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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 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])

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 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 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]

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 = 0.03, 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))

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 90, normalized size = 1.70 \[ \frac {b d^{2} x \log \relax (c) - b d^{2} n \log \left (\sqrt {x}\right ) + b d e n \sqrt {x} + a d^{2} x + {\left (b d^{2} - b e^{2}\right )} n \log \left (d \sqrt {x} + e\right ) + {\left (b d^{2} n x - b d^{2} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )}{d^{2}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.25, size = 56, normalized size = 1.06 \[ -{\left ({\left ({\left (\frac {e \log \left ({\left | d \sqrt {x} + e \right |}\right )}{d^{2}} - \frac {\sqrt {x}}{d}\right )} e - x \log \left (d + \frac {e}{\sqrt {x}}\right )\right )} n - x \log \relax (c)\right )} b + a x \]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 94, normalized size = 1.77 \[ -\frac {b \,e^{2} n \ln \left (d \sqrt {x}+e \right )}{2 d^{2}}+\frac {b \,e^{2} n \ln \left (d \sqrt {x}-e \right )}{2 d^{2}}-\frac {b \,e^{2} n \ln \left (d^{2} x -e^{2}\right )}{2 d^{2}}+b x \ln \left (c \left (\frac {d \sqrt {x}+e}{\sqrt {x}}\right )^{n}\right )+\frac {b e n \sqrt {x}}{d}+a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 48, normalized size = 0.91 \[ -{\left (e n {\left (\frac {e \log \left (d \sqrt {x} + e\right )}{d^{2}} - \frac {\sqrt {x}}{d}\right )} - x \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )\right )} b + a x \]

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

________________________________________________________________________________________

mupad [B]  time = 0.38, size = 44, normalized size = 0.83 \[ a\,x+b\,x\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )-\frac {b\,e\,n\,\left (e\,\ln \left (e+d\,\sqrt {x}\right )-d\,\sqrt {x}\right )}{d^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*log(c*(d + e/x^(1/2))^n),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 8.42, size = 76, normalized size = 1.43 \[ a x + b \left (\frac {e n \left (\frac {2 \sqrt {x}}{d} - \frac {2 e^{2} \left (\begin {cases} \frac {1}{d \sqrt {x}} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{\sqrt {x}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {2 e \log {\left (\frac {1}{\sqrt {x}} \right )}}{d^{2}}\right )}{2} + x \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right ) \]

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))

________________________________________________________________________________________