Optimal. Leaf size=43 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {b x-\frac {b (1-c)}{d}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, 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 = {63, 215} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {b x-\frac {b (1-c)}{d}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 215
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx &=\frac {2 \operatorname {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 \sinh ^{-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 = 0.03, size = 41, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {c+d x-1} \sinh ^{-1}\left (\sqrt {c+d x-1}\right )}{d \sqrt {\frac {b (c+d x-1)}{d}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 57, normalized size = 1.33 \begin {gather*} -\frac {2 \sqrt {\frac {b}{d}} \log \left (\sqrt {\frac {b (c+d x)}{d}-\frac {b}{d}}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 175, normalized size = 4.07 \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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giac [A] time = 0.97, size = 57, normalized size = 1.33 \begin {gather*} -\frac {2 \, b \log \left (-\sqrt {b d} \sqrt {\frac {b d x + b c - b}{d}} + \sqrt {{\left (b d x + b c - b\right )} b + b^{2}}\right )}{\sqrt {b d} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 100, normalized size = 2.33 \begin {gather*} \frac {\sqrt {\left (b x +\frac {\left (c -1\right ) b}{d}\right ) \left (d x +c \right )}\, \ln \left (\frac {b d x +\frac {b c}{2}+\frac {\left (c -1\right ) b}{2}}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\frac {\left (c -1\right ) b c}{d}+\left (b c +\left (c -1\right ) b \right ) x}\right )}{\sqrt {b x +\frac {\left (c -1\right ) b}{d}}\, \sqrt {d x +c}\, \sqrt {b d}} \end {gather*}
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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mupad [B] time = 0.50, size = 66, normalized size = 1.53 \begin {gather*} \frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {b\,x-\frac {b-b\,c}{d}}-\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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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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