3.58 \(\int \frac{\log (i (j (h x)^t)^u) \log (e (f (a+b x)^p (c+d x)^q)^r)}{x} \, dx\)

Optimal. Leaf size=194 \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right )+\frac{\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u} \]

[Out]

-(p*r*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (b*x)/a])/(2*t*u) + (Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x
)^q)^r])/(2*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (d*x)/c])/(2*t*u) - p*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2,
-((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2, -((d*x)/c)] + p*r*t*u*PolyLog[3, -((b*x)/a)] + q*r*t*u*Poly
Log[3, -((d*x)/c)]

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Rubi [A]  time = 0.603361, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {2499, 2317, 2374, 6589, 2445} \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right )+\frac{\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(p*r*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (b*x)/a])/(2*t*u) + (Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x
)^q)^r])/(2*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (d*x)/c])/(2*t*u) - p*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2,
-((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2, -((d*x)/c)] + p*r*t*u*PolyLog[3, -((b*x)/a)] + q*r*t*u*Poly
Log[3, -((d*x)/c)]

Rule 2499

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

Rule 2317

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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]

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 \frac{\log \left (58 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\log \left (58 j^u (h x)^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=\operatorname{Subst}\left (\operatorname{Subst}\left (\int \frac{\log \left (58 h^{t u} j^u x^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=\frac{\log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(b p r) \int \frac{\log ^2\left (58 h^{t u} j^u x^{t u}\right )}{a+b x} \, dx}{2 t u},58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(d q r) \int \frac{\log ^2\left (58 h^{t u} j^u x^{t u}\right )}{c+d x} \, dx}{2 t u},58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{2 t u}+\frac{\log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{q r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{2 t u}+\operatorname{Subst}\left (\operatorname{Subst}\left ((p r) \int \frac{\log \left (58 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{b x}{a}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((q r) \int \frac{\log \left (58 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{d x}{c}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{2 t u}+\frac{\log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{q r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{2 t u}-p r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((p r t u) \int \frac{\text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((q r t u) \int \frac{\text{Li}_2\left (-\frac{d x}{c}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{2 t u}+\frac{\log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{q r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{2 t u}-p r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+p r t u \text{Li}_3\left (-\frac{b x}{a}\right )+q r t u \text{Li}_3\left (-\frac{d x}{c}\right )\\ \end{align*}

Mathematica [B]  time = 0.448249, size = 451, normalized size = 2.32 \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right )+\log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac{1}{2} t u \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-t u \log (x) \log (h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (h x) \log (a+b x) \log \left (i \left (j (h x)^t\right )^u\right )-p r \log (h x) \log \left (\frac{b x}{a}+1\right ) \log \left (i \left (j (h x)^t\right )^u\right )-p r \log (x) \log (a+b x) \log \left (i \left (j (h x)^t\right )^u\right )-p r t u \log ^2(h x) \log (a+b x)+\frac{1}{2} p r t u \log ^2(h x) \log \left (\frac{b x}{a}+1\right )+p r t u \log (x) \log (h x) \log (a+b x)+q r \log (h x) \log (c+d x) \log \left (i \left (j (h x)^t\right )^u\right )-q r \log (h x) \log \left (\frac{d x}{c}+1\right ) \log \left (i \left (j (h x)^t\right )^u\right )-q r \log (x) \log (c+d x) \log \left (i \left (j (h x)^t\right )^u\right )-q r t u \log ^2(h x) \log (c+d x)+\frac{1}{2} q r t u \log ^2(h x) \log \left (\frac{d x}{c}+1\right )+q r t u \log (x) \log (h x) \log (c+d x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

p*r*t*u*Log[x]*Log[h*x]*Log[a + b*x] - p*r*t*u*Log[h*x]^2*Log[a + b*x] - p*r*Log[x]*Log[i*(j*(h*x)^t)^u]*Log[a
+ b*x] + p*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[a + b*x] + (p*r*t*u*Log[h*x]^2*Log[1 + (b*x)/a])/2 - p*r*Log[h
*x]*Log[i*(j*(h*x)^t)^u]*Log[1 + (b*x)/a] + q*r*t*u*Log[x]*Log[h*x]*Log[c + d*x] - q*r*t*u*Log[h*x]^2*Log[c +
d*x] - q*r*Log[x]*Log[i*(j*(h*x)^t)^u]*Log[c + d*x] + q*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[c + d*x] - t*u*Log
[x]*Log[h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (t*u*Log[h*x]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/2 +
Log[x]*Log[i*(j*(h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (q*r*t*u*Log[h*x]^2*Log[1 + (d*x)/c])/2 - q
*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[1 + (d*x)/c] - p*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2, -((b*x)/a)] - q*r*Log[
i*(j*(h*x)^t)^u]*PolyLog[2, -((d*x)/c)] + p*r*t*u*PolyLog[3, -((b*x)/a)] + q*r*t*u*PolyLog[3, -((d*x)/c)]

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Maple [F]  time = 1.784, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( i \left ( j \left ( hx \right ) ^{t} \right ) ^{u} \right ) \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="maxima")

[Out]

-1/2*(t*u*log(x)^2 - 2*(log((h^t)^u) + log(i) + log(j^u))*log(x) - 2*log(x)*log((x^t)^u))*log(((b*x + a)^p)^r)
- 1/2*(t*u*log(x)^2 - 2*(log((h^t)^u) + log(i) + log(j^u))*log(x) - 2*log(x)*log((x^t)^u))*log(((d*x + c)^q)^
r) - integrate(-1/2*(2*((log((h^t)^u) + log(i) + log(j^u))*log(e) + (log((h^t)^u) + log(i) + log(j^u))*log(f^r
))*b*d*x^2 + 2*((log((h^t)^u) + log(i) + log(j^u))*log(e) + (log((h^t)^u) + log(i) + log(j^u))*log(f^r))*a*c +
((p*r*t*u + q*r*t*u)*b*d*x^2 + (b*c*p*r*t*u + a*d*q*r*t*u)*x)*log(x)^2 + 2*(((log((h^t)^u) + log(i) + log(j^u
))*log(e) + (log((h^t)^u) + log(i) + log(j^u))*log(f^r))*b*c + ((log((h^t)^u) + log(i) + log(j^u))*log(e) + (l
og((h^t)^u) + log(i) + log(j^u))*log(f^r))*a*d)*x - 2*(((p*r + q*r)*log((h^t)^u) + (p*r + q*r)*log(i) + (p*r +
q*r)*log(j^u))*b*d*x^2 + ((p*r*log((h^t)^u) + p*r*log(i) + p*r*log(j^u))*b*c + (q*r*log((h^t)^u) + q*r*log(i)
+ q*r*log(j^u))*a*d)*x)*log(x) + 2*(b*d*x^2*(log(e) + log(f^r)) + a*c*(log(e) + log(f^r)) + (b*c*(log(e) + lo
g(f^r)) + a*d*(log(e) + log(f^r)))*x - ((p*r + q*r)*b*d*x^2 + (b*c*p*r + a*d*q*r)*x)*log(x))*log((x^t)^u))/(b*
d*x^3 + a*c*x + (b*c + a*d)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)*log(((h*x)^t*j)^u*i)/x, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(i*(j*(h*x)**t)**u)*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)*log(((h*x)^t*j)^u*i)/x, x)