3.5.43 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^4}{(g+h x)^2} \, dx\) [443]

Optimal. Leaf size=274 \[ \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {12 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {24 b^3 f p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac {24 b^4 f p^4 q^4 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)} \]

[Out]

(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^4/(-e*h+f*g)/(h*x+g)-4*b*f*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))^3*ln(f*(h*x+g)/
(-e*h+f*g))/h/(-e*h+f*g)-12*b^2*f*p^2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h/(-e
*h+f*g)+24*b^3*f*p^3*q^3*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(3,-h*(f*x+e)/(-e*h+f*g))/h/(-e*h+f*g)-24*b^4*f*p^
4*q^4*polylog(4,-h*(f*x+e)/(-e*h+f*g))/h/(-e*h+f*g)

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Rubi [A]
time = 0.35, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2444, 2443, 2481, 2421, 2430, 6724, 2495} \begin {gather*} \frac {24 b^3 f p^3 q^3 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)}-\frac {12 b^2 f p^2 q^2 \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h (f g-e h)}-\frac {24 b^4 f p^4 q^4 \text {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac {4 b f p q \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h (f g-e h)}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x)^2,x]

[Out]

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2443

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((
f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Dist[b*e*n*(p/g), Int[Log[(e*(f + g*x))/(e*f - d
*g)]*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2444

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[(d + e
*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x))), x] - Dist[b*e*n*(p/(e*f - d*g)), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2495

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]]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4}{(g+h x)^2} \, dx,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 g-e h) (g+h x)}-\text {Subst}\left (\frac {(4 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{f g-e h},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 g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}+\text {Subst}\left (\frac {\left (12 b^2 f^2 p^2 q^2\right ) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h (f g-e h)},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 g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}+\text {Subst}\left (\frac {\left (12 b^2 f p^2 q^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac {f \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)},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 g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {12 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\text {Subst}\left (\frac {\left (24 b^3 f p^3 q^3\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)},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 g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {12 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {24 b^3 f p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\text {Subst}\left (\frac {\left (24 b^4 f p^4 q^4\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)},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 g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {12 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {24 b^3 f p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac {24 b^4 f p^4 q^4 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1301\) vs. \(2(274)=548\).
time = 0.32, size = 1301, normalized size = 4.75 \begin {gather*} \frac {a^4 f g-a^4 e h-4 a^3 b f g p q \log (e+f x)-4 a^3 b f h p q x \log (e+f x)+6 a^2 b^2 f g p^2 q^2 \log ^2(e+f x)+6 a^2 b^2 f h p^2 q^2 x \log ^2(e+f x)-4 a b^3 f g p^3 q^3 \log ^3(e+f x)-4 a b^3 f h p^3 q^3 x \log ^3(e+f x)+b^4 f g p^4 q^4 \log ^4(e+f x)+b^4 f h p^4 q^4 x \log ^4(e+f x)+4 a^3 b f g \log \left (c \left (d (e+f x)^p\right )^q\right )-4 a^3 b e h \log \left (c \left (d (e+f x)^p\right )^q\right )-12 a^2 b^2 f g p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-12 a^2 b^2 f h p q x \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )+12 a b^3 f g p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )+12 a b^3 f h p^2 q^2 x \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f g p^3 q^3 \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f h p^3 q^3 x \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )+6 a^2 b^2 f g \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-6 a^2 b^2 e h \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-12 a b^3 f g p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-12 a b^3 f h p q x \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+6 b^4 f g p^2 q^2 \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+6 b^4 f h p^2 q^2 x \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+4 a b^3 f g \log ^3\left (c \left (d (e+f x)^p\right )^q\right )-4 a b^3 e h \log ^3\left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f g p q \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f h p q x \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right )+b^4 f g \log ^4\left (c \left (d (e+f x)^p\right )^q\right )-b^4 e h \log ^4\left (c \left (d (e+f x)^p\right )^q\right )+4 a^3 b f g p q \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 a^3 b f h p q x \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 a^2 b^2 f g p q \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 a^2 b^2 f h p q x \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 a b^3 f g p q \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 a b^3 f h p q x \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 b^4 f g p q \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 b^4 f h p q x \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 b^2 f p^2 q^2 (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-24 b^3 f p^3 q^3 (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (\frac {h (e+f x)}{-f g+e h}\right )+24 b^4 f g p^4 q^4 \text {Li}_4\left (\frac {h (e+f x)}{-f g+e h}\right )+24 b^4 f h p^4 q^4 x \text {Li}_4\left (\frac {h (e+f x)}{-f g+e h}\right )}{h (-f g+e h) (g+h x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x)^2,x]

[Out]

(a^4*f*g - a^4*e*h - 4*a^3*b*f*g*p*q*Log[e + f*x] - 4*a^3*b*f*h*p*q*x*Log[e + f*x] + 6*a^2*b^2*f*g*p^2*q^2*Log
[e + f*x]^2 + 6*a^2*b^2*f*h*p^2*q^2*x*Log[e + f*x]^2 - 4*a*b^3*f*g*p^3*q^3*Log[e + f*x]^3 - 4*a*b^3*f*h*p^3*q^
3*x*Log[e + f*x]^3 + b^4*f*g*p^4*q^4*Log[e + f*x]^4 + b^4*f*h*p^4*q^4*x*Log[e + f*x]^4 + 4*a^3*b*f*g*Log[c*(d*
(e + f*x)^p)^q] - 4*a^3*b*e*h*Log[c*(d*(e + f*x)^p)^q] - 12*a^2*b^2*f*g*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)
^q] - 12*a^2*b^2*f*h*p*q*x*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] + 12*a*b^3*f*g*p^2*q^2*Log[e + f*x]^2*Log[c*(
d*(e + f*x)^p)^q] + 12*a*b^3*f*h*p^2*q^2*x*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q] - 4*b^4*f*g*p^3*q^3*Log[e +
f*x]^3*Log[c*(d*(e + f*x)^p)^q] - 4*b^4*f*h*p^3*q^3*x*Log[e + f*x]^3*Log[c*(d*(e + f*x)^p)^q] + 6*a^2*b^2*f*g
*Log[c*(d*(e + f*x)^p)^q]^2 - 6*a^2*b^2*e*h*Log[c*(d*(e + f*x)^p)^q]^2 - 12*a*b^3*f*g*p*q*Log[e + f*x]*Log[c*(
d*(e + f*x)^p)^q]^2 - 12*a*b^3*f*h*p*q*x*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2 + 6*b^4*f*g*p^2*q^2*Log[e + f
*x]^2*Log[c*(d*(e + f*x)^p)^q]^2 + 6*b^4*f*h*p^2*q^2*x*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]^2 + 4*a*b^3*f*g
*Log[c*(d*(e + f*x)^p)^q]^3 - 4*a*b^3*e*h*Log[c*(d*(e + f*x)^p)^q]^3 - 4*b^4*f*g*p*q*Log[e + f*x]*Log[c*(d*(e
+ f*x)^p)^q]^3 - 4*b^4*f*h*p*q*x*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^3 + b^4*f*g*Log[c*(d*(e + f*x)^p)^q]^4
- b^4*e*h*Log[c*(d*(e + f*x)^p)^q]^4 + 4*a^3*b*f*g*p*q*Log[(f*(g + h*x))/(f*g - e*h)] + 4*a^3*b*f*h*p*q*x*Log[
(f*(g + h*x))/(f*g - e*h)] + 12*a^2*b^2*f*g*p*q*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a
^2*b^2*f*h*p*q*x*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a*b^3*f*g*p*q*Log[c*(d*(e + f*x)
^p)^q]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a*b^3*f*h*p*q*x*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*x))/(f*g
- e*h)] + 4*b^4*f*g*p*q*Log[c*(d*(e + f*x)^p)^q]^3*Log[(f*(g + h*x))/(f*g - e*h)] + 4*b^4*f*h*p*q*x*Log[c*(d*
(e + f*x)^p)^q]^3*Log[(f*(g + h*x))/(f*g - e*h)] + 12*b^2*f*p^2*q^2*(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])
^2*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - 24*b^3*f*p^3*q^3*(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])*Poly
Log[3, (h*(e + f*x))/(-(f*g) + e*h)] + 24*b^4*f*g*p^4*q^4*PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)] + 24*b^4*f*
h*p^4*q^4*x*PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)])/(h*(-(f*g) + e*h)*(g + h*x))

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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{4}}{\left (h x +g \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{4}}{\left (g + h x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**4/(h*x+g)**2,x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**4/(g + h*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^p*d)^q*c) + a)^4/(h*x + g)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^4}{{\left (g+h\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d*(e + f*x)^p)^q))^4/(g + h*x)^2,x)

[Out]

int((a + b*log(c*(d*(e + f*x)^p)^q))^4/(g + h*x)^2, x)

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