Optimal. Leaf size=170 \[ \frac {4}{3} (2 x+1)^{3/2}-\frac {3^{3/4} \log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}+\frac {3^{3/4} \log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}+\sqrt {2} 3^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )-\sqrt {2} 3^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {692, 694, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {4}{3} (2 x+1)^{3/2}-\frac {3^{3/4} \log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}+\frac {3^{3/4} \log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}+\sqrt {2} 3^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )-\sqrt {2} 3^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 692
Rule 694
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(1+2 x)^{5/2}}{1+x+x^2} \, dx &=\frac {4}{3} (1+2 x)^{3/2}-3 \int \frac {\sqrt {1+2 x}}{1+x+x^2} \, dx\\ &=\frac {4}{3} (1+2 x)^{3/2}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\frac {3}{4}+\frac {x^2}{4}} \, dx,x,1+2 x\right )\\ &=\frac {4}{3} (1+2 x)^{3/2}-3 \operatorname {Subst}\left (\int \frac {x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {4}{3} (1+2 x)^{3/2}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {\sqrt {3}-x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {4}{3} (1+2 x)^{3/2}-3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-\frac {3^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}+2 x}{-\sqrt {3}-\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {3^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{-\sqrt {3}+\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {2}}\\ &=\frac {4}{3} (1+2 x)^{3/2}-\frac {3^{3/4} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}+\frac {3^{3/4} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\left (\sqrt {2} 3^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )+\left (\sqrt {2} 3^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )\\ &=\frac {4}{3} (1+2 x)^{3/2}+\sqrt {2} 3^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )-\sqrt {2} 3^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )-\frac {3^{3/4} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}+\frac {3^{3/4} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 34, normalized size = 0.20 \begin {gather*} -\frac {4}{3} (2 x+1)^{3/2} \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {1}{3} (2 x+1)^2\right )-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.24, size = 113, normalized size = 0.66 \begin {gather*} \frac {4}{3} (2 x+1)^{3/2}-\sqrt {2} 3^{3/4} \tan ^{-1}\left (\frac {\frac {2 x+1}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3}}{\sqrt {2}}}{\sqrt {2 x+1}}\right )+\sqrt {2} 3^{3/4} \tanh ^{-1}\left (\frac {\sqrt {2} 3^{3/4} \sqrt {2 x+1}}{\sqrt {3} (2 x+1)+3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 199, normalized size = 1.17 \begin {gather*} 2 \cdot 27^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{9} \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 18 \, x + 9 \, \sqrt {3} + 9} - \frac {1}{3} \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} - 1\right ) + 2 \cdot 27^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{27} \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {-9 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 162 \, x + 81 \, \sqrt {3} + 81} - \frac {1}{3} \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 1\right ) + \frac {1}{2} \cdot 27^{\frac {1}{4}} \sqrt {2} \log \left (9 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 162 \, x + 81 \, \sqrt {3} + 81\right ) - \frac {1}{2} \cdot 27^{\frac {1}{4}} \sqrt {2} \log \left (-9 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 162 \, x + 81 \, \sqrt {3} + 81\right ) + \frac {4}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 129, normalized size = 0.76 \begin {gather*} \frac {4}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - 108^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{2} \cdot 108^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {1}{2} \cdot 108^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 120, normalized size = 0.71 \begin {gather*} -3^{\frac {3}{4}} \sqrt {2}\, \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )-3^{\frac {3}{4}} \sqrt {2}\, \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \ln \left (\frac {2 x +1+\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{2 x +1+\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )}{2}+\frac {4 \left (2 x +1\right )^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.01, size = 141, normalized size = 0.83 \begin {gather*} -3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - 3^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{2} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {1}{2} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {4}{3} \, {\left (2 \, x + 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 = 66, normalized size = 0.39 \begin {gather*} \frac {4\,{\left (2\,x+1\right )}^{3/2}}{3}+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-1+1{}\mathrm {i}\right )+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-1-\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 25.15, size = 163, normalized size = 0.96 \begin {gather*} \frac {4 \left (2 x + 1\right )^{\frac {3}{2}}}{3} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (2 x - \sqrt {2} \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{2} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (2 x + \sqrt {2} \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{2} - \sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} - 1 \right )} - \sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________