Integrand size = 41, antiderivative size = 56 \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \]
Time = 9.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \]
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.
\(\displaystyle \int \frac {\left (p x^6-2 q\right ) \sqrt {p x^6+q}}{x^3 \left (a p x^6+a q+b x^4\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \sqrt {p x^6+q} \left (3 a p x^2+2 b\right )}{a \left (a p x^6+a q+b x^4\right )}-\frac {2 \sqrt {p x^6+q}}{a x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \text {Subst}\left (\int \frac {\sqrt {p x^3+q}}{a p x^3+b x^2+a q}dx,x,x^2\right )}{a}+\frac {3}{2} p \text {Subst}\left (\int \frac {x \sqrt {p x^3+q}}{a p x^3+b x^2+a q}dx,x,x^2\right )-\frac {\sqrt {2} 3^{3/4} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right ) \sqrt {\frac {p^{2/3} x^4-\sqrt [3]{p} \sqrt [3]{q} x^2+q^{2/3}}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{p} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{q}}{\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}}\right ),-7-4 \sqrt {3}\right )}{a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right )}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \sqrt {p x^6+q}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right ) \sqrt {\frac {p^{2/3} x^4-\sqrt [3]{p} \sqrt [3]{q} x^2+q^{2/3}}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{p} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{q}}{\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}}\right )|-7-4 \sqrt {3}\right )}{2 a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right )}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \sqrt {p x^6+q}}-\frac {3 \sqrt [3]{p} \sqrt {p x^6+q}}{a \left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )}+\frac {\sqrt {p x^6+q}}{a x^2}\) |
3.8.30.3.1 Defintions of rubi rules used
Time = 4.62 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(\frac {-b \arctan \left (\frac {\sqrt {p \,x^{6}+q}\, a}{x^{2} \sqrt {a b}}\right ) x^{2}+\sqrt {p \,x^{6}+q}\, \sqrt {a b}}{a \,x^{2} \sqrt {a b}}\) | \(55\) |
(-b*arctan((p*x^6+q)^(1/2)/x^2*a/(a*b)^(1/2))*x^2+(p*x^6+q)^(1/2)*(a*b)^(1 /2))/a/x^2/(a*b)^(1/2)
Timed out. \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )}}{{\left (a p x^{6} + b x^{4} + a q\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )}}{{\left (a p x^{6} + b x^{4} + a q\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\int -\frac {\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{x^3\,\left (a\,p\,x^6+b\,x^4+a\,q\right )} \,d x \]