\(\int \frac {a+b \log (c x^n)}{(d x)^{3/2}} \, dx\) [93]

Optimal result

Integrand size = 18, antiderivative size = 37 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=-\frac {4 b n}{d \sqrt {d x}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2341} \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {4 b n}{d \sqrt {d x}} \]

[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 2341

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*} \text {integral}& = -\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] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=-\frac {2 x \left (a+2 b n+b \log \left (c x^n\right )\right )}{(d x)^{3/2}} \]

[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)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.30

method	result	size
risch	\(-\frac {2 b \ln \left (x^{n}\right )}{d \sqrt {d x}}-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+4 b n +2 a}{d \sqrt {d x}}\)	\(122\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (b n \log \left (x\right ) + 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {d x}}{d^{2} x} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=- \frac {2 a x}{\left (d x\right )^{\frac {3}{2}}} - \frac {4 b n x}{\left (d x\right )^{\frac {3}{2}}} - \frac {2 b x \log {\left (c x^{n} \right )}}{\left (d x\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=-\frac {4 \, b n}{\sqrt {d x} d} - \frac {2 \, b \log \left (c x^{n}\right )}{\sqrt {d x} d} - \frac {2 \, a}{\sqrt {d x} d} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {b n \log \left (d x\right )}{\sqrt {d x}} - \frac {b n \log \left (d\right ) - 2 \, b n - b \log \left (c\right ) - a}{\sqrt {d x}}\right )}}{d} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]

[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)