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]

-((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)

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Rubi [A]  time = 0.0577774, antiderivative size = 70, normalized size of antiderivative = 1., 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 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])

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 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])

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 = 0.0448363, 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))

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) }\, dx \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^2,x)

[Out]

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

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Maxima [A]  time = 1.025, size = 82, normalized size = 1.17 \begin{align*} \frac{1}{2} \, b e n{\left (\frac{2 \, e \log \left (e \sqrt{x} + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{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} \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^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

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Fricas [A]  time = 2.07748, size = 161, normalized size = 2.3 \begin{align*} -\frac{b e^{2} n x \log \left (\sqrt{x}\right ) + b d e n \sqrt{x} + b d^{2} \log \left (c\right ) + a d^{2} -{\left (b e^{2} n x - b d^{2} n\right )} \log \left (e \sqrt{x} + d\right )}{d^{2} 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^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)

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Sympy [A]  time = 96.1634, size = 493, normalized size = 7.04 \begin{align*} \begin{cases} - \frac{a d^{3} \sqrt{x}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{a d^{2} e x}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d^{3} n \sqrt{x} \log{\left (d + e \sqrt{x} \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d^{3} \sqrt{x} \log{\left (c \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d^{2} e n x \log{\left (d + e \sqrt{x} \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d^{2} e n x}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d^{2} e x \log{\left (c \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d e^{2} n x^{\frac{3}{2}} \log{\left (x \right )}}{2 \left (d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}\right )} + \frac{b d e^{2} n x^{\frac{3}{2}} \log{\left (d + e \sqrt{x} \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b d e^{2} n x^{\frac{3}{2}}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} + \frac{b d e^{2} x^{\frac{3}{2}} \log{\left (c \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} - \frac{b e^{3} n x^{2} \log{\left (x \right )}}{2 \left (d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}\right )} + \frac{b e^{3} n x^{2} \log{\left (d + e \sqrt{x} \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} + \frac{b e^{3} x^{2} \log{\left (c \right )}}{d^{3} x^{\frac{3}{2}} + d^{2} e x^{2}} & \text{for}\: d \neq 0 \\- \frac{a}{x} - \frac{b n \log{\left (e \right )}}{x} - \frac{b n \log{\left (x \right )}}{2 x} - \frac{b n}{2 x} - \frac{b \log{\left (c \right )}}{x} & \text{otherwise} \end{cases} \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**2,x)

[Out]

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

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Giac [B]  time = 1.29019, size = 252, normalized size = 3.6 \begin{align*} \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 \left (c\right ) - 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}} \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^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)