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

Optimal. Leaf size=283 \[ -\frac{4 b e^3 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 f^3}+\frac{2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{16 b e^2 k n \sqrt{x}}{9 f^2}-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right )}{9 f^3}-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^3}-\frac{5 b e k n x}{9 f}+\frac{8}{27} b k n x^{3/2} \]

[Out]

(16*b*e^2*k*n*Sqrt[x])/(9*f^2) - (5*b*e*k*n*x)/(9*f) + (8*b*k*n*x^(3/2))/27 - (4*b*e^3*k*n*Log[e + f*Sqrt[x]])
/(9*f^3) - (4*b*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k])/9 - (4*b*e^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])
/(3*f^3) - (2*e^2*k*Sqrt[x]*(a + b*Log[c*x^n]))/(3*f^2) + (e*k*x*(a + b*Log[c*x^n]))/(3*f) - (2*k*x^(3/2)*(a +
b*Log[c*x^n]))/9 + (2*e^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^3) + (2*x^(3/2)*Log[d*(e + f*Sqrt[x])
^k]*(a + b*Log[c*x^n]))/3 - (4*b*e^3*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^3)

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Rubi [A]  time = 0.228211, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2454, 2395, 43, 2376, 2394, 2315} \[ -\frac{4 b e^3 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 f^3}+\frac{2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac{4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{16 b e^2 k n \sqrt{x}}{9 f^2}-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right )}{9 f^3}-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^3}-\frac{5 b e k n x}{9 f}+\frac{8}{27} b k n x^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b*e^2*k*n*Sqrt[x])/(9*f^2) - (5*b*e*k*n*x)/(9*f) + (8*b*k*n*x^(3/2))/27 - (4*b*e^3*k*n*Log[e + f*Sqrt[x]])
/(9*f^3) - (4*b*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k])/9 - (4*b*e^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])
/(3*f^3) - (2*e^2*k*Sqrt[x]*(a + b*Log[c*x^n]))/(3*f^2) + (e*k*x*(a + b*Log[c*x^n]))/(3*f) - (2*k*x^(3/2)*(a +
b*Log[c*x^n]))/9 + (2*e^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^3) + (2*x^(3/2)*Log[d*(e + f*Sqrt[x])
^k]*(a + b*Log[c*x^n]))/3 - (4*b*e^3*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^3)

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 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

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 \sqrt{x} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac{2}{3} x^{3/2} \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{e k}{3 f}-\frac{2 e^2 k}{3 f^2 \sqrt{x}}-\frac{2 k \sqrt{x}}{9}+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right )}{3 f^3 x}+\frac{2}{3} \sqrt{x} \log \left (d \left (e+f \sqrt{x}\right )^k\right )\right ) \, dx\\ &=\frac{4 b e^2 k n \sqrt{x}}{3 f^2}-\frac{b e k n x}{3 f}+\frac{4}{27} b k n x^{3/2}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac{2}{3} x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (2 b n) \int \sqrt{x} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \, dx-\frac{\left (2 b e^3 k n\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{3 f^3}\\ &=\frac{4 b e^2 k n \sqrt{x}}{3 f^2}-\frac{b e k n x}{3 f}+\frac{4}{27} b k n x^{3/2}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac{2}{3} x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (4 b n) \operatorname{Subst}\left (\int x^2 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt{x}\right )-\frac{\left (4 b e^3 k n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{3 f^3}\\ &=\frac{4 b e^2 k n \sqrt{x}}{3 f^2}-\frac{b e k n x}{3 f}+\frac{4}{27} b k n x^{3/2}-\frac{4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^3}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac{2}{3} x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\left (4 b e^3 k n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{3 f^2}+\frac{1}{9} (4 b f k n) \operatorname{Subst}\left (\int \frac{x^3}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=\frac{4 b e^2 k n \sqrt{x}}{3 f^2}-\frac{b e k n x}{3 f}+\frac{4}{27} b k n x^{3/2}-\frac{4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^3}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac{2}{3} x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b e^3 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{3 f^3}+\frac{1}{9} (4 b f k n) \operatorname{Subst}\left (\int \left (\frac{e^2}{f^3}-\frac{e x}{f^2}+\frac{x^2}{f}-\frac{e^3}{f^3 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{16 b e^2 k n \sqrt{x}}{9 f^2}-\frac{5 b e k n x}{9 f}+\frac{8}{27} b k n x^{3/2}-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right )}{9 f^3}-\frac{4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{4 b e^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^3}-\frac{2 e^2 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac{2 e^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac{2}{3} x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b e^3 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{3 f^3}\\ \end{align*}

Mathematica [A]  time = 0.305871, size = 296, normalized size = 1.05 \[ \frac{36 b e^3 k n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )+6 e^3 k \log \left (e+f \sqrt{x}\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)-2 b n\right )+18 a f^3 x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )-18 a e^2 f k \sqrt{x}+9 a e f^2 k x-6 a f^3 k x^{3/2}+18 b f^3 x^{3/2} \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-18 b e^2 f k \sqrt{x} \log \left (c x^n\right )+9 b e f^2 k x \log \left (c x^n\right )-6 b f^3 k x^{3/2} \log \left (c x^n\right )-12 b f^3 n x^{3/2} \log \left (d \left (e+f \sqrt{x}\right )^k\right )+48 b e^2 f k n \sqrt{x}+18 b e^3 k n \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )-15 b e f^2 k n x+8 b f^3 k n x^{3/2}}{27 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*a*e^2*f*k*Sqrt[x] + 48*b*e^2*f*k*n*Sqrt[x] + 9*a*e*f^2*k*x - 15*b*e*f^2*k*n*x - 6*a*f^3*k*x^(3/2) + 8*b*f
^3*k*n*x^(3/2) + 18*a*f^3*x^(3/2)*Log[d*(e + f*Sqrt[x])^k] - 12*b*f^3*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k] + 18*
b*e^3*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 18*b*e^2*f*k*Sqrt[x]*Log[c*x^n] + 9*b*e*f^2*k*x*Log[c*x^n] - 6*b*f^3
*k*x^(3/2)*Log[c*x^n] + 18*b*f^3*x^(3/2)*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 6*e^3*k*Log[e + f*Sqrt[x]]*(3*a
- 2*b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + 36*b*e^3*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/(27*f^3)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \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(x^(1/2)*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k),x)

[Out]

int(x^(1/2)*(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{2}{9} \,{\left (3 \, b x \log \left (x^{n}\right ) -{\left (b{\left (2 \, n - 3 \, \log \left (c\right )\right )} - 3 \, a\right )} x\right )} \sqrt{x} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) + \frac{2}{9} \,{\left (3 \, b x \log \left (d\right ) \log \left (x^{n}\right ) -{\left ({\left (2 \, n \log \left (d\right ) - 3 \, \log \left (c\right ) \log \left (d\right )\right )} b - 3 \, a \log \left (d\right )\right )} x\right )} \sqrt{x} - \int \frac{3 \, b f k x \log \left (x^{n}\right ) +{\left (3 \, a f k -{\left (2 \, f k n - 3 \, f k \log \left (c\right )\right )} b\right )} x}{9 \,{\left (f \sqrt{x} + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sqrt{x} \log \left (c x^{n}\right ) + a \sqrt{x}\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(x^(1/2)*(a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k),x, algorithm="fricas")

[Out]

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