3.93 \(\int \frac{a+b \log (c x^n)}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{4 b n}{d \sqrt{d x}} \]

[Out]

(-4*b*n)/(d*Sqrt[d*x]) - (2*(a + b*Log[c*x^n]))/(d*Sqrt[d*x])

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Rubi [A]  time = 0.0159402, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2304} \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{4 b n}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d*x)^(3/2),x]

[Out]

(-4*b*n)/(d*Sqrt[d*x]) - (2*(a + b*Log[c*x^n]))/(d*Sqrt[d*x])

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx &=-\frac{4 b n}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}\\ \end{align*}

Mathematica [A]  time = 0.0067746, size = 24, normalized size = 0.65 \[ -\frac{2 x \left (a+b \log \left (c x^n\right )+2 b n\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(d*x)^(3/2),x]

[Out]

(-2*x*(a + 2*b*n + b*Log[c*x^n]))/(d*x)^(3/2)

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Maple [C]  time = 0.083, size = 122, normalized size = 3.3 \begin{align*} -2\,{\frac{b\ln \left ({x}^{n} \right ) }{d\sqrt{dx}}}-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +4\,bn+2\,a}{d}{\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/(d*x)^(3/2),x)

[Out]

-2/d*b/(d*x)^(1/2)*ln(x^n)-1/d*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-
I*b*Pi*csgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+4*b*n+2*a)/(d*x)^(1/2)

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Maxima [A]  time = 1.17758, size = 55, normalized size = 1.49 \begin{align*} -\frac{4 \, b n}{\sqrt{d x} d} - \frac{2 \, b \log \left (c x^{n}\right )}{\sqrt{d x} d} - \frac{2 \, a}{\sqrt{d x} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.864841, size = 78, normalized size = 2.11 \begin{align*} -\frac{2 \,{\left (b n \log \left (x\right ) + 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{d x}}{d^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.86456, size = 65, normalized size = 1.76 \begin{align*} - \frac{2 a}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b n \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 b n}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b \log{\left (c \right )}}{d^{\frac{3}{2}} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/(d*x)**(3/2),x)

[Out]

-2*a/(d**(3/2)*sqrt(x)) - 2*b*n*log(x)/(d**(3/2)*sqrt(x)) - 4*b*n/(d**(3/2)*sqrt(x)) - 2*b*log(c)/(d**(3/2)*sq
rt(x))

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Giac [A]  time = 1.36171, size = 58, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (\frac{b n \log \left (d x\right )}{\sqrt{d x}} - \frac{b n \log \left (d\right ) - 2 \, b n - b \log \left (c\right ) - a}{\sqrt{d x}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d*x)^(3/2),x, algorithm="giac")

[Out]

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