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

Optimal. Leaf size=172 \[ -\frac{2 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+\frac{b n \log \left (d f \sqrt{x}+1\right )}{d^2 f^2}-\frac{3 b n \sqrt{x}}{d f}-b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+b n x \]

[Out]

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

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Rubi [A]  time = 0.104873, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2448, 266, 43, 2370, 2391} \[ -\frac{2 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+\frac{b n \log \left (d f \sqrt{x}+1\right )}{d^2 f^2}-\frac{3 b n \sqrt{x}}{d f}-b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+b n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*n*Sqrt[x])/(d*f) + b*n*x - b*n*x*Log[d*(d^(-1) + f*Sqrt[x])] + (b*n*Log[1 + d*f*Sqrt[x]])/(d^2*f^2) + (S
qrt[x]*(a + b*Log[c*x^n]))/(d*f) - (x*(a + b*Log[c*x^n]))/2 + x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])
- (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(d^2*f^2) - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*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 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 \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 )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \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 )}{d^2 f^2}-(b n) \int \left (-\frac{1}{2}+\frac{1}{d f \sqrt{x}}+\log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right )}{d^2 f^2 x}\right ) \, dx\\ &=-\frac{2 b n \sqrt{x}}{d f}+\frac{b n x}{2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \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 )}{d^2 f^2}-(b n) \int \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \, dx+\frac{(b n) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{d^2 f^2}\\ &=-\frac{2 b n \sqrt{x}}{d f}+\frac{b n x}{2}-b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \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 )}{d^2 f^2}-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\frac{1}{2} (b f n) \int \frac{\sqrt{x}}{\frac{1}{d}+f \sqrt{x}} \, dx\\ &=-\frac{2 b n \sqrt{x}}{d f}+\frac{b n x}{2}-b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \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 )}{d^2 f^2}-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+(b f n) \operatorname{Subst}\left (\int \frac{x^2}{\frac{1}{d}+f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b n \sqrt{x}}{d f}+\frac{b n x}{2}-b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \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 )}{d^2 f^2}-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+(b f n) \operatorname{Subst}\left (\int \left (-\frac{1}{d f^2}+\frac{x}{f}+\frac{1}{d f^2 (1+d f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{3 b n \sqrt{x}}{d f}+b n x-b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+\frac{b n \log \left (1+d f \sqrt{x}\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \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 )}{d^2 f^2}-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}\\ \end{align*}

Mathematica [A]  time = 0.157862, size = 117, normalized size = 0.68 \[ -\frac{4 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )-2 \left (d^2 f^2 x-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )-b n\right )+d f \sqrt{x} \left (a d f \sqrt{x}-2 a+b \left (d f \sqrt{x}-2\right ) \log \left (c x^n\right )-2 b d f n \sqrt{x}+6 b n\right )}{2 d^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2))),x)

[Out]

int((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*}{\left (b x \log \left (x^{n}\right ) -{\left (b{\left (n - \log \left (c\right )\right )} - a\right )} x\right )} \log \left (d f \sqrt{x} + 1\right ) - \frac{3 \, b d f x^{2} \log \left (x^{n}\right ) +{\left (3 \, a d f -{\left (5 \, d f n - 3 \, d f \log \left (c\right )\right )} b\right )} x^{2}}{9 \, \sqrt{x}} + \int \frac{b d^{2} f^{2} x \log \left (x^{n}\right ) +{\left (a d^{2} f^{2} -{\left (d^{2} f^{2} n - d^{2} f^{2} \log \left (c\right )\right )} b\right )} x}{2 \,{\left (d f \sqrt{x} + 1\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*(1/d+f*x^(1/2))),x, algorithm="maxima")

[Out]

(b*x*log(x^n) - (b*(n - log(c)) - a)*x)*log(d*f*sqrt(x) + 1) - 1/9*(3*b*d*f*x^2*log(x^n) + (3*a*d*f - (5*d*f*n
- 3*d*f*log(c))*b)*x^2)/sqrt(x) + integrate(1/2*(b*d^2*f^2*x*log(x^n) + (a*d^2*f^2 - (d^2*f^2*n - d^2*f^2*log
(c))*b)*x)/(d*f*sqrt(x) + 1), 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 (d f \sqrt{x} + 1\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*(1/d+f*x^(1/2))),x, algorithm="fricas")

[Out]

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