Integrand size = 43, antiderivative size = 457 \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt {q+p x^4}}{-\sqrt [4]{a} q+\sqrt [4]{b} x^2-\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^4}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{a} q}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [8]{a} p x^4}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{a} q}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [8]{a} p x^4}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}} \]
-1/8*(2+2^(1/2))^(1/2)*arctan((2^(1/2)/(2-2^(1/2))^(1/2)*a^(1/8)*b^(1/8)-2 *a^(1/8)*b^(1/8)/(2-2^(1/2))^(1/2))*x*(p*x^4+q)^(1/2)/(-a^(1/4)*q+b^(1/4)* x^2-a^(1/4)*p*x^4))/a^(3/8)/b^(5/8)+1/8*(2-2^(1/2))^(1/2)*arctan((2+2^(1/2 ))^(1/2)*a^(1/8)*b^(1/8)*x*(p*x^4+q)^(1/2)/(a^(1/4)*q-b^(1/4)*x^2+a^(1/4)* p*x^4))/a^(3/8)/b^(5/8)-1/8*(2+2^(1/2))^(1/2)*arctanh((a^(1/8)*q/(2-2^(1/2 ))^(1/2)/b^(1/8)+b^(1/8)*x^2/(2-2^(1/2))^(1/2)/a^(1/8)+a^(1/8)*p*x^4/(2-2^ (1/2))^(1/2)/b^(1/8))/x/(p*x^4+q)^(1/2))/a^(3/8)/b^(5/8)+1/8*(2-2^(1/2))^( 1/2)*arctanh((a^(1/8)*q/(2+2^(1/2))^(1/2)/b^(1/8)+b^(1/8)*x^2/(2+2^(1/2))^ (1/2)/a^(1/8)+a^(1/8)*p*x^4/(2+2^(1/2))^(1/2)/b^(1/8))/x/(p*x^4+q)^(1/2))/ a^(3/8)/b^(5/8)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 17.59 (sec) , antiderivative size = 15065, normalized size of antiderivative = 32.96 \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Result too large to show} \]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^4 \left (p x^4-q\right ) \sqrt {p x^4+q}}{a \left (p x^4+q\right )^4+b x^8} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {p x^8 \sqrt {p x^4+q}}{b x^8 \left (\frac {6 a p^2 q^2}{b}+1\right )+a p^4 x^{16}+4 a p^3 q x^{12}+4 a p q^3 x^4+a q^4}+\frac {q x^4 \sqrt {p x^4+q}}{-b x^8 \left (\frac {6 a p^2 q^2}{b}+1\right )-a p^4 x^{16}-4 a p^3 q x^{12}-4 a p q^3 x^4-a q^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle q \int \frac {x^4 \sqrt {p x^4+q}}{-b x^8-a \left (p x^4+q\right )^4}dx+p \int \frac {x^8 \sqrt {p x^4+q}}{b x^8+a \left (p x^4+q\right )^4}dx\) |
3.31.47.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.09
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {p \,x^{4}+q}}{x}\right )}{\textit {\_R}^{5}}}{8 a}\) | \(40\) |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {p \,x^{4}+q}}{x}\right )}{\textit {\_R}^{5}}}{8 a}\) | \(40\) |
elliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right ) \sqrt {2}}{64 a}\) | \(47\) |
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\int { \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )} x^{4}}{b x^{8} + {\left (p x^{4} + q\right )}^{4} a} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=-\int \frac {x^4\,\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{a\,{\left (p\,x^4+q\right )}^4+b\,x^8} \,d x \]