3.6.37 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^3}{g+h x} \, dx\) [537]

Optimal. Leaf size=177 \[ \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2443, 2481, 2421, 2430, 6724, 2495} \begin {gather*} -\frac {6 b^2 p^2 q^2 \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}+\frac {3 b p q \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}+\frac {6 b^3 p^3 q^3 \text {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h + (3*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^
q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (6*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3,
-((h*(e + f*x))/(f*g - e*h))])/h + (6*b^3*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/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 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 )^3}{g+h x} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\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}-\text {Subst}\left (\frac {(3 b f p q) \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},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\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}-\text {Subst}\left (\frac {(3 b p q) \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},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\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}+\frac {3 b p q \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}-\text {Subst}\left (\frac {\left (6 b^2 p^2 q^2\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},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\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}+\frac {3 b p q \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}-\frac {6 b^2 p^2 q^2 \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}+\text {Subst}\left (\frac {\left (6 b^3 p^3 q^3\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},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\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}+\frac {3 b p q \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}-\frac {6 b^2 p^2 q^2 \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}+\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(646\) vs. \(2(177)=354\).
time = 0.12, size = 646, normalized size = 3.65 \begin {gather*} \frac {a^3 \log (g+h x)-3 a^2 b p q \log (e+f x) \log (g+h x)+3 a b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^3 p^3 q^3 \log ^3(e+f x) \log (g+h x)+3 a^2 b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-6 a b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 b^3 p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 a b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 b^3 p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b^3 \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 a^2 b p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 a b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+b^3 p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+6 a b^2 p q \log (e+f 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 )-3 b^3 p^2 q^2 \log ^2(e+f 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 )+3 b^3 p q \log (e+f 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 )+3 b p q \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 )-6 b^2 p^2 q^2 \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 )+6 b^3 p^3 q^3 \text {Li}_4\left (\frac {h (e+f x)}{-f g+e h}\right )}{h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{3}}{h x +g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^3/(h*x+g),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^3/(h*x+g),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))^3/(h*x+g),x, algorithm="maxima")

[Out]

a^3*log(h*x + g)/h + integrate((b^3*log(((f*x + e)^p)^q)^3 + 3*(q*log(d) + log(c))*a^2*b + 3*(q^2*log(d)^2 + 2
*q*log(c)*log(d) + log(c)^2)*a*b^2 + (q^3*log(d)^3 + 3*q^2*log(c)*log(d)^2 + 3*q*log(c)^2*log(d) + log(c)^3)*b
^3 + 3*((q*log(d) + log(c))*b^3 + a*b^2)*log(((f*x + e)^p)^q)^2 + 3*(2*(q*log(d) + log(c))*a*b^2 + (q^2*log(d)
^2 + 2*q*log(c)*log(d) + log(c)^2)*b^3 + a^2*b)*log(((f*x + e)^p)^q))/(h*x + g), 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))^3/(h*x+g),x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**3/(g + h*x), 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))^3/(h*x+g),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{g+h\,x} \,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))^3/(g + h*x),x)

[Out]

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

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