Optimal. Leaf size=42 \[ \frac {2 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {65, 222}
\begin {gather*} \frac {2 \text {ArcSin}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 222
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {b-b c}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {\frac {b-b c}{d}+b x}\right )}{b}\\ &=\frac {2 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 58, normalized size = 1.38 \begin {gather*} -\frac {2 \sqrt {1-c+d x} \tan ^{-1}\left (\frac {\sqrt {c-d x}}{\sqrt {1-c+d x}}\right )}{d \sqrt {\frac {b (1-c+d x)}{d}}} \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. \(108\) vs.
\(2(32)=64\).
time = 0.18, size = 109, normalized size = 2.60
method | result | size |
default | \(\frac {\sqrt {\left (b x -\frac {b \left (c -1\right )}{d}\right ) \left (-d x +c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {b \left (c -1\right )+b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+\left (b \left (c -1\right )+b c \right ) x -\frac {b \left (c -1\right ) c}{d}}}\right )}{\sqrt {b x -\frac {b \left (c -1\right )}{d}}\, \sqrt {-d x +c}\, \sqrt {b d}}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (31) = 62\).
time = 1.36, size = 176, normalized size = 4.19 \begin {gather*} \left [-\frac {\sqrt {-b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} - 8 \, {\left (2 \, b c - b\right )} d x - 4 \, \sqrt {-b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}}}{2 \, {\left (b d^{2} x^{2} + b c^{2} - {\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \left (- \frac {c}{d} + x + \frac {1}{d}\right )} \sqrt {c - d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (31) = 62\).
time = 1.75, size = 69, normalized size = 1.64 \begin {gather*} -\frac {2 \, {\left | b \right |} \log \left (-\sqrt {b d^{2} x - b c d + b d} \sqrt {-b d} + \sqrt {b^{2} d^{2} - {\left (b d^{2} x - b c d + b d\right )} b d}\right )}{\sqrt {-b d} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.51, size = 63, normalized size = 1.50 \begin {gather*} -\frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {\frac {b-b\,c}{d}+b\,x}-\sqrt {\frac {b-b\,c}{d}}\right )}{\sqrt {b\,d}\,\left (\sqrt {c-d\,x}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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