3.6.33 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^2}{(g+h x) (i+j x)} \, dx\) [533]

Optimal. Leaf size=288 \[ \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}-\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]

[Out]

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

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Rubi [A]
time = 0.60, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2465, 2443, 2481, 2421, 6724, 2495} \begin {gather*} \frac {2 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 )}{h i-g j}-\frac {2 b p q \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac {2 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac {2 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}+\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 )^2}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 )^2}{(g+h x) (533+j x)} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x) (533+j x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\int \left (\frac {h \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(533 h-g j) (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(533 h-g j) (533+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {h \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {j \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{533+j x} \, dx}{533 h-g j},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 )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{533 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (533+j x)}{533 f-e j}\right )}{533 h-g j}-\text {Subst}\left (\frac {(2 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (533+j x)}{533 f-e j}\right )}{e+f x} \, dx}{533 h-g j},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 )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{533 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (533+j x)}{533 f-e j}\right )}{533 h-g j}-\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \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 )}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac {f \left (\frac {533 f-e j}{f}+\frac {j x}{f}\right )}{533 f-e j}\right )}{x} \, dx,x,e+f x\right )}{533 h-g j},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 )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{533 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (533+j x)}{533 f-e j}\right )}{533 h-g j}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{533 h-g j}-\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{533 f-e j}\right )}{533 h-g j}-\text {Subst}\left (\frac {\left (2 b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (2 b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{533 f-e j}\right )}{x} \, dx,x,e+f x\right )}{533 h-g j},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 )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{533 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (533+j x)}{533 f-e j}\right )}{533 h-g j}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{533 h-g j}-\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{533 f-e j}\right )}{533 h-g j}-\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{533 h-g j}+\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{533 f-e j}\right )}{533 h-g j}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(652\) vs. \(2(288)=576\).
time = 0.19, size = 652, normalized size = 2.26 \begin {gather*} \frac {a^2 \log (g+h x)-2 a b p q \log (e+f x) \log (g+h x)+b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)+2 a b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+2 a b p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 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 )-a^2 \log (i+j x)+2 a b p q \log (e+f x) \log (i+j x)-b^2 p^2 q^2 \log ^2(e+f x) \log (i+j x)-2 a b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)+2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)-b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)-2 a b p q \log (e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )-2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )+2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (\frac {j (e+f x)}{-f i+e j}\right )-2 b^2 p^2 q^2 \text {Li}_3\left (\frac {h (e+f x)}{-f g+e h}\right )+2 b^2 p^2 q^2 \text {Li}_3\left (\frac {j (e+f x)}{-f i+e j}\right )}{h i-g j} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}}{\left (h x +g \right ) \left (j x +i \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)/(j*x+i),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))^2/(h*x+g)/(j*x+i),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

[Out]

integrate((b*log(((f*x + e)^p*d)^q*c) + a)^2/((h*x + g)*(j*x + I)), 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 )}^2}{\left (g+h\,x\right )\,\left (i+j\,x\right )} \,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))^2/((g + h*x)*(i + j*x)),x)

[Out]

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

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