Integrand size = 14, antiderivative size = 85 \[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=-\frac {1}{2} \sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+x \sqrt {a+b \log \left (c x^n\right )} \] Output:
-1/2*b^(1/2)*n^(1/2)*Pi^(1/2)*x*erfi((a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2) )/exp(a/b/n)/((c*x^n)^(1/n))+x*(a+b*ln(c*x^n))^(1/2)
Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=-\frac {1}{2} \sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+x \sqrt {a+b \log \left (c x^n\right )} \] Input:
Integrate[Sqrt[a + b*Log[c*x^n]],x]
Output:
-1/2*(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt [n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) + x*Sqrt[a + b*Log[c*x^n]]
Time = 0.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2733, 2737, 2611, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a+b \log \left (c x^n\right )} \, dx\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} b n \int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}}dx\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} b x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a+b \log \left (c x^n\right )}}d\log \left (c x^n\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle x \sqrt {a+b \log \left (c x^n\right )}-x \left (c x^n\right )^{-1/n} \int e^{\frac {a+b \log \left (c x^n\right )}{b n}-\frac {a}{b n}}d\sqrt {a+b \log \left (c x^n\right )}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {n} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )\) |
Input:
Int[Sqrt[a + b*Log[c*x^n]],x]
Output:
-1/2*(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt [n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) + x*Sqrt[a + b*Log[c*x^n]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x ^n)^(1/n)) Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ [{a, b, c, n, p}, x]
\[\int \sqrt {a +b \ln \left (c \,x^{n}\right )}d x\]
Input:
int((a+b*ln(c*x^n))^(1/2),x)
Output:
int((a+b*ln(c*x^n))^(1/2),x)
Exception generated. \[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*log(c*x^n))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=\int \sqrt {a + b \log {\left (c x^{n} \right )}}\, dx \] Input:
integrate((a+b*ln(c*x**n))**(1/2),x)
Output:
Integral(sqrt(a + b*log(c*x**n)), x)
\[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=\int { \sqrt {b \log \left (c x^{n}\right ) + a} \,d x } \] Input:
integrate((a+b*log(c*x^n))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*log(c*x^n) + a), x)
\[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=\int { \sqrt {b \log \left (c x^{n}\right ) + a} \,d x } \] Input:
integrate((a+b*log(c*x^n))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*log(c*x^n) + a), x)
Timed out. \[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=\int \sqrt {a+b\,\ln \left (c\,x^n\right )} \,d x \] Input:
int((a + b*log(c*x^n))^(1/2),x)
Output:
int((a + b*log(c*x^n))^(1/2), x)
\[ \int \sqrt {a+b \log \left (c x^n\right )} \, dx=\frac {2 \sqrt {\mathrm {log}\left (x^{n} c \right ) b +a}\, a x +2 \left (\int \frac {\sqrt {\mathrm {log}\left (x^{n} c \right ) b +a}\, \mathrm {log}\left (x^{n} c \right )}{2 \,\mathrm {log}\left (x^{n} c \right ) a b +\mathrm {log}\left (x^{n} c \right ) b^{2} n +2 a^{2}+a b n}d x \right ) a \,b^{2} n +\left (\int \frac {\sqrt {\mathrm {log}\left (x^{n} c \right ) b +a}\, \mathrm {log}\left (x^{n} c \right )}{2 \,\mathrm {log}\left (x^{n} c \right ) a b +\mathrm {log}\left (x^{n} c \right ) b^{2} n +2 a^{2}+a b n}d x \right ) b^{3} n^{2}}{b n +2 a} \] Input:
int((a+b*log(c*x^n))^(1/2),x)
Output:
(2*sqrt(log(x**n*c)*b + a)*a*x + 2*int((sqrt(log(x**n*c)*b + a)*log(x**n*c ))/(2*log(x**n*c)*a*b + log(x**n*c)*b**2*n + 2*a**2 + a*b*n),x)*a*b**2*n + int((sqrt(log(x**n*c)*b + a)*log(x**n*c))/(2*log(x**n*c)*a*b + log(x**n*c )*b**2*n + 2*a**2 + a*b*n),x)*b**3*n**2)/(2*a + b*n)