Integrand size = 43, antiderivative size = 97 \[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\frac {(-b-2 k) \arctan \left (\frac {(-1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{2 (-1+k) k}+\frac {(b-2 k) \arctan \left (\frac {(1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{2 k (1+k)} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.80 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.77, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {6850, 6857, 116, 174, 552, 551} \[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\frac {\sqrt {1-x} \sqrt {x} (b+2 k) \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \operatorname {EllipticPi}\left (-\frac {k}{1-k},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{(1-k) k \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\sqrt {1-x} \sqrt {x} \left (2-\frac {b}{k}\right ) \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \operatorname {EllipticPi}\left (\frac {k}{k+1},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{(k+1) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \]
[In]
[Out]
Rule 116
Rule 174
Rule 551
Rule 552
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 {1+b x+k^2 x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+k^2 x^2\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 {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}}+\frac {2+b x}{\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 {\left (\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 {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {2+b x}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \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 {2+\frac {b}{k}}{2 \sqrt {1-x} \sqrt {x} (1-k x) \sqrt {1-k^2 x}}-\frac {2-\frac {b}{k}}{2 \sqrt {1-x} \sqrt {x} (1+k x) \sqrt {1-k^2 x}}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \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 (-2-\frac {b}{k}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} (1-k x) \sqrt {1-k^2 x}} \, dx}{2 \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (-2+\frac {b}{k}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} (1+k x) \sqrt {1-k^2 x}} \, dx}{2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \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 (-2-\frac {b}{k}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1-k+k x^2\right ) \sqrt {1-k^2+k^2 x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (-2+\frac {b}{k}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1+k-k x^2\right ) \sqrt {1-k^2+k^2 x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \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 (-2-\frac {b}{k}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1-k+k x^2\right ) \sqrt {1+\frac {k^2 x^2}{1-k^2}}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (-2+\frac {b}{k}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1+k-k x^2\right ) \sqrt {1+\frac {k^2 x^2}{1-k^2}}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \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 {(b+2 k) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \operatorname {EllipticPi}\left (-\frac {k}{1-k},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{(1-k) k \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2-\frac {b}{k}\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \operatorname {EllipticPi}\left (\frac {k}{1+k},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{(1+k) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ \end{align*}
Time = 10.97 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\frac {-\frac {(b+2 k) \arctan \left (\frac {(-1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{-1+k}+\frac {(b-2 k) \arctan \left (\frac {(1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{1+k}}{2 k} \]
[In]
[Out]
Time = 1.75 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\left (1+k \right ) \left (b +2 k \right ) \arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )-\arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right ) \left (-1+k \right ) \left (b -2 k \right )}{2 k^{3}-2 k}\) | \(81\) |
pseudoelliptic | \(\frac {\left (1+k \right ) \left (b +2 k \right ) \arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )-\arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right ) \left (-1+k \right ) \left (b -2 k \right )}{2 k^{3}-2 k}\) | \(81\) |
elliptic | \(-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\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 {x}{\frac {1}{k^{2}}-1}-\frac {1}{\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}}-\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\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}}-\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\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}}+\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\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}}+\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}\) | \(575\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (85) = 170\).
Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.03 \[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=-\frac {{\left ({\left (b + 2\right )} k - 2 \, k^{2} - b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right ) - {\left ({\left (b + 2\right )} k + 2 \, k^{2} + b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} - k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} - k^{2}\right )} x^{3} - {\left (k^{3} - k^{2} + k - 1\right )} x^{2} + {\left (k - 1\right )} x\right )}}\right )}{4 \, {\left (k^{3} - k\right )}} \]
[In]
[Out]
\[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\int \frac {b x + k^{2} x^{2} + 1}{\sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k x - 1\right ) \left (k x + 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\int { \frac {k^{2} x^{2} + b x + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
[In]
[Out]
\[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\int { \frac {k^{2} x^{2} + b x + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1+b x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\text {Hanged} \]
[In]
[Out]