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

Optimal. Leaf size=97 \[ \frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{4 b n \sqrt{e f-d g} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} g}-\frac{4 b n \sqrt{f+g x}}{g} \]

[Out]

(-4*b*n*Sqrt[f + g*x])/g + (4*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(Sqrt[e]*g
) + (2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/g

________________________________________________________________________________________

Rubi [A]  time = 0.0586838, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2395, 50, 63, 208} \[ \frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{4 b n \sqrt{e f-d g} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} g}-\frac{4 b n \sqrt{f+g x}}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*n*Sqrt[f + g*x])/g + (4*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(Sqrt[e]*g
) + (2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/g

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{f+g x}} \, dx &=\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{(2 b e n) \int \frac{\sqrt{f+g x}}{d+e x} \, dx}{g}\\ &=-\frac{4 b n \sqrt{f+g x}}{g}+\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{(2 b (e f-d g) n) \int \frac{1}{(d+e x) \sqrt{f+g x}} \, dx}{g}\\ &=-\frac{4 b n \sqrt{f+g x}}{g}+\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{(4 b (e f-d g) n) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e f}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g^2}\\ &=-\frac{4 b n \sqrt{f+g x}}{g}+\frac{4 b \sqrt{e f-d g} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} g}+\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}\\ \end{align*}

Mathematica [A]  time = 0.0799041, size = 83, normalized size = 0.86 \[ \frac{2 \left (\sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )-2 b n\right )+\frac{2 b n \sqrt{e f-d g} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e}}\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((2*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/Sqrt[e] + Sqrt[f + g*x]*(a - 2*b*
n + b*Log[c*(d + e*x)^n])))/g

________________________________________________________________________________________

Maple [A]  time = 0.331, size = 148, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{gx+f}a}{g}}+2\,{\frac{b\sqrt{gx+f}}{g}\ln \left ( c \left ({\frac{ \left ( gx+f \right ) e+dg-fe}{g}} \right ) ^{n} \right ) }-4\,{\frac{bn\sqrt{gx+f}}{g}}+4\,{\frac{bdn}{\sqrt{ \left ( dg-fe \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-fe \right ) e}}} \right ) }-4\,{\frac{befn}{g\sqrt{ \left ( dg-fe \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-fe \right ) e}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/g*(g*x+f)^(1/2)*a+2/g*b*ln(c*(((g*x+f)*e+d*g-f*e)/g)^n)*(g*x+f)^(1/2)-4*b*n*(g*x+f)^(1/2)/g+4*b*n/((d*g-e*f)
*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*d-4/g*b*e*n/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/(
(d*g-e*f)*e)^(1/2))*f

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.92591, size = 432, normalized size = 4.45 \begin{align*} \left [\frac{2 \,{\left (b n \sqrt{\frac{e f - d g}{e}} \log \left (\frac{e g x + 2 \, e f - d g + 2 \, \sqrt{g x + f} e \sqrt{\frac{e f - d g}{e}}}{e x + d}\right ) +{\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{g x + f}\right )}}{g}, \frac{2 \,{\left (2 \, b n \sqrt{-\frac{e f - d g}{e}} \arctan \left (-\frac{\sqrt{g x + f} e \sqrt{-\frac{e f - d g}{e}}}{e f - d g}\right ) +{\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{g x + f}\right )}}{g}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*(b*n*sqrt((e*f - d*g)/e)*log((e*g*x + 2*e*f - d*g + 2*sqrt(g*x + f)*e*sqrt((e*f - d*g)/e))/(e*x + d)) + (b*
n*log(e*x + d) - 2*b*n + b*log(c) + a)*sqrt(g*x + f))/g, 2*(2*b*n*sqrt(-(e*f - d*g)/e)*arctan(-sqrt(g*x + f)*e
*sqrt(-(e*f - d*g)/e)/(e*f - d*g)) + (b*n*log(e*x + d) - 2*b*n + b*log(c) + a)*sqrt(g*x + f))/g]

________________________________________________________________________________________

Sympy [A]  time = 24.5964, size = 326, normalized size = 3.36 \begin{align*} \begin{cases} - \frac{\frac{2 a f}{\sqrt{f + g x}} + 2 a \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right ) + 2 b f \left (\frac{2 e n \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} + \frac{\log{\left (c \left (d + e x\right )^{n} \right )}}{\sqrt{f + g x}}\right ) + 2 b \left (- \frac{2 e n \left (- \frac{g \sqrt{f + g x}}{e} - \frac{g \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{e \sqrt{\frac{e}{d g - e f}}}\right )}{g} - f \left (\frac{2 e n \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} + \frac{\log{\left (c \left (d - \frac{e f}{g} + \frac{e \left (f + g x\right )}{g}\right )^{n} \right )}}{\sqrt{f + g x}}\right ) - \sqrt{f + g x} \log{\left (c \left (d - \frac{e f}{g} + \frac{e \left (f + g x\right )}{g}\right )^{n} \right )}\right )}{g} & \text{for}\: g \neq 0 \\\frac{a x + b \left (- e n \left (- \frac{d \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{e} + \frac{x}{e}\right ) + x \log{\left (c \left (d + e x\right )^{n} \right )}\right )}{\sqrt{f}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a*f/sqrt(f + g*x) + 2*a*(-f/sqrt(f + g*x) - sqrt(f + g*x)) + 2*b*f*(2*e*n*atan(1/(sqrt(e/(d*g -
e*f))*sqrt(f + g*x)))/(sqrt(e/(d*g - e*f))*(d*g - e*f)) + log(c*(d + e*x)**n)/sqrt(f + g*x)) + 2*b*(-2*e*n*(-
g*sqrt(f + g*x)/e - g*atan(1/(sqrt(e/(d*g - e*f))*sqrt(f + g*x)))/(e*sqrt(e/(d*g - e*f))))/g - f*(2*e*n*atan(1
/(sqrt(e/(d*g - e*f))*sqrt(f + g*x)))/(sqrt(e/(d*g - e*f))*(d*g - e*f)) + log(c*(d - e*f/g + e*(f + g*x)/g)**n
)/sqrt(f + g*x)) - sqrt(f + g*x)*log(c*(d - e*f/g + e*(f + g*x)/g)**n)))/g, Ne(g, 0)), ((a*x + b*(-e*n*(-d*Pie
cewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e + x/e) + x*log(c*(d + e*x)**n)))/sqrt(f), True))

________________________________________________________________________________________

Giac [A]  time = 1.26109, size = 149, normalized size = 1.54 \begin{align*} \frac{2 \,{\left ({\left (2 \,{\left (\frac{{\left (d g - f e\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right ) e^{\left (-1\right )}}{\sqrt{d g e - f e^{2}}} - \sqrt{g x + f} e^{\left (-1\right )}\right )} e + \sqrt{g x + f} \log \left (x e + d\right )\right )} b n + \sqrt{g x + f} b \log \left (c\right ) + \sqrt{g x + f} a\right )}}{g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((2*((d*g - f*e)*arctan(sqrt(g*x + f)*e/sqrt(d*g*e - f*e^2))*e^(-1)/sqrt(d*g*e - f*e^2) - sqrt(g*x + f)*e^(-
1))*e + sqrt(g*x + f)*log(x*e + d))*b*n + sqrt(g*x + f)*b*log(c) + sqrt(g*x + f)*a)/g