Integrand size = 41, antiderivative size = 173 \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2 \sqrt {q+p x^6}}{\sqrt {a} q-\sqrt {b} x^4+\sqrt {a} p x^6}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^4}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^6}{\sqrt {2} \sqrt [4]{b}}}{x^2 \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \]
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Time = 0.36 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.47, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6846, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}} \]
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Rule 210
Rule 217
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6846
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{2 \sqrt {a}}-\frac {\text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{2 \sqrt {a}} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {a} \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}} \\ & = \frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \\ & = \frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}} \\ \end{align*}
\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx \]
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Time = 6.10 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {p \,x^{6}-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{6}+q}\, \sqrt {2}\, x^{2}+\sqrt {\frac {b}{a}}\, x^{4}+q}{p \,x^{6}+\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {p \,x^{6}+q}\, \sqrt {2}\, x^{2}+\sqrt {\frac {b}{a}}\, x^{4}+q}\right )+2 \arctan \left (\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{2}+\sqrt {2}\, \sqrt {p \,x^{6}+q}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{2}}\right )+2 \arctan \left (\frac {-\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{2}+\sqrt {2}\, \sqrt {p \,x^{6}+q}}{\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{2}}\right )\right )}{8 \left (\frac {b}{a}\right )^{\frac {1}{4}} a}\) | \(183\) |
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Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.74 \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} + 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) - \frac {1}{8} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (i \, a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (-i \, a^{3} b p x^{8} - i \, a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} + 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) + \frac {1}{8} i \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (-i \, a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (i \, a^{3} b p x^{8} + i \, a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} + 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) \]
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\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int \frac {x \left (p x^{6} - 2 q\right ) \sqrt {p x^{6} + q}}{a p^{2} x^{12} + 2 a p q x^{6} + a q^{2} + b x^{8}}\, dx \]
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\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )} x}{b x^{8} + {\left (p x^{6} + q\right )}^{2} a} \,d x } \]
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\[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )} x}{b x^{8} + {\left (p x^{6} + q\right )}^{2} a} \,d x } \]
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Timed out. \[ \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx=\int -\frac {x\,\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{a\,{\left (p\,x^6+q\right )}^2+b\,x^8} \,d x \]
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