Integrand size = 20, antiderivative size = 98 \[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {7 e^{-\frac {7 a}{2 b n}} (d x)^{7/2} \left (c x^n\right )^{\left .-\frac {7}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2343, 2347, 2209} \[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {7 (d x)^{7/2} e^{-\frac {7 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {7}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
[In]
[Out]
Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {7 \int \frac {(d x)^{5/2}}{a+b \log \left (c x^n\right )} \, dx}{2 b n} \\ & = -\frac {(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (7 (d x)^{7/2} \left (c x^n\right )^{\left .-\frac {7}{2}\right /n}\right ) \text {Subst}\left (\int \frac {e^{\frac {7 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2} \\ & = \frac {7 e^{-\frac {7 a}{2 b n}} (d x)^{7/2} \left (c x^n\right )^{\left .-\frac {7}{2}\right /n} \text {Ei}\left (\frac {7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x (d x)^{5/2} \left (7 e^{-\frac {7 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {7}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.94 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.41
method | result | size |
risch | \(-\frac {2 x^{4} d^{3}}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right )}-\frac {7 \,{\mathrm e}^{\frac {7 i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \operatorname {Ei}_{1}\left (-\frac {7 \ln \left (d x \right )}{2}+\frac {7 i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 d \,b^{2} n^{2}}\) | \(432\) |
[In]
[Out]
\[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{\frac {5}{2}}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{\frac {5}{2}}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{\frac {5}{2}}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^{5/2}}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
[In]
[Out]