3.405 \(\int \frac {a+b \log (c (d+e \sqrt {x})^n)}{x^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {b e^2 n \log (x)}{2 d^2}-\frac {b e n}{d \sqrt {x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2454, 2395, 44} \[ -\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {b e^2 n \log (x)}{2 d^2}-\frac {b e n}{d \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+(b e n) \operatorname {Subst}\left (\int \frac {1}{x^2 (d+e x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+(b e n) \operatorname {Subst}\left (\int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {b e n}{d \sqrt {x}}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}-\frac {b e^2 n \log (x)}{2 d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 67, normalized size = 0.96 \[ -\frac {a}{x}-\frac {b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+b e n \left (\frac {e \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {e \log (x)}{2 d^2}-\frac {1}{d \sqrt {x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 65, normalized size = 0.93 \[ -\frac {b e^{2} n x \log \left (\sqrt {x}\right ) + b d e n \sqrt {x} + b d^{2} \log \relax (c) + a d^{2} - {\left (b e^{2} n x - b d^{2} n\right )} \log \left (e \sqrt {x} + d\right )}{d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.18, size = 187, normalized size = 2.67 \[ \frac {{\left ({\left (\sqrt {x} e + d\right )}^{2} b n e^{3} \log \left (\sqrt {x} e + d\right ) - 2 \, {\left (\sqrt {x} e + d\right )} b d n e^{3} \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} b n e^{3} \log \left (\sqrt {x} e\right ) + 2 \, {\left (\sqrt {x} e + d\right )} b d n e^{3} \log \left (\sqrt {x} e\right ) - b d^{2} n e^{3} \log \left (\sqrt {x} e\right ) - {\left (\sqrt {x} e + d\right )} b d n e^{3} + b d^{2} n e^{3} - b d^{2} e^{3} \log \relax (c) - a d^{2} e^{3}\right )} e^{\left (-1\right )}}{{\left (\sqrt {x} e + d\right )}^{2} d^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d^{3} + d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((sqrt(x)*e + d)^2*b*n*e^3*log(sqrt(x)*e + d) - 2*(sqrt(x)*e + d)*b*d*n*e^3*log(sqrt(x)*e + d) - (sqrt(x)*e +
d)^2*b*n*e^3*log(sqrt(x)*e) + 2*(sqrt(x)*e + d)*b*d*n*e^3*log(sqrt(x)*e) - b*d^2*n*e^3*log(sqrt(x)*e) - (sqrt(
x)*e + d)*b*d*n*e^3 + b*d^2*n*e^3 - b*d^2*e^3*log(c) - a*d^2*e^3)*e^(-1)/((sqrt(x)*e + d)^2*d^2 - 2*(sqrt(x)*e
+ d)*d^3 + d^4)

________________________________________________________________________________________

maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 61, normalized size = 0.87 \[ \frac {1}{2} \, b e n {\left (\frac {2 \, e \log \left (e \sqrt {x} + d\right )}{d^{2}} - \frac {e \log \relax (x)}{d^{2}} - \frac {2}{d \sqrt {x}}\right )} - \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{x} - \frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.80, size = 58, normalized size = 0.83 \[ \frac {2\,b\,e^2\,n\,\mathrm {atanh}\left (\frac {2\,e\,\sqrt {x}}{d}+1\right )}{d^2}-\frac {b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x}-\frac {b\,e\,n}{d\,\sqrt {x}}-\frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 63.06, size = 554, normalized size = 7.91 \[ \begin {cases} - \frac {2 a d^{3} \sqrt {x}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 a d^{2} e x}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{3} n \sqrt {x} \log {\left (d + e \sqrt {x} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{3} \sqrt {x} \log {\relax (c )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e n x \log {\left (d + e \sqrt {x} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e n x}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e x \log {\relax (c )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {b d e^{2} n x^{\frac {3}{2}} \log {\relax (x )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b d e^{2} n x^{\frac {3}{2}} \log {\left (d + e \sqrt {x} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d e^{2} n x^{\frac {3}{2}}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b d e^{2} x^{\frac {3}{2}} \log {\relax (c )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {b e^{3} n x^{2} \log {\relax (x )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b e^{3} n x^{2} \log {\left (d + e \sqrt {x} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b e^{3} x^{2} \log {\relax (c )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} & \text {for}\: d \neq 0 \\- \frac {a}{x} - \frac {b n \log {\relax (e )}}{x} - \frac {b n \log {\relax (x )}}{2 x} - \frac {b n}{2 x} - \frac {b \log {\relax (c )}}{x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a*d**3*sqrt(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*a*d**2*e*x/(2*d**3*x**(3/2) + 2*d**2*e*x**2
) - 2*b*d**3*n*sqrt(x)*log(d + e*sqrt(x))/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d**3*sqrt(x)*log(c)/(2*d**3*
x**(3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*n*x*log(d + e*sqrt(x))/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*n
*x/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*x*log(c)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - b*d*e**2*n*x**(
3/2)*log(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) + 2*b*d*e**2*n*x**(3/2)*log(d + e*sqrt(x))/(2*d**3*x**(3/2) + 2*
d**2*e*x**2) - 2*b*d*e**2*n*x**(3/2)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) + 2*b*d*e**2*x**(3/2)*log(c)/(2*d**3*x*
*(3/2) + 2*d**2*e*x**2) - b*e**3*n*x**2*log(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) + 2*b*e**3*n*x**2*log(d + e*s
qrt(x))/(2*d**3*x**(3/2) + 2*d**2*e*x**2) + 2*b*e**3*x**2*log(c)/(2*d**3*x**(3/2) + 2*d**2*e*x**2), Ne(d, 0)),
(-a/x - b*n*log(e)/x - b*n*log(x)/(2*x) - b*n/(2*x) - b*log(c)/x, True))

________________________________________________________________________________________