Integrand size = 45, antiderivative size = 244 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {2 \sqrt {-b^2 x+a^2 x^3}}{3 \left (b^2-a^2 x^2\right )}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-\sqrt {3} b^2-3 a b x+\sqrt {3} a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt [4]{3} \sqrt {a}}+\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt [4]{3} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.81 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.13, number of steps used = 36, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2081, 6847, 6857, 230, 227, 2098, 1166, 425, 21, 434, 438, 437, 435, 259, 6860, 1233, 1232} \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {4 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {x (b-a x)}{3 b \sqrt {a^2 x^3-b^2 x}}-\frac {x (a x+b)}{3 b \sqrt {a^2 x^3-b^2 x}} \]
[In]
[Out]
Rule 21
Rule 227
Rule 230
Rule 259
Rule 425
Rule 434
Rule 435
Rule 437
Rule 438
Rule 1166
Rule 1232
Rule 1233
Rule 2081
Rule 2098
Rule 6847
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \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+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 ) \text {Subst}\left (\int \frac {b^6+a^6 x^{12}}{\sqrt {-b^2+a^2 x^4} \left (-b^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 ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {-b^2+a^2 x^4}}+\frac {2 b^6}{\sqrt {-b^2+a^2 x^4} \left (-b^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 ) \text {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 (4 b^6 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (-b^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}} \\ & = \frac {\left (4 b^6 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{6 b^5 \left (b-a x^2\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{6 b^5 \left (b+a x^2\right ) \sqrt {-b^2+a^2 x^4}}+\frac {-2 b+a x^2}{6 b^5 \sqrt {-b^2+a^2 x^4} \left (b^2-a b x^2+a^2 x^4\right )}+\frac {-2 b-a x^2}{6 b^5 \sqrt {-b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\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 ) \text {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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b-a 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 {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b+a 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 {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-2 b+a x^2}{\sqrt {-b^2+a^2 x^4} \left (b^2-a b x^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-2 b-a x^2}{\sqrt {-b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )} \, 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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-a x^2} \left (b-a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b+a x} \sqrt {b+a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+a x^2} \left (b+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {a+\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}+\frac {a-\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}+\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {-a b+a^2 x^2}{\sqrt {-b-a x^2} \sqrt {b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 a b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b+a x} \sqrt {b+a x}\right ) \text {Subst}\left (\int \frac {a b+a^2 x^2}{\sqrt {-b+a x^2} \sqrt {b+a x^2}} \, dx,x,\sqrt {x}\right )}{3 a b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+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 {\left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+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 {\left (2 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+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 {\left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+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 {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {\sqrt {b-a x^2}}{\sqrt {-b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b+a x} \sqrt {b+a x}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a x^2}}{\sqrt {-b+a x^2}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+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 {\left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+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 {\left (2 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+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 {\left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+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 {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-a x^2} \sqrt {b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {\sqrt {-b-a x^2}}{\sqrt {b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x} \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a x^2}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {-b-a x^2}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} (b+a x) \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {1+\frac {a x}{b}} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {x} (b+a x) \sqrt {1-\frac {a x}{b}} E\left (\left .\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {b} \sqrt {1+\frac {a x}{b}} \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} (-b-a x) \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {1+\frac {a x}{b}} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {4 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\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}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {x} \left (6 \sqrt {a} \sqrt {b} \sqrt {x}+3^{3/4} \sqrt {-2 b^2+2 a^2 x^2} \arctan \left (\frac {\sqrt {3} b^2+3 a b x-\sqrt {3} a^2 x^2}{\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )-3^{3/4} \sqrt {-2 b^2+2 a^2 x^2} \text {arctanh}\left (\frac {\sqrt {3} b^2-3 a b x-\sqrt {3} a^2 x^2}{\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )\right )}{9 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]
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Time = 2.00 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {\left (\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{3}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x -3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}}{2}-2 \,3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{9 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}\) | \(297\) |
pseudoelliptic | \(\frac {\left (\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{3}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x -3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}}{2}-2 \,3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{9 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}\) | \(297\) |
elliptic | \(-\frac {2 x}{3 \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 b \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {2 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {a \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {4 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {4 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 a \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\) | \(678\) |
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 771, normalized size of antiderivative = 3.16 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, a^{2} x^{2} - i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b^{2} x^{2} + i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{2} x^{2} + i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (-i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (i \, a^{4} b^{2} x^{2} - i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} + 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + 4 \, \sqrt {a^{2} x^{3} - b^{2} x}}{6 \, {\left (a^{2} x^{2} - b^{2}\right )}} \]
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\[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \]
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\[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
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\[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
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Time = 13.24 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {2\,\sqrt {a^2\,x^3-b^2\,x}}{3\,\left (b^2-a^2\,x^2\right )}+\frac {3^{1/4}\,\sqrt {-\frac {1}{27}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{1/4}\,3^{3/4}\,b^2-{\left (-1\right )}^{1/4}\,3^{3/4}\,a^2\,x^2-3\,{\left (-1\right )}^{3/4}\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,6{}\mathrm {i}}{-a^2\,x^2+1{}\mathrm {i}\,\sqrt {3}\,a\,b\,x+b^2}\right )}{\sqrt {a}\,\sqrt {b}}+\frac {3^{1/4}\,\sqrt {\frac {1}{27}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{3/4}\,3^{3/4}\,b^2-{\left (-1\right )}^{3/4}\,3^{3/4}\,a^2\,x^2-3\,{\left (-1\right )}^{1/4}\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+1{}\mathrm {i}\,\sqrt {3}\,a\,b\,x-b^2}\right )}{\sqrt {a}\,\sqrt {b}} \]
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