Optimal. Leaf size=41 \[ \frac {2 \sqrt {c+d x}}{b d}-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {247, 190, 43} \[ \frac {2 \sqrt {c+d x}}{b d}-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{a+b \sqrt {c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b \sqrt {x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \sqrt {c+d x}}{b d}-\frac {2 a \log \left (a+b \sqrt {c+d x}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.95 \[ \frac {2 \left (\frac {\sqrt {c+d x}}{b}-\frac {a \log \left (a+b \sqrt {c+d x}\right )}{b^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 33, normalized size = 0.80 \[ -\frac {2 \, {\left (a \log \left (\sqrt {d x + c} b + a\right ) - \sqrt {d x + c} b\right )}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 38, normalized size = 0.93 \[ -\frac {2 \, a \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{2} d} + \frac {2 \, \sqrt {d x + c}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 87, normalized size = 2.12 \[ \frac {a \ln \left (-a +\sqrt {d x +c}\, b \right )}{b^{2} d}-\frac {a \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d}-\frac {a \ln \left (b^{2} d x +b^{2} c -a^{2}\right )}{b^{2} d}+\frac {2 \sqrt {d x +c}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 35, normalized size = 0.85 \[ -\frac {2 \, {\left (\frac {a \log \left (\sqrt {d x + c} b + a\right )}{b^{2}} - \frac {\sqrt {d x + c}}{b}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 33, normalized size = 0.80 \[ -\frac {2\,\left (a\,\ln \left (a+b\,\sqrt {c+d\,x}\right )-b\,\sqrt {c+d\,x}\right )}{b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 49, normalized size = 1.20 \[ \begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x}{a + b \sqrt {c}} & \text {for}\: d = 0 \\- \frac {2 a \log {\left (\frac {a}{b} + \sqrt {c + d x} \right )}}{b^{2} d} + \frac {2 \sqrt {c + d x}}{b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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