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]
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]
[Out]
Rule 2341
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*}
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]