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

Optimal. Leaf size=53 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{b e^2 n \log \left (d \sqrt{x}+e\right )}{d^2}+\frac{b e n \sqrt{x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.0343658, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2448, 263, 190, 43} \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{b e^2 n \log \left (d \sqrt{x}+e\right )}{d^2}+\frac{b e n \sqrt{x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e/Sqrt[x])^n],x]

[Out]

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

Rule 2448

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0314587, size = 62, normalized size = 1.17 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b e n \left (\frac{e \log \left (d+\frac{e}{\sqrt{x}}\right )}{d^2}+\frac{e \log (x)}{2 d^2}-\frac{\sqrt{x}}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.093, size = 94, normalized size = 1.8 \begin{align*} ax+xb\ln \left ( c \left ({ \left ( e+d\sqrt{x} \right ){\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) +{\frac{enb}{d}\sqrt{x}}+{\frac{b{e}^{2}n}{2\,{d}^{2}}\ln \left ( d\sqrt{x}-e \right ) }-{\frac{b{e}^{2}n}{2\,{d}^{2}}\ln \left ( e+d\sqrt{x} \right ) }-{\frac{b{e}^{2}n\ln \left ( x{d}^{2}-{e}^{2} \right ) }{2\,{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0404, size = 65, normalized size = 1.23 \begin{align*} -{\left (e n{\left (\frac{e \log \left (d \sqrt{x} + e\right )}{d^{2}} - \frac{\sqrt{x}}{d}\right )} - x \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.79058, size = 217, normalized size = 4.09 \begin{align*} \frac{b d^{2} x \log \left (c\right ) - b d^{2} n \log \left (\sqrt{x}\right ) + b d e n \sqrt{x} + a d^{2} x +{\left (b d^{2} - b e^{2}\right )} n \log \left (d \sqrt{x} + e\right ) +{\left (b d^{2} n x - b d^{2} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 16.9777, size = 76, normalized size = 1.43 \begin{align*} a x + b \left (\frac{e n \left (\frac{2 \sqrt{x}}{d} - \frac{2 e^{2} \left (\begin{cases} \frac{1}{d \sqrt{x}} & \text{for}\: e = 0 \\\frac{\log{\left (d + \frac{e}{\sqrt{x}} \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{2}} + \frac{2 e \log{\left (\frac{1}{\sqrt{x}} \right )}}{d^{2}}\right )}{2} + x \log{\left (c \left (d + \frac{e}{\sqrt{x}}\right )^{n} \right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.34533, size = 76, normalized size = 1.43 \begin{align*} -{\left ({\left ({\left (\frac{e \log \left ({\left | d \sqrt{x} + e \right |}\right )}{d^{2}} - \frac{\sqrt{x}}{d}\right )} e - x \log \left (d + \frac{e}{\sqrt{x}}\right )\right )} n - x \log \left (c\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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