3.187 \(\int \frac{(a+b \log (c \sqrt{x}))^p}{x} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 \left (a+b \log \left (c \sqrt{x}\right )\right )^{p+1}}{b (p+1)} \]

[Out]

(2*(a + b*Log[c*Sqrt[x]])^(1 + p))/(b*(1 + p))

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Rubi [A]  time = 0.0284044, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2302, 30} \[ \frac{2 \left (a+b \log \left (c \sqrt{x}\right )\right )^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*Sqrt[x]])^p/x,x]

[Out]

(2*(a + b*Log[c*Sqrt[x]])^(1 + p))/(b*(1 + p))

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0066608, size = 26, normalized size = 1. \[ \frac{2 \left (a+b \log \left (c \sqrt{x}\right )\right )^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*Log[c*Sqrt[x]])^(1 + p))/(b*(1 + p))

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Maple [A]  time = 0.04, size = 25, normalized size = 1. \begin{align*} 2\,{\frac{ \left ( a+b\ln \left ( c\sqrt{x} \right ) \right ) ^{1+p}}{b \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^(1/2)))^p/x,x)

[Out]

2*(a+b*ln(c*x^(1/2)))^(1+p)/b/(1+p)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04289, size = 82, normalized size = 3.15 \begin{align*} \frac{2 \,{\left (b \log \left (c \sqrt{x}\right ) + a\right )}{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p}}{b p + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*log(c*sqrt(x)) + a)*(b*log(c*sqrt(x)) + a)^p/(b*p + b)

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Sympy [A]  time = 16.2973, size = 48, normalized size = 1.85 \begin{align*} - \begin{cases} - a^{p} \log{\left (x \right )} & \text{for}\: b = 0 \\- \frac{2 \left (\begin{cases} \frac{\left (a + b \log{\left (c \sqrt{x} \right )}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b \log{\left (c \sqrt{x} \right )} \right )} & \text{otherwise} \end{cases}\right )}{b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**(1/2)))**p/x,x)

[Out]

-Piecewise((-a**p*log(x), Eq(b, 0)), (-2*Piecewise(((a + b*log(c*sqrt(x)))**(p + 1)/(p + 1), Ne(p, -1)), (log(
a + b*log(c*sqrt(x))), True))/b, True))

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Giac [A]  time = 1.26272, size = 34, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (b \log \left (c\right ) + \frac{1}{2} \, b \log \left (x\right ) + a\right )}^{p + 1}}{b{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*log(c) + 1/2*b*log(x) + a)^(p + 1)/(b*(p + 1))