Optimal. Leaf size=180 \[ -\frac {4}{9 (2 x+1)^{3/2}}+\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{3\ 3^{3/4}} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 180, 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 = {693, 694, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {4}{9 (2 x+1)^{3/2}}+\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{3\ 3^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 693
Rule 694
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx &=-\frac {4}{9 (1+2 x)^{3/2}}-\frac {1}{3} \int \frac {1}{\sqrt {1+2 x} \left (1+x+x^2\right )} \, dx\\ &=-\frac {4}{9 (1+2 x)^{3/2}}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (\frac {3}{4}+\frac {x^2}{4}\right )} \, dx,x,1+2 x\right )\\ &=-\frac {4}{9 (1+2 x)^{3/2}}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{9 (1+2 x)^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}-x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )}{6 \sqrt {3}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )}{6 \sqrt {3}}\\ &=-\frac {4}{9 (1+2 x)^{3/2}}+\frac {\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 )}{3 \sqrt {2} 3^{3/4}}+\frac {\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 )}{3 \sqrt {2} 3^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3 \sqrt {3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3 \sqrt {3}}\\ &=-\frac {4}{9 (1+2 x)^{3/2}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}+\frac {\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}\\ &=-\frac {4}{9 (1+2 x)^{3/2}}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 32, normalized size = 0.18 \begin {gather*} -\frac {4 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\frac {1}{3} (2 x+1)^2\right )}{9 (2 x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.29, size = 118, normalized size = 0.66 \begin {gather*} -\frac {4}{9 (2 x+1)^{3/2}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\frac {2 x+1}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3}}{\sqrt {2}}}{\sqrt {2 x+1}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} 3^{3/4} \sqrt {2 x+1}}{\sqrt {3} (2 x+1)+3}\right )}{3\ 3^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 252, normalized size = 1.40 \begin {gather*} \frac {4 \cdot 27^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \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 ) + 4 \cdot 27^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \arctan \left (\frac {1}{54} \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {-36 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 648 \, x + 324 \, \sqrt {3} + 324} - \frac {1}{3} \cdot 27^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 1\right ) - 27^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (36 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 648 \, x + 324 \, \sqrt {3} + 324\right ) + 27^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-36 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 648 \, x + 324 \, \sqrt {3} + 324\right ) - 72 \, \sqrt {2 \, x + 1}}{162 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 129, normalized size = 0.72 \begin {gather*} -\frac {1}{9} \cdot 12^{\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}{9} \cdot 12^{\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}{18} \cdot 12^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{18} \cdot 12^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{9 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 120, normalized size = 0.67 \begin {gather*} -\frac {3^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )}{9}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )}{9}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \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 )}{18}-\frac {4}{9 \left (2 x +1\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.00, size = 141, normalized size = 0.78 \begin {gather*} -\frac {1}{9} \cdot 3^{\frac {1}{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}{9} \cdot 3^{\frac {1}{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}{18} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{18} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{9 \, {\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 = 0.10, size = 66, normalized size = 0.37 \begin {gather*} -\frac {4}{9\,{\left (2\,x+1\right )}^{3/2}}+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}-\frac {1}{9}{}\mathrm {i}\right )+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}+\frac {1}{9}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________