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

Optimal. Leaf size=65 \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{x}+\frac{b d^2 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^2}-\frac{b d n}{e \sqrt{x}}+\frac{b n}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.051102, antiderivative size = 65, 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, 43} \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{x}+\frac{b d^2 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^2}-\frac{b d n}{e \sqrt{x}}+\frac{b n}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0331369, size = 68, normalized size = 1.05 \[ -\frac{a}{x}-\frac{b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{x}+\frac{b d^2 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^2}-\frac{b d n}{e \sqrt{x}}+\frac{b n}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.103, size = 63, normalized size = 1. \begin{align*} -{\frac{a}{x}}-{\frac{b}{x}\ln \left ( c{{\rm e}^{n\ln \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) }} \right ) }+{\frac{bn}{2\,x}}+{\frac{b{d}^{2}n}{{e}^{2}}\ln \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) }-{\frac{bdn}{e}{\frac{1}{\sqrt{x}}}} \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]

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

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Maxima [A]  time = 1.06207, size = 101, normalized size = 1.55 \begin{align*} \frac{1}{2} \, b e n{\left (\frac{2 \, d^{2} \log \left (d \sqrt{x} + e\right )}{e^{3}} - \frac{d^{2} \log \left (x\right )}{e^{3}} - \frac{2 \, d \sqrt{x} - e}{e^{2} x}\right )} - \frac{b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\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*d^2*log(d*sqrt(x) + e)/e^3 - d^2*log(x)/e^3 - (2*d*sqrt(x) - e)/(e^2*x)) - b*log(c*(d + e/sqrt(x)
)^n)/x - a/x

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Fricas [A]  time = 1.93247, size = 165, normalized size = 2.54 \begin{align*} -\frac{2 \, b d e n \sqrt{x} - b e^{2} n + 2 \, b e^{2} \log \left (c\right ) + 2 \, a e^{2} - 2 \,{\left (b d^{2} n x - b e^{2} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right )}{2 \, e^{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]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

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Giac [A]  time = 1.31088, size = 120, normalized size = 1.85 \begin{align*} \frac{{\left (2 \, b d^{2} n x \log \left (d \sqrt{x} + e\right ) - 2 \, b d^{2} n x \log \left (\sqrt{x}\right ) - 2 \, b d n \sqrt{x} e - 2 \, b n e^{2} \log \left (d \sqrt{x} + e\right ) + 2 \, b n e^{2} \log \left (\sqrt{x}\right ) + b n e^{2} - 2 \, b e^{2} \log \left (c\right ) - 2 \, a e^{2}\right )} e^{\left (-2\right )}}{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="giac")

[Out]

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