Optimal. Leaf size=33 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {b} \sqrt {c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {116} \[ \frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 116
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx &=\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 27, normalized size = 0.82 \[ \frac {2 x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )}{\sqrt {b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}}{b c^{2} x^{3} - b x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 32, normalized size = 0.97 \[ \frac {\sqrt {2}\, \sqrt {-c x}\, \EllipticF \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b x}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {b\,x}\,\sqrt {1-c\,x}\,\sqrt {c\,x+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 6.46, size = 94, normalized size = 2.85 \[ \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {c}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________