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

Optimal. Leaf size=69 \[ \frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{4 b n \sqrt{d+e x}}{e}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0337705, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2319, 50, 63, 208} \[ \frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{4 b n \sqrt{d+e x}}{e}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && 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 x^n\right )}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{(2 b n) \int \frac{\sqrt{d+e x}}{x} \, dx}{e}\\ &=-\frac{4 b n \sqrt{d+e x}}{e}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{(2 b d n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{e}\\ &=-\frac{4 b n \sqrt{d+e x}}{e}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{(4 b d n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e^2}\\ &=-\frac{4 b n \sqrt{d+e x}}{e}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.0471237, size = 55, normalized size = 0.8 \[ \frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )-2 b n\right )+4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 70, normalized size = 1. \begin{align*} 4\,{\frac{\sqrt{d}bn}{e}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+2\,{\frac{\sqrt{ex+d}b\ln \left ( c{x}^{n} \right ) }{e}}-4\,{\frac{bn\sqrt{ex+d}}{e}}+2\,{\frac{\sqrt{ex+d}a}{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*x^n))/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.41235, size = 308, normalized size = 4.46 \begin{align*} \left [\frac{2 \,{\left (b \sqrt{d} n \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) +{\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{e x + d}\right )}}{e}, -\frac{2 \,{\left (2 \, b \sqrt{-d} n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) -{\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{e x + d}\right )}}{e}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 17.7907, size = 252, normalized size = 3.65 \begin{align*} \begin{cases} - \frac{\frac{2 a d}{\sqrt{d + e x}} + 2 a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 2 b d \left (\frac{\log{\left (c x^{n} \right )}}{\sqrt{d + e x}} - \frac{2 n \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{d \sqrt{- \frac{1}{d}}}\right ) + 2 b \left (- d \left (\frac{\log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )}}{\sqrt{d + e x}} - \frac{2 n \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{d \sqrt{- \frac{1}{d}}}\right ) - \sqrt{d + e x} \log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )} - \frac{2 n \left (- e \sqrt{d + e x} - \frac{e \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{1}{d}}}\right )}{e}\right )}{e} & \text{for}\: e \neq 0 \\\frac{a x + b \left (- n x + x \log{\left (c x^{n} \right )}\right )}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a*d/sqrt(d + e*x) + 2*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 2*b*d*(log(c*x**n)/sqrt(d + e*x) -
2*n*atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/(d*sqrt(-1/d))) + 2*b*(-d*(log(c*(-d/e + (d + e*x)/e)**n)/sqrt(d + e*x
) - 2*n*atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/(d*sqrt(-1/d))) - sqrt(d + e*x)*log(c*(-d/e + (d + e*x)/e)**n) - 2*
n*(-e*sqrt(d + e*x) - e*atan(1/(sqrt(-1/d)*sqrt(d + e*x)))/sqrt(-1/d))/e))/e, Ne(e, 0)), ((a*x + b*(-n*x + x*l
og(c*x**n)))/sqrt(d), True))

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Giac [A]  time = 1.25156, size = 105, normalized size = 1.52 \begin{align*} -2 \,{\left ({\left (\frac{2 \, d \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{\sqrt{-d}} - \sqrt{x e + d} \log \left (x\right ) + 2 \, \sqrt{x e + d}\right )} b n - \sqrt{x e + d} b \log \left (c\right ) - \sqrt{x e + d} a\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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