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

Optimal. Leaf size=172 \[ -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {3 b n \sqrt {x}}{d f}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+b n x \]

[Out]

b*n*x-1/2*x*(a+b*ln(c*x^n))-b*n*x*ln(d*(1/d+f*x^(1/2)))+x*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2)))+b*n*ln(1+d*f*x
^(1/2))/d^2/f^2-(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/d^2/f^2-2*b*n*polylog(2,-d*f*x^(1/2))/d^2/f^2-3*b*n*x^(1/2)/
d/f+(a+b*ln(c*x^n))*x^(1/2)/d/f

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Rubi [A]  time = 0.10, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 0.16, size = 117, normalized size = 0.68 \[ -\frac {-2 \left (d^2 f^2 x-1\right ) \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )-b n\right )+d f \sqrt {x} \left (a d f \sqrt {x}-2 a+b \left (d f \sqrt {x}-2\right ) \log \left (c x^n\right )-2 b d f n \sqrt {x}+6 b n\right )+4 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(-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*Sq
rt[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])])/(d^2*f^2)

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fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (d f \sqrt {x} + 1\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]

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)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (b x \log \left (x^{n}\right ) - {\left (b {\left (n - \log \relax (c)\right )} - a\right )} x\right )} \log \left (d f \sqrt {x} + 1\right ) - \frac {3 \, b d f x^{2} \log \left (x^{n}\right ) + {\left (3 \, a d f - {\left (5 \, d f n - 3 \, d f \log \relax (c)\right )} b\right )} x^{2}}{9 \, \sqrt {x}} + \int \frac {b d^{2} f^{2} x \log \left (x^{n}\right ) + {\left (a d^{2} f^{2} - {\left (d^{2} f^{2} n - d^{2} f^{2} \log \relax (c)\right )} b\right )} x}{2 \, {\left (d f \sqrt {x} + 1\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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