Optimal. Leaf size=33 \[ \frac {2 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]
[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 = {111}
\begin {gather*} \frac {2 E\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 111
Rubi steps
\begin {align*} \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx &=\frac {2 E\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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.80, size = 52, normalized size = 1.58 \begin {gather*} \frac {2 x \left (3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )+c x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right )\right )}{3 \sqrt {b x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs.
\(2(23)=46\).
time = 0.09, size = 49, normalized size = 1.48
method | result | size |
default | \(\frac {2 \sqrt {2}\, \sqrt {-c x}\, \left (\EllipticF \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )-\EllipticE \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )\right )}{c \sqrt {b x}}\) | \(49\) |
elliptic | \(\frac {\sqrt {-b x \left (c^{2} x^{2}-1\right )}\, \left (\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {-b \,c^{2} x^{3}+b x}}+\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \left (-\frac {2 \EllipticE \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {-b \,c^{2} x^{3}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {c x +1}}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 52, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (\sqrt {-b c^{2}} c {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) - \sqrt {-b c^{2}} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right )}}{b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x + 1}}{\sqrt {b x} \sqrt {- c x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {c\,x+1}}{\sqrt {b\,x}\,\sqrt {1-c\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________