3.47 \(\int x \log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=268 \[ -\frac{b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^4 f^4}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{3 b n x}{8 d^2 f^2}-\frac{5 b n \sqrt{x}}{4 d^3 f^3}+\frac{b n \log \left (d f \sqrt{x}+1\right )}{4 d^4 f^4}-\frac{7 b n x^{3/2}}{36 d f}-\frac{1}{4} b n x^2 \log \left (d f \sqrt{x}+1\right )+\frac{1}{8} b n x^2 \]

[Out]

(-5*b*n*Sqrt[x])/(4*d^3*f^3) + (3*b*n*x)/(8*d^2*f^2) - (7*b*n*x^(3/2))/(36*d*f) + (b*n*x^2)/8 + (b*n*Log[1 + d
*f*Sqrt[x]])/(4*d^4*f^4) - (b*n*x^2*Log[1 + d*f*Sqrt[x]])/4 + (Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) - (x*(a
+ b*Log[c*x^n]))/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n]))/(6*d*f) - (x^2*(a + b*Log[c*x^n]))/8 - (Log[1 + d
*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (b*n*PolyLog[2
, -(d*f*Sqrt[x])])/(d^4*f^4)

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Rubi [A]  time = 0.192155, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2454, 2395, 43, 2376, 2391} \[ -\frac{b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^4 f^4}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{3 b n x}{8 d^2 f^2}-\frac{5 b n \sqrt{x}}{4 d^3 f^3}+\frac{b n \log \left (d f \sqrt{x}+1\right )}{4 d^4 f^4}-\frac{7 b n x^{3/2}}{36 d f}-\frac{1}{4} b n x^2 \log \left (d f \sqrt{x}+1\right )+\frac{1}{8} b n x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]),x]

[Out]

(-5*b*n*Sqrt[x])/(4*d^3*f^3) + (3*b*n*x)/(8*d^2*f^2) - (7*b*n*x^(3/2))/(36*d*f) + (b*n*x^2)/8 + (b*n*Log[1 + d
*f*Sqrt[x]])/(4*d^4*f^4) - (b*n*x^2*Log[1 + d*f*Sqrt[x]])/4 + (Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) - (x*(a
+ b*Log[c*x^n]))/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n]))/(6*d*f) - (x^2*(a + b*Log[c*x^n]))/8 - (Log[1 + d
*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (b*n*PolyLog[2
, -(d*f*Sqrt[x])])/(d^4*f^4)

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 2391

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

Rubi steps

\begin{align*} \int x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{1}{4 d^2 f^2}+\frac{1}{2 d^3 f^3 \sqrt{x}}+\frac{\sqrt{x}}{6 d f}-\frac{x}{8}-\frac{\log \left (1+d f \sqrt{x}\right )}{2 d^4 f^4 x}+\frac{1}{2} x \log \left (1+d f \sqrt{x}\right )\right ) \, dx\\ &=-\frac{b n \sqrt{x}}{d^3 f^3}+\frac{b n x}{4 d^2 f^2}-\frac{b n x^{3/2}}{9 d f}+\frac{1}{16} b n x^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int x \log \left (1+d f \sqrt{x}\right ) \, dx+\frac{(b n) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{2 d^4 f^4}\\ &=-\frac{b n \sqrt{x}}{d^3 f^3}+\frac{b n x}{4 d^2 f^2}-\frac{b n x^{3/2}}{9 d f}+\frac{1}{16} b n x^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-(b n) \operatorname{Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt{x}\right )\\ &=-\frac{b n \sqrt{x}}{d^3 f^3}+\frac{b n x}{4 d^2 f^2}-\frac{b n x^{3/2}}{9 d f}+\frac{1}{16} b n x^2-\frac{1}{4} b n x^2 \log \left (1+d f \sqrt{x}\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{1}{4} (b d f n) \operatorname{Subst}\left (\int \frac{x^4}{1+d f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{b n \sqrt{x}}{d^3 f^3}+\frac{b n x}{4 d^2 f^2}-\frac{b n x^{3/2}}{9 d f}+\frac{1}{16} b n x^2-\frac{1}{4} b n x^2 \log \left (1+d f \sqrt{x}\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{1}{4} (b d f n) \operatorname{Subst}\left (\int \left (-\frac{1}{d^4 f^4}+\frac{x}{d^3 f^3}-\frac{x^2}{d^2 f^2}+\frac{x^3}{d f}+\frac{1}{d^4 f^4 (1+d f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{5 b n \sqrt{x}}{4 d^3 f^3}+\frac{3 b n x}{8 d^2 f^2}-\frac{7 b n x^{3/2}}{36 d f}+\frac{1}{8} b n x^2+\frac{b n \log \left (1+d f \sqrt{x}\right )}{4 d^4 f^4}-\frac{1}{4} b n x^2 \log \left (1+d f \sqrt{x}\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}\\ \end{align*}

Mathematica [A]  time = 0.210753, size = 191, normalized size = 0.71 \[ \frac{-72 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+18 \left (d^4 f^4 x^2-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (2 a+2 b \log \left (c x^n\right )-b n\right )+d f \sqrt{x} \left (-3 a \left (3 d^3 f^3 x^{3/2}-4 d^2 f^2 x+6 d f \sqrt{x}-12\right )-3 b \left (3 d^3 f^3 x^{3/2}-4 d^2 f^2 x+6 d f \sqrt{x}-12\right ) \log \left (c x^n\right )+b n \left (9 d^3 f^3 x^{3/2}-14 d^2 f^2 x+27 d f \sqrt{x}-90\right )\right )}{72 d^4 f^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]),x]

[Out]

(18*(-1 + d^4*f^4*x^2)*Log[1 + d*f*Sqrt[x]]*(2*a - b*n + 2*b*Log[c*x^n]) + d*f*Sqrt[x]*(-3*a*(-12 + 6*d*f*Sqrt
[x] - 4*d^2*f^2*x + 3*d^3*f^3*x^(3/2)) + b*n*(-90 + 27*d*f*Sqrt[x] - 14*d^2*f^2*x + 9*d^3*f^3*x^(3/2)) - 3*b*(
-12 + 6*d*f*Sqrt[x] - 4*d^2*f^2*x + 3*d^3*f^3*x^(3/2))*Log[c*x^n]) - 72*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(72*d^
4*f^4)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*x*log((f*sqrt(x) + 1/d)*d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x*log(c*x^n) + a*x)*log(d*f*sqrt(x) + 1), 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*(a+b*ln(c*x**n))*ln(d*(1/d+f*x**(1/2))),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 )} x \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*x*log((f*sqrt(x) + 1/d)*d), x)