Integrand size = 53, antiderivative size = 293 \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\frac {\left (2 i b+2 i a k^2-i b k^2-2 b \sqrt {-1+k^2}-2 a k^2 \sqrt {-1+k^2}\right ) \arctan \left (\frac {\sqrt {-2+k^2-2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{k^2 (-1+x) x}\right )}{2 k^2 \sqrt {-1+k^2} \sqrt {-2+k^2-2 i \sqrt {-1+k^2}}}+\frac {\left (-2 i b-2 i a k^2+i b k^2-2 b \sqrt {-1+k^2}-2 a k^2 \sqrt {-1+k^2}\right ) \arctan \left (\frac {\sqrt {-2+k^2+2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{k^2 (-1+x) x}\right )}{2 k^2 \sqrt {-1+k^2} \sqrt {-2+k^2+2 i \sqrt {-1+k^2}}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.18 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6850, 6820, 6860, 6, 728, 116, 948, 12, 174, 551} \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=-\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (-2 \sqrt {1-k^2} \left (a k^2+b\right )+2 a k^2+b \left (2-k^2\right )\right ) \operatorname {EllipticPi}\left (\frac {1}{1-\sqrt {1-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \left (1-\sqrt {1-k^2}\right ) \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (2 \sqrt {1-k^2} \left (a k^2+b\right )+2 a k^2+b \left (2-k^2\right )\right ) \operatorname {EllipticPi}\left (\frac {1}{\sqrt {1-k^2}+1},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \left (\sqrt {1-k^2}+1\right ) \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (a+\frac {b}{k^2}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \]
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Rule 6
Rule 12
Rule 116
Rule 174
Rule 551
Rule 728
Rule 948
Rule 6820
Rule 6850
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1-2 x+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 \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+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 {a}{\sqrt {1-k^2 x} \sqrt {x-x^2}}+\frac {b}{k^2 \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {b+2 a k^2-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{k^2 \sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+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 \left (\frac {a+\frac {b}{k^2}}{\sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {b+2 a k^2-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{k^2 \sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+k^2 x^2\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {\left (\left (a+\frac {b}{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}} \, 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 {b+2 a k^2-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+k^2 x^2\right )} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {\left (\left (a+\frac {b}{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}} \, 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 {-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )}{\sqrt {1-k^2 x} \left (-2-2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}}+\frac {-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}}\right ) \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \left (a+\frac {b}{k^2}\right ) \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 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (-2-2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \left (a+\frac {b}{k^2}\right ) \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 {2} \left (-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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 (-2-2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\sqrt {2} \left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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 (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ & = \frac {2 \left (a+\frac {b}{k^2}\right ) \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 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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 (-2-2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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 (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ & = \frac {2 \left (a+\frac {b}{k^2}\right ) \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 (2 \left (-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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 (-2 \left (1+\sqrt {1-k^2}\right )-2 k^2 x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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 (-2 \left (1-\sqrt {1-k^2}\right )-2 k^2 x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ & = \frac {2 \left (a+\frac {b}{k^2}\right ) \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 \left (2-k^2-2 \sqrt {1-k^2}\right )+2 a k^2 \left (1-\sqrt {1-k^2}\right )\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (\frac {1}{1-\sqrt {1-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \left (1-\sqrt {1-k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (2 a k^2 \left (1+\sqrt {1-k^2}\right )+b \left (2-k^2+2 \sqrt {1-k^2}\right )\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (\frac {1}{1+\sqrt {1-k^2}},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \left (1+\sqrt {1-k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 28.01 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.83 \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\frac {i \sqrt {1+\frac {1}{-1+x}} \sqrt {1+\frac {1-\frac {1}{k^2}}{-1+x}} (-1+x)^{3/2} \left (2 a k^2 \sqrt {1-k^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {-1+x}}\right ),1-\frac {1}{k^2}\right )+\left (-2 a k^2+b \left (-1+\sqrt {1-k^2}\right )\right ) \operatorname {EllipticPi}\left (\frac {-1+k^2}{-1+k^2-\sqrt {1-k^2}},i \text {arcsinh}\left (\frac {1}{\sqrt {-1+x}}\right ),1-\frac {1}{k^2}\right )+\left (b+2 a k^2+b \sqrt {1-k^2}\right ) \operatorname {EllipticPi}\left (\frac {-1+k^2}{-1+k^2+\sqrt {1-k^2}},i \text {arcsinh}\left (\frac {1}{\sqrt {-1+x}}\right ),1-\frac {1}{k^2}\right )\right )}{k^2 \sqrt {1-k^2} \sqrt {(-1+x) x \left (-1+k^2 x\right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 1907, normalized size of antiderivative = 6.51
method | result | size |
default | \(\text {Expression too large to display}\) | \(1907\) |
elliptic | \(\text {Expression too large to display}\) | \(2014\) |
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Time = 0.66 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.68 \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\left [-\frac {{\left (2 \, a k^{2} + b\right )} \sqrt {-k^{2} + 1} \log \left (\frac {k^{4} x^{4} - 4 \, {\left (2 \, k^{4} - k^{2}\right )} x^{3} + 2 \, {\left (4 \, k^{4} + k^{2} - 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {-k^{2} + 1} - 4 \, {\left (2 \, k^{2} - 1\right )} x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) - {\left (b k^{2} - b\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}, \frac {2 \, {\left (2 \, a k^{2} + b\right )} \sqrt {k^{2} - 1} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {k^{2} - 1}}{2 \, {\left ({\left (k^{4} - k^{2}\right )} x^{3} - {\left (k^{4} - 1\right )} x^{2} + {\left (k^{2} - 1\right )} x\right )}}\right ) + {\left (b k^{2} - b\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\int { \frac {{\left (a k^{2} + b\right )} x^{2} - b x - a}{{\left (k^{2} x^{2} - 2 \, x + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
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Timed out. \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\int -\frac {\left (-a\,k^2-b\right )\,x^2+b\,x+a}{\left (k^2\,x^2-2\,x+1\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}} \,d x \]
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