Optimal. Leaf size=35 \[ \sqrt {1+x} \sqrt {2+3 x}-\frac {\sinh ^{-1}\left (\sqrt {2+3 x}\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 35, 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 = {52, 56, 221}
\begin {gather*} \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{\sqrt {1+x}} \, dx &=\sqrt {1+x} \sqrt {2+3 x}-\frac {1}{2} \int \frac {1}{\sqrt {1+x} \sqrt {2+3 x}} \, dx\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {2+3 x}\right )}{\sqrt {3}}\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\sinh ^{-1}\left (\sqrt {2+3 x}\right )}{\sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 43, normalized size = 1.23 \begin {gather*} \sqrt {1+x} \sqrt {2+3 x}-\frac {\tanh ^{-1}\left (\sqrt {\frac {2+3 x}{3+3 x}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs.
\(2(27)=54\).
time = 0.56, size = 67, normalized size = 1.91
method | result | size |
default | \(\sqrt {1+x}\, \sqrt {2+3 x}-\frac {\sqrt {\left (1+x \right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{6 \sqrt {2+3 x}\, \sqrt {1+x}}\) | \(67\) |
risch | \(\sqrt {1+x}\, \sqrt {2+3 x}-\frac {\sqrt {\left (1+x \right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{6 \sqrt {2+3 x}\, \sqrt {1+x}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 41, normalized size = 1.17 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 52, normalized size = 1.49 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (-4 \, \sqrt {3} {\left (6 \, x + 5\right )} \sqrt {3 \, x + 2} \sqrt {x + 1} + 72 \, x^{2} + 120 \, x + 49\right ) + \sqrt {3 \, x + 2} \sqrt {x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.92, size = 100, normalized size = 2.86 \begin {gather*} \begin {cases} \sqrt {x + 1} \sqrt {3 x + 2} - \frac {\sqrt {3} \operatorname {acosh}{\left (\sqrt {3} \sqrt {x + 1} \right )}}{3} & \text {for}\: \left |{x + 1}\right | > \frac {1}{3} \\\frac {\sqrt {3} i \operatorname {asin}{\left (\sqrt {3} \sqrt {x + 1} \right )}}{3} - \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {- 3 x - 2}} + \frac {i \sqrt {x + 1}}{\sqrt {- 3 x - 2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.40, size = 39, normalized size = 1.11 \begin {gather*} \frac {1}{3} \, \sqrt {3} {\left (\sqrt {3 \, x + 3} \sqrt {3 \, x + 2} + \log \left (\sqrt {3 \, x + 3} - \sqrt {3 \, x + 2}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.14, size = 172, normalized size = 4.91 \begin {gather*} \frac {2\,\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}{3\,\left (\sqrt {x+1}-1\right )}\right )}{3}-\frac {\frac {30\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}{\sqrt {x+1}-1}+\frac {10\,{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {24\,\sqrt {2}\,{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}}{\frac {{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {6\,{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________