∫ b6+a6x6 _______________________________________________

Optimal. Leaf size=60 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6+3 \text {$\#$1}^2 a^2 b^2+c\& ,\frac {\log \left (\sqrt {a^2 x^3-b^2 x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

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Rubi [B] time = 10.57, antiderivative size = 720, normalized size of antiderivative = 12.00, number of steps used = 92, number of rules used = 13, integrand size = 50, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.260, Rules used = {2056, 6715, 6728, 224, 221, 6725, 1725, 1219, 1218, 1248, 725, 204, 206} \begin {gather*} \frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

\begin {align*} \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {-b^2+a^2 x^2} \left (-b^6+c x^3+a^6 x^6\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {b^6+a^6 x^{12}}{\sqrt {-b^2+a^2 x^4} \left (-b^6+c x^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-b^2+a^2 x^4}}+\frac {2 b^6-c x^6}{\sqrt {-b^2+a^2 x^4} \left (-b^6+c x^6+a^6 x^{12}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {2 b^6-c x^6}{\sqrt {-b^2+a^2 x^4} \left (-b^6+c x^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-c+\sqrt {4 a^6 b^6+c^2}}{\sqrt {-b^2+a^2 x^4} \left (c-\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )}+\frac {-c-\sqrt {4 a^6 b^6+c^2}}{\sqrt {-b^2+a^2 x^4} \left (c+\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c-\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (c+\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c+\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (c-\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c-\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}}{2 \left (c+\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}+\frac {\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}}{2 \left (c+\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c+\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}}{2 \left (c-\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}+\frac {\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}}{2 \left (c-\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-2} a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{2} a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{2} a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-2} a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{2} a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{2} a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-2} a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{2} a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{2} a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-2} a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{2} a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{2} a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\\ \end {align*}

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Mathematica [C] time = 1.48, size = 544, normalized size = 9.07 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {1-\frac {b^2}{a^2 x^2}} \left (-\Pi \left (\frac {\sqrt [3]{2} a}{b \sqrt [3]{\frac {c-\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt [3]{2} a}{b \sqrt [3]{\frac {c+\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i \left (-i+\sqrt {3}\right ) a}{2^{2/3} b \sqrt [3]{\frac {c+\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \left (i+\sqrt {3}\right ) a}{2^{2/3} b \sqrt [3]{\frac {c+\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\left (1-i \sqrt {3}\right ) a b^5 \left (\frac {c-\sqrt {4 a^6 b^6+c^2}}{b^6}\right )^{2/3}}{2^{2/3} \left (\sqrt {4 a^6 b^6+c^2}-c\right )};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\left (1+i \sqrt {3}\right ) a b^5 \left (\frac {c-\sqrt {4 a^6 b^6+c^2}}{b^6}\right )^{2/3}}{2^{2/3} \left (\sqrt {4 a^6 b^6+c^2}-c\right )};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )+3 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{3 \sqrt {-\frac {b}{a}} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

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IntegrateAlgebraic [A] time = 0.00, size = 60, normalized size = 1.00 \begin {gather*} \frac {1}{3} \text {RootSum}\left [c+3 a^2 b^2 \text {$\#$1}^2+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt {-b^2 x+a^2 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6} + c x^{3}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}

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method	result	size

default	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{6} \textit {\_Z}^{6}-b^{6}+c \,\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-2 b^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3} c \right ) \left (a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c}{b^{2} c}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{6}+c \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b^{2} c}\)	$371$
elliptic	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{6} \textit {\_Z}^{6}-b^{6}+c \,\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-2 b^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3} c \right ) \left (a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c}{b^{2} c}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{6}+c \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b^{2} c}\)	$371$

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6} + c x^{3}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}

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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a^6\,x^6+b^6}{\sqrt {a^2\,x^3-b^2\,x}\,\left (a^6\,x^6-b^6+c\,x^3\right )} \,d x \end {gather*}

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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3.8.87 \(\int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (-b^6+c x^3+a^6 x^6)} \, dx\)