3.117 \(\int \log (d (e+f \sqrt{x})^k) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=209 \[ \frac{2 b e^2 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^2}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{b e^2 k n \log \left (e+f \sqrt{x}\right )}{f^2}+\frac{2 b e^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{3 b e k n \sqrt{x}}{f}+b k n x \]

[Out]

(-3*b*e*k*n*Sqrt[x])/f + b*k*n*x + (b*e^2*k*n*Log[e + f*Sqrt[x]])/f^2 - b*n*x*Log[d*(e + f*Sqrt[x])^k] + (2*b*
e^2*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + (e*k*Sqrt[x]*(a + b*Log[c*x^n]))/f - (k*x*(a + b*Log[c
*x^n]))/2 - (e^2*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + x*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])
+ (2*b*e^2*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2

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Rubi [A]  time = 0.149565, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2448, 266, 43, 2370, 2454, 2394, 2315} \[ \frac{2 b e^2 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^2}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{b e^2 k n \log \left (e+f \sqrt{x}\right )}{f^2}+\frac{2 b e^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{3 b e k n \sqrt{x}}{f}+b k n x \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]),x]

[Out]

(-3*b*e*k*n*Sqrt[x])/f + b*k*n*x + (b*e^2*k*n*Log[e + f*Sqrt[x]])/f^2 - b*n*x*Log[d*(e + f*Sqrt[x])^k] + (2*b*
e^2*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + (e*k*Sqrt[x]*(a + b*Log[c*x^n]))/f - (k*x*(a + b*Log[c
*x^n]))/2 - (e^2*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + x*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])
+ (2*b*e^2*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2

Rule 2448

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2394

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{k}{2}+\frac{e k}{f \sqrt{x}}-\frac{e^2 k \log \left (e+f \sqrt{x}\right )}{f^2 x}+\log \left (d \left (e+f \sqrt{x}\right )^k\right )\right ) \, dx\\ &=-\frac{2 b e k n \sqrt{x}}{f}+\frac{1}{2} b k n x+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \, dx+\frac{\left (b e^2 k n\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{f^2}\\ &=-\frac{2 b e k n \sqrt{x}}{f}+\frac{1}{2} b k n x-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\left (2 b e^2 k n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{f^2}+\frac{1}{2} (b f k n) \int \frac{\sqrt{x}}{e+f \sqrt{x}} \, dx\\ &=-\frac{2 b e k n \sqrt{x}}{f}+\frac{1}{2} b k n x-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 b e^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (2 b e^2 k n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{f}+(b f k n) \operatorname{Subst}\left (\int \frac{x^2}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b e k n \sqrt{x}}{f}+\frac{1}{2} b k n x-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 b e^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^2}+(b f k n) \operatorname{Subst}\left (\int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{3 b e k n \sqrt{x}}{f}+b k n x+\frac{b e^2 k n \log \left (e+f \sqrt{x}\right )}{f^2}-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 b e^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{e k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^2}\\ \end{align*}

Mathematica [A]  time = 0.232269, size = 218, normalized size = 1.04 \[ -\frac{2 b e^2 k n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{e^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)-b n\right )}{f^2}+a x \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{a e k \sqrt{x}}{f}-\frac{a k x}{2}+b x \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{b e k \sqrt{x} \log \left (c x^n\right )}{f}-\frac{1}{2} b k x \log \left (c x^n\right )-b n x \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{b e^2 k n \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )}{f^2}-\frac{3 b e k n \sqrt{x}}{f}+b k n x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]),x]

[Out]

(a*e*k*Sqrt[x])/f - (3*b*e*k*n*Sqrt[x])/f - (a*k*x)/2 + b*k*n*x + a*x*Log[d*(e + f*Sqrt[x])^k] - b*n*x*Log[d*(
e + f*Sqrt[x])^k] - (b*e^2*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x])/f^2 + (b*e*k*Sqrt[x]*Log[c*x^n])/f - (b*k*x*Log[
c*x^n])/2 + b*x*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] - (e^2*k*Log[e + f*Sqrt[x]]*(a - b*n - b*n*Log[x] + b*Log[
c*x^n]))/f^2 - (2*b*e^2*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/f^2

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f\sqrt{x} \right ) ^{k} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k),x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{9 \, b e x \log \left (d\right ) \log \left (x^{n}\right ) + 9 \,{\left (a e \log \left (d\right ) -{\left (e n \log \left (d\right ) - e \log \left (c\right ) \log \left (d\right )\right )} b\right )} x + 9 \,{\left (b e x \log \left (x^{n}\right ) -{\left ({\left (e n - e \log \left (c\right )\right )} b - a e\right )} x\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) - \frac{3 \, b f k x^{2} \log \left (x^{n}\right ) +{\left (3 \, a f k -{\left (5 \, f k n - 3 \, f k \log \left (c\right )\right )} b\right )} x^{2}}{\sqrt{x}}}{9 \, e} + \int \frac{b f^{2} k x \log \left (x^{n}\right ) +{\left (a f^{2} k -{\left (f^{2} k n - f^{2} k \log \left (c\right )\right )} b\right )} x}{2 \,{\left (e f \sqrt{x} + e^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k),x, algorithm="maxima")

[Out]

1/9*(9*b*e*x*log(d)*log(x^n) + 9*(a*e*log(d) - (e*n*log(d) - e*log(c)*log(d))*b)*x + 9*(b*e*x*log(x^n) - ((e*n
- e*log(c))*b - a*e)*x)*log((f*sqrt(x) + e)^k) - (3*b*f*k*x^2*log(x^n) + (3*a*f*k - (5*f*k*n - 3*f*k*log(c))*
b)*x^2)/sqrt(x))/e + integrate(1/2*(b*f^2*k*x*log(x^n) + (a*f^2*k - (f^2*k*n - f^2*k*log(c))*b)*x)/(e*f*sqrt(x
) + e^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k),x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d), 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((a+b*ln(c*x**n))*ln(d*(e+f*x**(1/2))**k),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d), x)