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]

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

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.48, size = 32, normalized size = 0.78 \[ -\frac {2 \, {\left (3 \, b n \log \relax (x) + 2 \, b n + 3 \, b \log \relax (c) + 3 \, a\right )} \sqrt {d x}}{9 \, d^{3} x^{2}} \]

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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giac [B]  time = 0.35, size = 67, normalized size = 1.63 \[ -\frac {2 \, {\left (\frac {3 \, b d n \log \left (d x\right )}{\sqrt {d x} x} - \frac {3 \, b d^{2} n \log \relax (d) - 2 \, b d^{2} n - 3 \, b d^{2} \log \relax (c) - 3 \, a d^{2}}{\sqrt {d x} d x}\right )}}{9 \, d^{3}} \]

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

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maple [C]  time = 0.13, size = 128, normalized size = 3.12 \[ -\frac {2 b \ln \left (x^{n}\right )}{3 \sqrt {d x}\, d^{2} x}-\frac {-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +6 b \ln \relax (c )+6 a}{9 \sqrt {d x}\, d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)/(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 = 0.63, size = 41, normalized size = 1.00 \[ -\frac {4 \, b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a}{3 \, \left (d x\right )^{\frac {3}{2}} d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d*x)^(5/2),x)

[Out]

int((a + b*log(c*x^n))/(d*x)^(5/2), x)

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sympy [A]  time = 28.21, size = 71, normalized size = 1.73 \[ - \frac {2 a}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b n \log {\relax (x )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 b n}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b \log {\relax (c )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} \]

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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