\(\int \sqrt {a+b \log (c (d+e x)^n)} \, dx\) [26]

Optimal result

Integrand size = 18, antiderivative size = 111 \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=-\frac {\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e}+\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e} \]

[Out]

-1/2*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*Pi^(1/2)/e/exp(a/b/n)/((c*(e*x+
d)^n)^(1/n))+(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(1/2)/e

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2436, 2333, 2337, 2211, 2235} \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e} \]

[In]

Int[Sqrt[a + b*Log[c*(d + e*x)^n]],x]

[Out]

-1/2*(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(e*E^(a/(b*n)
)*(c*(d + e*x)^n)^n^(-1)) + ((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[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]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e} \\ & = \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {(b n) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e} \\ & = \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\left (b (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 e} \\ & = \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{e} \\ & = -\frac {\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e}+\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\frac {(d+e x) \left (-\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{2 e} \]

[In]

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

[Out]

((d + e*x)*(-((Sqrt[b]*Sqrt[n]*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(
c*(d + e*x)^n)^n^(-1))) + 2*Sqrt[a + b*Log[c*(d + e*x)^n]]))/(2*e)

Maple [F]

\[\int \sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]

[In]

integrate((a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

Integral(sqrt(a + b*log(c*(d + e*x)**n)), x)

Maxima [F]

\[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*log((e*x + d)^n*c) + a), x)

Giac [F]

\[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*log((e*x + d)^n*c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

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

[Out]

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