3.94 \(\int \frac{a+b \log (c x^n)}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac{4 b n}{9 d (d x)^{3/2}} \]

[Out]

(-4*b*n)/(9*d*(d*x)^(3/2)) - (2*(a + b*Log[c*x^n]))/(3*d*(d*x)^(3/2))

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Rubi [A]  time = 0.0151008, antiderivative size = 41, 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} \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac{4 b n}{9 d (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d*x)^(5/2),x]

[Out]

(-4*b*n)/(9*d*(d*x)^(3/2)) - (2*(a + b*Log[c*x^n]))/(3*d*(d*x)^(3/2))

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)^{5/2}} \, dx &=-\frac{4 b n}{9 d (d x)^{3/2}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0086047, size = 29, normalized size = 0.71 \[ -\frac{2 x \left (3 a+3 b \log \left (c x^n\right )+2 b n\right )}{9 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(d*x)^(5/2),x]

[Out]

(-2*x*(3*a + 2*b*n + 3*b*Log[c*x^n]))/(9*(d*x)^(5/2))

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Maple [C]  time = 0.088, size = 128, normalized size = 3.1 \begin{align*} -{\frac{2\,b\ln \left ({x}^{n} \right ) }{3\,x{d}^{2}}{\frac{1}{\sqrt{dx}}}}-{\frac{3\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -3\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+3\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +6\,b\ln \left ( c \right ) +4\,bn+6\,a}{9\,x{d}^{2}}{\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)^(5/2),x)

[Out]

-2/3/d^2*b/x/(d*x)^(1/2)*ln(x^n)-1/9/d^2*(3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)-3*I*b*Pi*csgn(I*c*x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)+4*b*n+6*a)/x/(d*x)^(1/2)

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Maxima [A]  time = 1.04975, size = 55, normalized size = 1.34 \begin{align*} -\frac{4 \, b n}{9 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{2 \, b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{2 \, a}{3 \, \left (d x\right )^{\frac{3}{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d*x)^(5/2),x, algorithm="maxima")

[Out]

-4/9*b*n/((d*x)^(3/2)*d) - 2/3*b*log(c*x^n)/((d*x)^(3/2)*d) - 2/3*a/((d*x)^(3/2)*d)

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Fricas [A]  time = 0.866572, size = 92, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (3 \, b n \log \left (x\right ) + 2 \, b n + 3 \, b \log \left (c\right ) + 3 \, a\right )} \sqrt{d x}}{9 \, d^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d*x)^(5/2),x, algorithm="fricas")

[Out]

-2/9*(3*b*n*log(x) + 2*b*n + 3*b*log(c) + 3*a)*sqrt(d*x)/(d^3*x^2)

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Sympy [A]  time = 39.234, size = 71, normalized size = 1.73 \begin{align*} - \frac{2 a}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b n \log{\left (x \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{4 b n}{9 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b \log{\left (c \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/(d*x)**(5/2),x)

[Out]

-2*a/(3*d**(5/2)*x**(3/2)) - 2*b*n*log(x)/(3*d**(5/2)*x**(3/2)) - 4*b*n/(9*d**(5/2)*x**(3/2)) - 2*b*log(c)/(3*
d**(5/2)*x**(3/2))

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Giac [A]  time = 1.40789, size = 90, normalized size = 2.2 \begin{align*} -\frac{2 \,{\left (\frac{3 \, b d n \log \left (d x\right )}{\sqrt{d x} x} - \frac{3 \, b d^{2} n \log \left (d\right ) - 2 \, b d^{2} n - 3 \, b d^{2} \log \left (c\right ) - 3 \, a d^{2}}{\sqrt{d x} d x}\right )}}{9 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d*x)^(5/2),x, algorithm="giac")

[Out]

-2/9*(3*b*d*n*log(d*x)/(sqrt(d*x)*x) - (3*b*d^2*n*log(d) - 2*b*d^2*n - 3*b*d^2*log(c) - 3*a*d^2)/(sqrt(d*x)*d*
x))/d^3