log(d) + 12*a*b**3*p**2*q**2*x*log(c)*log(e + f*x)**2 - 24*a*b**3*p
**2*q**2*x*log(c)*log(e + f*x) + 24*a*b**3*p**2*q**2*x*log(c) + 12*a*b**3*p*q**3*x*log(d)**2*log(e + f*x) - 12
*a*b**3*p*q**3*x*log(d)**2 + 24*a*b**3*p*q**2*x*log(c)*log(d)*log(e + f*x) - 24*a*b**3*p*q**2*x*log(c)*log(d)
+ 12*a*b**3*p*q*x*log(c)**2*log(e + f*x) - 12*a*b**3*p*q*x*log(c)**2 + 4*a*b**3*q**3*x*log(d)**3 + 12*a*b**3*q
**2*x*log(c)*log(d)**2 + 12*a*b**3*q*x*log(c)**2*log(d) + 4*a*b**3*x*log(c)**3 + b**4*e*p**4*q**4*log(e + f*x)
**4/f - 4*b**4*e*p**4*q**4*log(e + f*x)**3/f + 12*b**4*e*p**4*q**4*log(e + f*x)**2/f - 24*b**4*e*p**4*q**4*log
(e + f*x)/f + 4*b**4*e*p**3*q**4*log(d)*log(e + f*x)**3/f - 12*b**4*e*p**3*q**4*log(d)*log(e + f*x)**2/f + 24*
b**4*e*p**3*q**4*log(d)*log(e + f*x)/f + 4*b**4*e*p**3*q**3*log(c)*log(e + f*x)**3/f - 12*b**4*e*p**3*q**3*log
(c)*log(e + f*x)**2/f + 24*b**4*e*p**3*q**3*log(c)*log(e + f*x)/f + 6*b**4*e*p**2*q**4*log(d)**2*log(e + f*x)*
*2/f - 12*b**4*e*p**2*q**4*log(d)**2*log(e + f*x)/f + 12*b**4*e*p**2*q**3*log(c)*log(d)*log(e + f*x)**2/f - 24
*b**4*e*p**2*q**3*log(c)*log(d)*log(e + f*x)/f + 6*b**4*e*p**2*q**2*log(c)**2*log(e + f*x)**2/f - 12*b**4*e*p*
*2*q**2*log(c)**2*log(e + f*x)/f + 4*b**4*e*p*q**4*log(d)**3*log(e + f*x)/f + 12*b**4*e*p*q**3*log(c)*log(d)**
2*log(e + f*x)/f + 12*b**4*e*p*q**2*log(c)**2*log(d)*log(e + f*x)/f + 4*b**4*e*p*q*log(c)**3*log(e + f*x)/f +
b**4*p**4*q**4*x*log(e + f*x)**4 - 4*b**4*p**4*q**4*x*log(e + f*x)**3 + 12*b**4*p**4*q**4*x*log(e + f*x)**2 -
24*b**4*p**4*q**4*x*log(e + f*x) + 24*b**4*p**4*q**4*x + 4*b**4*p**3*q**4*x*log(d)*log(e + f*x)**3 - 12*b**4*p
**3*q**4*x*log(d)*log(e + f*x)**2 + 24*b**4*p**3*q**4*x*log(d)*log(e + f*x) - 24*b**4*p**3*q**4*x*log(d) + 4*b
**4*p**3*q**3*x*log(c)*log(e + f*x)**3 - 12*b**4*p**3*q**3*x*log(c)*log(e + f*x)**2 + 24*b**4*p**3*q**3*x*log(
c)*log(e + f*x) - 24*b**4*p**3*q**3*x*log(c) + 6*b**4*p**2*q**4*x*log(d)**2*log(e + f*x)**2 - 12*b**4*p**2*q**
4*x*log(d)**2*log(e + f*x) + 12*b**4*p**2*q**4*x*log(d)**2 + 12*b**4*p**2*q**3*x*log(c)*log(d)*log(e + f*x)**2
- 24*b**4*p**2*q**3*x*log(c)*log(d)*log(e + f*x) + 24*b**4*p**2*q**3*x*log(c)*log(d) + 6*b**4*p**2*q**2*x*log
(c)**2*log(e + f*x)**2 - 12*b**4*p**2*q**2*x*log(c)**2*log(e + f*x) + 12*b**4*p**2*q**2*x*log(c)**2 + 4*b**4*p
*q**4*x*log(d)**3*log(e + f*x) - 4*b**4*p*q**4*x*log(d)**3 + 12*b**4*p*q**3*x*log(c)*log(d)**2*log(e + f*x) -
12*b**4*p*q**3*x*log(c)*log(d)**2 + 12*b**4*p*q**2*x*log(c)**2*log(d)*log(e + f*x) - 12*b**4*p*q**2*x*log(c)**
2*log(d) + 4*b**4*p*q*x*log(c)**3*log(e + f*x) - 4*b**4*p*q*x*log(c)**3 + b**4*q**4*x*log(d)**4 + 4*b**4*q**3*
x*log(c)*log(d)**3 + 6*b**4*q**2*x*log(c)**2*log(d)**2 + 4*b**4*q*x*log(c)**3*log(d) + b**4*x*log(c)**4, Ne(f,
0)), (x*(a + b*log(c*(d*e**p)**q))**4, True))

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