Integrand size = 18, antiderivative size = 193 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {b^2 p^2}{6 a^2 x^2}-\frac {b^3 p^2 \log (x)}{a^3}+\frac {b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {b^3 p \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{3 a^3}-\frac {b^3 p^2 \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{3 a^3} \]
[Out]
Time = 0.24 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\frac {b^3 p \log \left (1-\frac {a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3}-\frac {b^3 p^2 \operatorname {PolyLog}\left (2,\frac {a}{b x^2+a}\right )}{3 a^3}+\frac {b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac {b^3 p^2 \log (x)}{a^3}+\frac {b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac {b^2 p^2}{6 a^2 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4} \]
[In]
[Out]
Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {1}{3} (b p) \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^3 (a+b x)} \, dx,x,x^2\right ) \\ & = -\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {1}{3} p \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x^2\right ) \\ & = -\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {p \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x^2\right )}{3 a}-\frac {(b p) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{3 a} \\ & = -\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {(b p) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{3 a^2}+\frac {\left (b^2 p\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x^2\right )}{3 a^2}+\frac {\left (b p^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{6 a} \\ & = -\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {b^3 p \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{3 a^3}+\frac {\left (b p^2\right ) \text {Subst}\left (\int \left (\frac {b^2}{a (a-x)^2}+\frac {b^2}{a^2 (a-x)}+\frac {b^2}{a^2 x}\right ) \, dx,x,a+b x^2\right )}{6 a}-\frac {\left (b^2 p^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x^2\right )}{3 a^3}-\frac {\left (b^3 p^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {a}{x}\right )}{x} \, dx,x,a+b x^2\right )}{3 a^3} \\ & = -\frac {b^2 p^2}{6 a^2 x^2}-\frac {b^3 p^2 \log (x)}{a^3}+\frac {b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {b^3 p \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{3 a^3}-\frac {b^3 p^2 \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{3 a^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {a b^2 p^2 x^4+6 b^3 p^2 x^6 \log (x)-3 b^3 p^2 x^6 \log \left (a+b x^2\right )+a^2 b p x^2 \log \left (c \left (a+b x^2\right )^p\right )-2 a b^2 p x^4 \log \left (c \left (a+b x^2\right )^p\right )-2 b^3 p x^6 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+a^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+b^3 x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-2 b^3 p^2 x^6 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{6 a^3 x^6} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.30 (sec) , antiderivative size = 607, normalized size of antiderivative = 3.15
method | result | size |
risch | \(-\frac {{\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}^{2}}{6 x^{6}}-\frac {p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{6 a \,x^{4}}+\frac {2 p \,b^{3} \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{3 a^{3}}+\frac {p \,b^{2} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{3 a^{2} x^{2}}-\frac {p \,b^{3} \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (b \,x^{2}+a \right )}{3 a^{3}}-\frac {b^{2} p^{2}}{6 a^{2} x^{2}}-\frac {b^{3} p^{2} \ln \left (x \right )}{a^{3}}+\frac {b^{3} p^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}-\frac {2 p^{2} b^{3} \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}-\frac {2 p^{2} b^{3} \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}-\frac {2 p^{2} b^{3} \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}-\frac {2 p^{2} b^{3} \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}+\frac {p^{2} b^{3} \ln \left (b \,x^{2}+a \right )^{2}}{6 a^{3}}+\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{6 x^{6}}+\frac {p b \left (-\frac {1}{4 a \,x^{4}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}+\frac {b}{2 a^{2} x^{2}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\right )}{3}\right )-\frac {{\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2}}{24 x^{6}}\) | \(607\) |
[In]
[Out]
\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{7}} \,d x } \]
[In]
[Out]
\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{7}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.90 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {1}{6} \, b^{2} p^{2} {\left (\frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )} b}{a^{3}} - \frac {3 \, b \log \left (b x^{2} + a\right )}{a^{3}} - \frac {b x^{2} \log \left (b x^{2} + a\right )^{2} - 6 \, b x^{2} \log \left (x\right ) - a}{a^{3} x^{2}}\right )} - \frac {1}{6} \, b p {\left (\frac {2 \, b^{2} \log \left (b x^{2} + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x^{2}\right )}{a^{3}} - \frac {2 \, b x^{2} - a}{a^{2} x^{4}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{6 \, x^{6}} \]
[In]
[Out]
\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{7}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^7} \,d x \]
[In]
[Out]