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

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Rubi [A]  time = 0.02, antiderivative size = 37, 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} \[ -\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {4 b n}{d \sqrt {d x}} \]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.47, size = 28, normalized size = 0.76 \[ -\frac {2 \, {\left (b n \log \relax (x) + 2 \, b n + b \log \relax (c) + a\right )} \sqrt {d x}}{d^{2} x} \]

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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giac [A]  time = 0.38, size = 43, normalized size = 1.16 \[ -\frac {2 \, {\left (\frac {b n \log \left (d x\right )}{\sqrt {d x}} - \frac {b n \log \relax (d) - 2 \, b n - b \log \relax (c) - a}{\sqrt {d x}}\right )}}{d} \]

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

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maple [C]  time = 0.13, size = 122, normalized size = 3.30 \[ -\frac {2 b \ln \left (x^{n}\right )}{\sqrt {d x}\, d}-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +2 b \ln \relax (c )+2 a}{\sqrt {d x}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 41, normalized size = 1.11 \[ -\frac {4 \, b n}{\sqrt {d x} d} - \frac {2 \, b \log \left (c x^{n}\right )}{\sqrt {d x} d} - \frac {2 \, a}{\sqrt {d x} d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d*x)^(3/2),x)

[Out]

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

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sympy [A]  time = 2.86, size = 65, normalized size = 1.76 \[ - \frac {2 a}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b n \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 b n}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b \log {\relax (c )}}{d^{\frac {3}{2}} \sqrt {x}} \]

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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