Optimal. Leaf size=77 \[ -\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {-x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {118, 117}
\begin {gather*} -\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b} \sqrt {-x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 117
Rule 118
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-x} \sqrt {a-b x} \sqrt {a+b x}} \, dx &=\frac {\left (\sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}}\right ) \int \frac {1}{\sqrt {-x} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}}} \, dx}{\sqrt {a-b x} \sqrt {a+b x}}\\ &=-\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {-x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.02, size = 66, normalized size = 0.86 \begin {gather*} \frac {2 x \sqrt {1-\frac {b^2 x^2}{a^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b^2 x^2}{a^2}\right )}{\sqrt {-x} \sqrt {a-b x} \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 92, normalized size = 1.19
method | result | size |
default | \(\frac {\sqrt {-b x +a}\, \sqrt {b x +a}\, a \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \EllipticF \left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-x}\, \left (-b^{2} x^{2}+a^{2}\right )}\) | \(92\) |
elliptic | \(\frac {\sqrt {-x \left (-b^{2} x^{2}+a^{2}\right )}\, a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x}\, \sqrt {-b x +a}\, \sqrt {b x +a}\, b \sqrt {b^{2} x^{3}-a^{2} x}}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 16, normalized size = 0.21 \begin {gather*} \frac {2 \, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.96, size = 95, normalized size = 1.23 \begin {gather*} \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} - \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {-x}\,\sqrt {a+b\,x}\,\sqrt {a-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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