Integrand size = 36, antiderivative size = 74 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=-\frac {2 \sqrt {\frac {f (c+d x)}{d+c f}} \operatorname {EllipticPi}\left (\frac {2 b}{b+a f},\arcsin \left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right ),\frac {2 d}{d+c f}\right )}{(b+a f) \sqrt {c+d x}} \]
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Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {174, 552, 551} \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=-\frac {2 \sqrt {\frac {f (c+d x)}{c f+d}} \operatorname {EllipticPi}\left (\frac {2 b}{b+a f},\arcsin \left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right ),\frac {2 d}{d+c f}\right )}{(a f+b) \sqrt {c+d x}} \]
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Rule 174
Rule 551
Rule 552
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (b+a f-b x^2\right ) \sqrt {c+\frac {d}{f}-\frac {d x^2}{f}}} \, dx,x,\sqrt {1-f x}\right )\right ) \\ & = -\frac {\left (2 \sqrt {\frac {f (c+d x)}{d+c f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (b+a f-b x^2\right ) \sqrt {1-\frac {d x^2}{\left (c+\frac {d}{f}\right ) f}}} \, dx,x,\sqrt {1-f x}\right )}{\sqrt {c+d x}} \\ & = -\frac {2 \sqrt {\frac {f (c+d x)}{d+c f}} \Pi \left (\frac {2 b}{b+a f};\sin ^{-1}\left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right )|\frac {2 d}{d+c f}\right )}{(b+a f) \sqrt {c+d x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 22.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.74 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=\frac {2 i (c+d x) \sqrt {\frac {d (-1+f x)}{f (c+d x)}} \sqrt {\frac {d+d f x}{c f+d f x}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d+c f}{f}}}{\sqrt {c+d x}}\right ),\frac {-d+c f}{d+c f}\right )-\operatorname {EllipticPi}\left (\frac {b c f-a d f}{b d+b c f},i \text {arcsinh}\left (\frac {\sqrt {-\frac {d+c f}{f}}}{\sqrt {c+d x}}\right ),\frac {-d+c f}{d+c f}\right )\right )}{(-b c+a d) \sqrt {-\frac {d+c f}{f}} \sqrt {1-f^2 x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs. \(2(71)=142\).
Time = 3.41 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.49
method | result | size |
default | \(-\frac {2 \left (c f -d \right ) \Pi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d}}, -\frac {\left (c f -d \right ) b}{f \left (a d -b c \right )}, \sqrt {\frac {c f -d}{c f +d}}\right ) \sqrt {-\frac {\left (f x +1\right ) d}{c f -d}}\, \sqrt {-\frac {\left (f x -1\right ) d}{c f +d}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d}}\, \sqrt {f x +1}\, \sqrt {-f x +1}\, \sqrt {d x +c}}{f \left (a d -b c \right ) \left (d \,f^{2} x^{3}+c \,f^{2} x^{2}-d x -c \right )}\) | \(184\) |
elliptic | \(\frac {2 \sqrt {-\left (f^{2} x^{2}-1\right ) \left (d x +c \right )}\, \left (\frac {c}{d}-\frac {1}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {1}{f}}}\, \sqrt {\frac {x -\frac {1}{f}}{-\frac {c}{d}-\frac {1}{f}}}\, \sqrt {\frac {x +\frac {1}{f}}{-\frac {c}{d}+\frac {1}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {1}{f}}}, \frac {-\frac {c}{d}+\frac {1}{f}}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {-\frac {c}{d}+\frac {1}{f}}{-\frac {c}{d}-\frac {1}{f}}}\right )}{\sqrt {d x +c}\, \sqrt {-f x +1}\, \sqrt {f x +1}\, b \sqrt {-d \,f^{2} x^{3}-c \,f^{2} x^{2}+d x +c}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\) | \(239\) |
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Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt {c + d x} \sqrt {- f x + 1} \sqrt {f x + 1}}\, dx \]
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\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + 1} \sqrt {-f x + 1}} \,d x } \]
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\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + 1} \sqrt {-f x + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx=\int \frac {1}{\sqrt {1-f\,x}\,\sqrt {f\,x+1}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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