3.413 \(\int \sqrt{a+b \log (c (d (e+f x)^m)^n)} \, dx\)

Optimal. Leaf size=139 \[ \frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{m} \sqrt{n} (e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{2 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.219506, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2389, 2296, 2300, 2180, 2204, 2445} \[ \frac{(e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{m} \sqrt{n} (e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2389

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]

Rule 2296

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 2300

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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

Mathematica [A]  time = 0.062111, size = 134, normalized size = 0.96 \[ \frac{(e+f x) \left (2 \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}-\sqrt{\pi } \sqrt{b} \sqrt{m} \sqrt{n} e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.081, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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