\(\int \frac {a+b \log (c x^n)}{(d x)^{5/2}} \, dx\) [94]

Optimal result

Integrand size = 18, antiderivative size = 41 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=-\frac {4 b n}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, 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} \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac {4 b n}{9 d (d x)^{3/2}} \]

[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 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}{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] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=-\frac {2 x \left (3 a+2 b n+3 b \log \left (c x^n\right )\right )}{9 (d x)^{5/2}} \]

[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))

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 = 128, normalized size of antiderivative = 3.12

method	result	size
risch	\(-\frac {2 b \ln \left (x^{n}\right )}{3 d^{2} x \sqrt {d x}}-\frac {-3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+6 b \ln \left (c \right )+4 b n +6 a}{9 d^{2} x \sqrt {d x}}\)	\(128\)

[In]

int((a+b*ln(c*x^n))/(d*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b n \log \left (x\right ) + 2 \, b n + 3 \, b \log \left (c\right ) + 3 \, a\right )} \sqrt {d x}}{9 \, d^{3} x^{2}} \]

[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)

Sympy [A] (verification not implemented)

Time = 2.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=- \frac {2 a x}{3 \left (d x\right )^{\frac {5}{2}}} - \frac {4 b n x}{9 \left (d x\right )^{\frac {5}{2}}} - \frac {2 b x \log {\left (c x^{n} \right )}}{3 \left (d x\right )^{\frac {5}{2}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=-\frac {4 \, b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a}{3 \, \left (d x\right )^{\frac {3}{2}} d} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (33) = 66\).

Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {3 \, b d n \log \left (d x\right )}{\sqrt {d x} x} - \frac {3 \, b d^{2} n \log \left (d\right ) - 2 \, b d^{2} n - 3 \, b d^{2} \log \left (c\right ) - 3 \, a d^{2}}{\sqrt {d x} d x}\right )}}{9 \, d^{3}} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]

[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)