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 \[ \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-p r \text {Li}_2\left (-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {p r \log \left (\frac {b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}+p r t u \text {Li}_3\left (-\frac {b x}{a}\right )-q r \text {Li}_2\left (-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {q r \log \left (\frac {d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}+q r t u \text {Li}_3\left (-\frac {d x}{c}\right ) \]

[Out]

-1/2*p*r*ln(i*(j*(h*x)^t)^u)^2*ln(1+b*x/a)/t/u+1/2*ln(i*(j*(h*x)^t)^u)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/t/u-1
/2*q*r*ln(i*(j*(h*x)^t)^u)^2*ln(1+d*x/c)/t/u-p*r*ln(i*(j*(h*x)^t)^u)*polylog(2,-b*x/a)-q*r*ln(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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Rubi [A]  time = 0.60, antiderivative size = 194, normalized size of antiderivative = 1.00, 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 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 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]]

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 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]

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*}

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Mathematica [B]  time = 0.43, size = 451, normalized size = 2.32 \[ \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {1}{2} t u \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-t u \log (x) \log (h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \text {Li}_2\left (-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r \log (h x) \log (a+b x) \log \left (i \left (j (h x)^t\right )^u\right )-p r \log (h x) \log \left (\frac {b x}{a}+1\right ) \log \left (i \left (j (h x)^t\right )^u\right )-p r \log (x) \log (a+b x) \log \left (i \left (j (h x)^t\right )^u\right )-p r t u \log ^2(h x) \log (a+b x)+\frac {1}{2} p r t u \log ^2(h x) \log \left (\frac {b x}{a}+1\right )+p r t u \log (x) \log (h x) \log (a+b x)+p r t u \text {Li}_3\left (-\frac {b x}{a}\right )-q r \text {Li}_2\left (-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r \log (h x) \log (c+d x) \log \left (i \left (j (h x)^t\right )^u\right )-q r \log (h x) \log \left (\frac {d x}{c}+1\right ) \log \left (i \left (j (h x)^t\right )^u\right )-q r \log (x) \log (c+d x) \log \left (i \left (j (h x)^t\right )^u\right )-q r t u \log ^2(h x) \log (c+d x)+\frac {1}{2} q r t u \log ^2(h x) \log \left (\frac {d x}{c}+1\right )+q r t u \log (x) \log (h x) \log (c+d x)+q r t u \text {Li}_3\left (-\frac {d x}{c}\right ) \]

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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fricas [F]  time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x}\,{d x} \]

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)

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maple [F]  time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) \ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right )}{x}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (t u \log \relax (x)^{2} - 2 \, {\left (t u \log \relax (h) + u \log \relax (j) + \log \relax (i)\right )} \log \relax (x) - 2 \, \log \relax (x) \log \left ({\left (x^{t}\right )}^{u}\right )\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right ) - \frac {1}{2} \, {\left (t u \log \relax (x)^{2} - 2 \, {\left (t u \log \relax (h) + u \log \relax (j) + \log \relax (i)\right )} \log \relax (x) - 2 \, \log \relax (x) \log \left ({\left (x^{t}\right )}^{u}\right )\right )} \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right ) - \int -\frac {2 \, {\left ({\left (t u \log \relax (h) + u \log \relax (j) + \log \relax (i)\right )} \log \relax (e) + {\left (r t u \log \relax (h) + r u \log \relax (j) + r \log \relax (i)\right )} \log \relax (f)\right )} b d x^{2} + 2 \, {\left ({\left (t u \log \relax (h) + u \log \relax (j) + \log \relax (i)\right )} \log \relax (e) + {\left (r t u \log \relax (h) + r u \log \relax (j) + r \log \relax (i)\right )} \log \relax (f)\right )} a c + {\left ({\left (p r t u + q r t u\right )} b d x^{2} + {\left (b c p r t u + a d q r t u\right )} x\right )} \log \relax (x)^{2} + 2 \, {\left ({\left ({\left (t u \log \relax (h) + u \log \relax (j) + \log \relax (i)\right )} \log \relax (e) + {\left (r t u \log \relax (h) + r u \log \relax (j) + r \log \relax (i)\right )} \log \relax (f)\right )} b c + {\left ({\left (t u \log \relax (h) + u \log \relax (j) + \log \relax (i)\right )} \log \relax (e) + {\left (r t u \log \relax (h) + r u \log \relax (j) + r \log \relax (i)\right )} \log \relax (f)\right )} a d\right )} x + 2 \, {\left ({\left (r \log \relax (f) + \log \relax (e)\right )} b d x^{2} + {\left (r \log \relax (f) + \log \relax (e)\right )} a c + {\left ({\left (r \log \relax (f) + \log \relax (e)\right )} b c + {\left (r \log \relax (f) + \log \relax (e)\right )} a d\right )} x - {\left ({\left (p r + q r\right )} b d x^{2} + {\left (b c p r + a d q r\right )} x\right )} \log \relax (x)\right )} \log \left ({\left (x^{t}\right )}^{u}\right ) - 2 \, {\left ({\left ({\left (p r t u + q r t u\right )} \log \relax (h) + {\left (p r + q r\right )} \log \relax (i) + {\left (p r u + q r u\right )} \log \relax (j)\right )} b d x^{2} + {\left ({\left (p r t u \log \relax (h) + p r u \log \relax (j) + p r \log \relax (i)\right )} b c + {\left (q r t u \log \relax (h) + q r u \log \relax (j) + q r \log \relax (i)\right )} a d\right )} x\right )} \log \relax (x)}{2 \, {\left (b d x^{3} + a c x + {\left (b c + a d\right )} x^{2}\right )}}\,{d x} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\ln \left (i\,{\left (j\,{\left (h\,x\right )}^t\right )}^u\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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