Integrand size = 20, antiderivative size = 73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {16 b^2 n^2}{27 d (d x)^{3/2}}-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2342, 2341} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac {16 b^2 n^2}{27 d (d x)^{3/2}} \]
[In]
[Out]
Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}+\frac {1}{3} (4 b n) \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx \\ & = -\frac {16 b^2 n^2}{27 d (d x)^{3/2}}-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {2 x \left (9 a^2+12 a b n+8 b^2 n^2+6 b (3 a+2 b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )\right )}{27 (d x)^{5/2}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 716, normalized size of antiderivative = 9.81
method | result | size |
risch | \(-\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{3 d^{2} x \sqrt {d x}}-\frac {2 b \left (-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 \right ) \ln \left (x^{n}\right )}{9 d^{2} x \sqrt {d x}}-\frac {36 a^{2}-24 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+32 b^{2} n^{2}+72 \ln \left (c \right ) a b +36 \ln \left (c \right )^{2} b^{2}+48 b^{2} \ln \left (c \right ) n +48 a b n -9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+36 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{54 d^{2} x \sqrt {d x}}\) | \(716\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (9 \, b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + 9 \, b^{2} \log \left (c\right )^{2} + 12 \, a b n + 9 \, a^{2} + 6 \, {\left (2 \, b^{2} n + 3 \, a b\right )} \log \left (c\right ) + 6 \, {\left (2 \, b^{2} n^{2} + 3 \, b^{2} n \log \left (c\right ) + 3 \, a b n\right )} \log \left (x\right )\right )} \sqrt {d x}}{27 \, d^{3} x^{2}} \]
[In]
[Out]
Time = 2.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=- \frac {2 a^{2} x}{3 \left (d x\right )^{\frac {5}{2}}} - \frac {8 a b n x}{9 \left (d x\right )^{\frac {5}{2}}} - \frac {4 a b x \log {\left (c x^{n} \right )}}{3 \left (d x\right )^{\frac {5}{2}}} - \frac {16 b^{2} n^{2} x}{27 \left (d x\right )^{\frac {5}{2}}} - \frac {8 b^{2} n x \log {\left (c x^{n} \right )}}{9 \left (d x\right )^{\frac {5}{2}}} - \frac {2 b^{2} x \log {\left (c x^{n} \right )}^{2}}{3 \left (d x\right )^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {8}{27} \, b^{2} {\left (\frac {2 \, n^{2}}{\left (d x\right )^{\frac {3}{2}} d} + \frac {3 \, n \log \left (c x^{n}\right )}{\left (d x\right )^{\frac {3}{2}} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {8 \, a b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (61) = 122\).
Time = 0.33 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.92 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {9 \, b^{2} d n^{2} \log \left (d x\right )^{2}}{\sqrt {d x} x} - \frac {6 \, {\left (3 \, b^{2} d^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} d^{2} n^{2} - 3 \, b^{2} d^{2} n \log \left (c\right ) - 3 \, a b d^{2} n\right )} \log \left (d x\right )}{\sqrt {d x} d x} + \frac {9 \, b^{2} d^{2} n^{2} \log \left (d\right )^{2} - 12 \, b^{2} d^{2} n^{2} \log \left (d\right ) - 18 \, b^{2} d^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} d^{2} n^{2} + 12 \, b^{2} d^{2} n \log \left (c\right ) + 9 \, b^{2} d^{2} \log \left (c\right )^{2} - 18 \, a b d^{2} n \log \left (d\right ) + 12 \, a b d^{2} n + 18 \, a b d^{2} \log \left (c\right ) + 9 \, a^{2} d^{2}}{\sqrt {d x} d x}\right )}}{27 \, d^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \]
[In]
[Out]