Optimal. Leaf size=38 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b \sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {65, 223, 209}
\begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 48, normalized size = 1.26 \begin {gather*} \frac {2 \sqrt {a-b x} \tan ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b \sqrt {c (a-b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs.
\(2(30)=60\).
time = 0.19, size = 71, normalized size = 1.87
method | result | size |
default | \(\frac {\sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {b x +a}\, \sqrt {-b c x +a c}\, \sqrt {b^{2} c}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 14, normalized size = 0.37 \begin {gather*} \frac {\arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 108, normalized size = 2.84 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right )}{2 \, b c}, -\frac {\arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right )}{b \sqrt {c}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 13.23, size = 90, normalized size = 2.37 \begin {gather*} - \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b \sqrt {c}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.88, size = 42, normalized size = 1.11 \begin {gather*} -\frac {2 \, \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{b \sqrt {-c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 53, normalized size = 1.39 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a\,c-b\,c\,x}-\sqrt {a\,c}\right )}{\sqrt {b^2\,c}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {b^2\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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