Optimal. Leaf size=27 \[ -\frac {2}{9} \left (1+x^3\right )^{3/2}+\frac {2}{15} \left (1+x^3\right )^{5/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} \frac {2}{15} \left (x^3+1\right )^{5/2}-\frac {2}{9} \left (x^3+1\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{3} \text {Subst}\left (\int x \sqrt {1+x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2}{9} \left (1+x^3\right )^{3/2}+\frac {2}{15} \left (1+x^3\right )^{5/2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 20, normalized size = 0.74 \begin {gather*} \frac {2}{45} \left (1+x^3\right )^{3/2} \left (-2+3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 35, normalized size = 1.30
method | result | size |
pseudoelliptic | \(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}} \left (3 x^{3}-2\right )}{45}\) | \(17\) |
risch | \(\frac {2 \left (3 x^{6}+x^{3}-2\right ) \sqrt {x^{3}+1}}{45}\) | \(20\) |
trager | \(\left (\frac {2}{15} x^{6}+\frac {2}{45} x^{3}-\frac {4}{45}\right ) \sqrt {x^{3}+1}\) | \(21\) |
gosper | \(\frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (3 x^{3}-2\right ) \sqrt {x^{3}+1}}{45}\) | \(28\) |
meijerg | \(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (x^{3}+1\right )^{\frac {3}{2}} \left (-3 x^{3}+2\right )}{15}}{6 \sqrt {\pi }}\) | \(31\) |
default | \(\frac {2 x^{6} \sqrt {x^{3}+1}}{15}+\frac {2 x^{3} \sqrt {x^{3}+1}}{45}-\frac {4 \sqrt {x^{3}+1}}{45}\) | \(35\) |
elliptic | \(\frac {2 x^{6} \sqrt {x^{3}+1}}{15}+\frac {2 x^{3} \sqrt {x^{3}+1}}{45}-\frac {4 \sqrt {x^{3}+1}}{45}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.39, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{15} \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.57, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{45} \, {\left (3 \, x^{6} + x^{3} - 2\right )} \sqrt {x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 41, normalized size = 1.52 \begin {gather*} \frac {2 x^{6} \sqrt {x^{3} + 1}}{15} + \frac {2 x^{3} \sqrt {x^{3} + 1}}{45} - \frac {4 \sqrt {x^{3} + 1}}{45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{15} \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 19, normalized size = 0.70 \begin {gather*} \frac {2\,{\left (x^3+1\right )}^{5/2}}{15}-\frac {2\,{\left (x^3+1\right )}^{3/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Chatgpt [F] Failed to verify
time = 1.00, size = 5, normalized size = 0.19 \begin {gather*} \frac {2 x^{\frac {21}{2}}}{21} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________