Optimal. Leaf size=19 \[ 4 \sqrt {x}+x+2 \log \left (4+\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {28, 1411, 785}
\begin {gather*} x+4 \sqrt {x}+2 \log \left (\sqrt {x}+4\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 785
Rule 1411
Rubi steps
\begin {align*} \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx &=\int \frac {\left (3+\sqrt {x}\right )^2}{4 \sqrt {x}+x} \, dx\\ &=2 \text {Subst}\left (\int \frac {x (3+x)^2}{4 x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (2+x+\frac {1}{4+x}\right ) \, dx,x,\sqrt {x}\right )\\ &=4 \sqrt {x}+x+2 \log \left (4+\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} 4 \sqrt {x}+x+2 \log \left (4+\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.41, size = 16, normalized size = 0.84
method | result | size |
derivativedivides | \(x +2 \ln \left (4+\sqrt {x}\right )+4 \sqrt {x}\) | \(16\) |
default | \(x +2 \ln \left (4+\sqrt {x}\right )+4 \sqrt {x}\) | \(16\) |
trager | \(-1+x +4 \sqrt {x}+\ln \left (8 \sqrt {x}+16+x \right )\) | \(18\) |
meijerg | \(2 \ln \left (1+\frac {\sqrt {x}}{4}\right )-\frac {4 \sqrt {x}\, \left (-\frac {3 \sqrt {x}}{4}+6\right )}{3}+12 \sqrt {x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 15, normalized size = 0.79 \begin {gather*} x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 15, normalized size = 0.79 \begin {gather*} x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.06, size = 17, normalized size = 0.89 \begin {gather*} 4 \sqrt {x} + x + 2 \log {\left (\sqrt {x} + 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.20, size = 15, normalized size = 0.79 \begin {gather*} x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.04, size = 15, normalized size = 0.79 \begin {gather*} x+2\,\ln \left (\sqrt {x}+4\right )+4\,\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________