Integrand size = 46, antiderivative size = 178 \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\frac {\left (-b-2 a k^2\right ) \arctan \left (\frac {(-1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{4 (-1+k) k^2}+\frac {\left (-b-2 a k^2\right ) \arctan \left (\frac {(1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{4 k^2 (1+k)}+\frac {\left (b-2 a k^2\right ) \arctan \left (\frac {\sqrt {1+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{2 k^2 \sqrt {1+k^2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.82 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.78, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6850, 6857, 116, 6820, 948, 12, 174, 551} \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=-\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (2 a k^2+b\right ) \operatorname {EllipticPi}\left (-\frac {1}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (2 a k^2+b\right ) \operatorname {EllipticPi}\left (\frac {1}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (b-2 a k^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (b-2 a k^2\right ) \operatorname {EllipticPi}\left (\frac {1}{\sqrt {-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \]
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Rule 12
Rule 116
Rule 174
Rule 551
Rule 948
Rule 6820
Rule 6850
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {a+b x^2+a k^4 x^4}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+k^4 x^4\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {a}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}}+\frac {2 a+b x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+k^4 x^4\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {2 a+b x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+k^4 x^4\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (-\frac {b+2 a k^2}{2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1-k^2 x^2\right )}+\frac {b-2 a k^2}{2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1+k^2 x^2\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1+k^2 x^2\right )} \, dx}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1-k^2 x^2\right )} \, dx}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1+k^2 x^2\right )} \, dx}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-k^2 x^2\right )} \, dx}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {1}{2 \sqrt {1-k^2 x} \left (1-\sqrt {-k^2} x\right ) \sqrt {x-x^2}}+\frac {1}{2 \sqrt {1-k^2 x} \left (1+\sqrt {-k^2} x\right ) \sqrt {x-x^2}}\right ) \, dx}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {1}{2 (1-k x) \sqrt {1-k^2 x} \sqrt {x-x^2}}+\frac {1}{2 (1+k x) \sqrt {1-k^2 x} \sqrt {x-x^2}}\right ) \, dx}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (1-\sqrt {-k^2} x\right ) \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (1+\sqrt {-k^2} x\right ) \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{(1-k x) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{(1+k x) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (1-\sqrt {-k^2} x\right )} \, dx}{2 \sqrt {2} k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (1+\sqrt {-k^2} x\right )} \, dx}{2 \sqrt {2} k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} (1-k x) \sqrt {1-k^2 x}} \, dx}{2 \sqrt {2} k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} (1+k x) \sqrt {1-k^2 x}} \, dx}{2 \sqrt {2} k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (1-\sqrt {-k^2} x\right )} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b-2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (1+\sqrt {-k^2} x\right )} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} (1-k x) \sqrt {1-k^2 x}} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} (1+k x) \sqrt {1-k^2 x}} \, dx}{4 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b-2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (1-\sqrt {-k^2} x^2\right )} \, dx,x,\sqrt {-x}\right )}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b-2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (1+\sqrt {-k^2} x^2\right )} \, dx,x,\sqrt {-x}\right )}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b+2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (1-k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (\left (b+2 a k^2\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (1+k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{2 k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ & = \frac {2 a \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (b+2 a k^2\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (-\frac {1}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (b+2 a k^2\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (\frac {1}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (b-2 a k^2\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (b-2 a k^2\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ \end{align*}
Time = 12.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=-\frac {\frac {\left (b+2 a k^2\right ) \arctan \left (\frac {(-1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{-1+k}+\frac {\left (b+2 a k^2\right ) \arctan \left (\frac {(1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{1+k}-\frac {2 \left (b-2 a k^2\right ) \arctan \left (\frac {\sqrt {1+k^2} x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {1+k^2}}}{4 k^2} \]
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Time = 1.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right )}{2+2 k}+\frac {\left (2 a \,k^{2}-b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {k^{2}+1}}\right )}{\sqrt {k^{2}+1}}+\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )}{-2+2 k}}{2 k^{2}}\) | \(129\) |
pseudoelliptic | \(\frac {\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right )}{2+2 k}+\frac {\left (2 a \,k^{2}-b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {k^{2}+1}}\right )}{\sqrt {k^{2}+1}}+\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )}{-2+2 k}}{2 k^{2}}\) | \(129\) |
elliptic | \(-\frac {2 a \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-1\right )}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{2 k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-1\right )}-\frac {\left (2 a \,k^{2}-b \right ) \sqrt {-k^{2} x +1}\, \sqrt {-\frac {\left (-1+x \right ) k^{2}}{k^{2}-1}}\, \sqrt {k^{2} x}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2} k^{2}+1\right )}{\sum }\operatorname {EllipticPi}\left (\sqrt {-k^{2} x +1}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{2}+1}, \sqrt {-\frac {1}{k^{2}-1}}\right )\right )}{2 k^{2} \left (k^{2}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}-\frac {\left (2 a \,k^{2}-b \right ) \sqrt {-k^{2} x +1}\, \sqrt {-\frac {\left (-1+x \right ) k^{2}}{k^{2}-1}}\, \sqrt {k^{2} x}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2} k^{2}+1\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {-k^{2} x +1}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{2}+1}, \sqrt {-\frac {1}{k^{2}-1}}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2 k^{4} \left (k^{2}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}+\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+1\right )}+\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{2 k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+1\right )}\) | \(813\) |
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Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (157) = 314\).
Time = 0.73 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.00 \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\frac {2 \, {\left (2 \, a k^{4} - {\left (2 \, a + b\right )} k^{2} + b\right )} \sqrt {k^{2} + 1} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + 1\right )} x + 1\right )} \sqrt {k^{2} + 1}}{2 \, {\left ({\left (k^{4} + k^{2}\right )} x^{3} - {\left (k^{4} + 2 \, k^{2} + 1\right )} x^{2} + {\left (k^{2} + 1\right )} x\right )}}\right ) + {\left (2 \, a k^{5} - 2 \, a k^{4} + {\left (2 \, a + b\right )} k^{3} - {\left (2 \, a + b\right )} k^{2} + b k - b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right ) + {\left (2 \, a k^{5} + 2 \, a k^{4} + {\left (2 \, a + b\right )} k^{3} + {\left (2 \, a + b\right )} k^{2} + b k + b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} - k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} - k^{2}\right )} x^{3} - {\left (k^{3} - k^{2} + k - 1\right )} x^{2} + {\left (k - 1\right )} x\right )}}\right )}{8 \, {\left (k^{6} - k^{2}\right )}} \]
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\[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\int \frac {a k^{4} x^{4} + a + b x^{2}}{\sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k x - 1\right ) \left (k x + 1\right ) \left (k^{2} x^{2} + 1\right )}\, dx \]
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\[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\int { \frac {a k^{4} x^{4} + b x^{2} + a}{{\left (k^{4} x^{4} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
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\[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\int { \frac {a k^{4} x^{4} + b x^{2} + a}{{\left (k^{4} x^{4} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
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Timed out. \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\text {Hanged} \]
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