Optimal. Leaf size=76 \[ \frac {(c+d x) \sqrt {\frac {a+b x}{c+d x}}}{d}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1959, 288, 208} \[ \frac {(c+d x) \sqrt {\frac {a+b x}{c+d x}}}{d}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 288
Rule 1959
Rubi steps
\begin {align*} \int \sqrt {\frac {a+b x}{c+d x}} \, dx &=(2 (b c-a d)) \operatorname {Subst}\left (\int \frac {x^2}{\left (b-d x^2\right )^2} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right )\\ &=\frac {\sqrt {\frac {a+b x}{c+d x}} (c+d x)}{d}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{b-d x^2} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right )}{d}\\ &=\frac {\sqrt {\frac {a+b x}{c+d x}} (c+d x)}{d}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 123, normalized size = 1.62 \[ \frac {\sqrt {\frac {a+b x}{c+d x}} \left (b \sqrt {d} (a+b x) (c+d x)-\sqrt {a+b x} (b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{b d^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 180, normalized size = 2.37 \[ \left [-\frac {{\left (b c - a d\right )} \sqrt {b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt {b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {\frac {b x + a}{d x + c}}}{2 \, b d^{2}}, \frac {{\left (b c - a d\right )} \sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d x + a d}\right ) + {\left (b d^{2} x + b c d\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 119, normalized size = 1.57 \[ \frac {\sqrt {b d x^{2} + b c x + a d x + a c} \mathrm {sgn}\left (d x + c\right )}{d} + \frac {{\left (b c \mathrm {sgn}\left (d x + c\right ) - a d \mathrm {sgn}\left (d x + c\right )\right )} \sqrt {b d} \log \left ({\left | -2 \, {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt {b d} b c - \sqrt {b d} a d \right |}\right )}{2 \, b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 152, normalized size = 2.00 \[ \frac {\sqrt {\frac {b x +a}{d x +c}}\, \left (d x +c \right ) \left (a d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-b c \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 118, normalized size = 1.55 \[ \frac {{\left (b c - a d\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d - \frac {{\left (b x + a\right )} d^{2}}{d x + c}} + \frac {{\left (b c - a d\right )} \log \left (\frac {d \sqrt {\frac {b x + a}{d x + c}} - \sqrt {b d}}{d \sqrt {\frac {b x + a}{d x + c}} + \sqrt {b d}}\right )}{2 \, \sqrt {b d} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 90, normalized size = 1.18 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{\sqrt {b}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {b}\,d^{3/2}}+\frac {\left (a\,d-b\,c\right )\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{b\,d\,\left (\frac {d\,\left (a+b\,x\right )}{b\,\left (c+d\,x\right )}-1\right )} \]
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 \[ \int \sqrt {\frac {a + b x}{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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