Optimal. Leaf size=33 \[ x-\frac {4 \sqrt {a+b x}}{b}+\frac {4 \log \left (1+\sqrt {a+b x}\right )}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {442, 383, 78}
\begin {gather*} -\frac {4 \sqrt {a+b x}}{b}+\frac {4 \log \left (\sqrt {a+b x}+1\right )}{b}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 383
Rule 442
Rubi steps
\begin {align*} \int \frac {-1+\sqrt {a+b x}}{1+\sqrt {a+b x}} \, dx &=\frac {\text {Subst}\left (\int \frac {-1+\sqrt {x}}{1+\sqrt {x}} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 \text {Subst}\left (\int \frac {(-1+x) x}{1+x} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \text {Subst}\left (\int \left (-2+x+\frac {2}{1+x}\right ) \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=x-\frac {4 \sqrt {a+b x}}{b}+\frac {4 \log \left (1+\sqrt {a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 1.09 \begin {gather*} \frac {a+b x-4 \sqrt {a+b x}+4 \log \left (b \left (1+\sqrt {a+b x}\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 35, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {b x +a -4 \sqrt {b x +a}+4 \ln \left (1+\sqrt {b x +a}\right )}{b}\) | \(35\) |
default | \(\frac {b x +a -4 \sqrt {b x +a}+4 \ln \left (1+\sqrt {b x +a}\right )}{b}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 0.91 \begin {gather*} \frac {b x + a - 4 \, \sqrt {b x + a} + 4 \, \log \left (\sqrt {b x + a} + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 29, normalized size = 0.88 \begin {gather*} \frac {b x - 4 \, \sqrt {b x + a} + 4 \, \log \left (\sqrt {b x + a} + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 42, normalized size = 1.27 \begin {gather*} \begin {cases} x - \frac {4 \sqrt {a + b x}}{b} + \frac {4 \log {\left (\sqrt {a + b x} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x \left (\sqrt {a} - 1\right )}{\sqrt {a} + 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.41, size = 38, normalized size = 1.15 \begin {gather*} \frac {4 \, \log \left (\sqrt {b x + a} + 1\right )}{b} + \frac {{\left (b x + a\right )} b - 4 \, \sqrt {b x + a} b}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.02, size = 29, normalized size = 0.88 \begin {gather*} x+\frac {4\,\ln \left (\sqrt {a+b\,x}+1\right )}{b}-\frac {4\,\sqrt {a+b\,x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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