Optimal. Leaf size=80 \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2}-2 \sqrt{x+1}+(1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1-i}}\right )+(1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1+i}}\right ) \]
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Rubi [B] time = 0.285622, antiderivative size = 224, normalized size of antiderivative = 2.8, number of steps used = 16, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1625, 1629, 825, 12, 708, 1094, 634, 618, 204, 628} \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2}-2 \sqrt{x+1}-\frac{\log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}}}+\frac{\log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}}}-\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]
Warning: Unable to verify antiderivative.
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Rule 1625
Rule 1629
Rule 825
Rule 12
Rule 708
Rule 1094
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x} \left (1+x^3\right )}{1+x^2} \, dx &=\int \frac{(1+x)^{3/2} \left (1-x+x^2\right )}{1+x^2} \, dx\\ &=\int \left ((1+x)^{3/2}-\frac{x (1+x)^{3/2}}{1+x^2}\right ) \, dx\\ &=\frac{2}{5} (1+x)^{5/2}-\int \frac{x (1+x)^{3/2}}{1+x^2} \, dx\\ &=-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}-\int \frac{(-1+x) \sqrt{1+x}}{1+x^2} \, dx\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}-\int -\frac{2}{\sqrt{1+x} \left (1+x^2\right )} \, dx\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}+2 \int \frac{1}{\sqrt{1+x} \left (1+x^2\right )} \, dx\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}+4 \operatorname{Subst}\left (\int \frac{1}{2-2 x^2+x^4} \, dx,x,\sqrt{1+x}\right )\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{2}\right )}-x}{\sqrt{2}-\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+x}\right )}{\sqrt{1+\sqrt{2}}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{2}\right )}+x}{\sqrt{2}+\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+x}\right )}{\sqrt{1+\sqrt{2}}}\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+x}\right )}{\sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+x}\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt{2 \left (1+\sqrt{2}\right )}+2 x}{\sqrt{2}-\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+x}\right )}{2 \sqrt{1+\sqrt{2}}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{2}\right )}+2 x}{\sqrt{2}+\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+x}\right )}{2 \sqrt{1+\sqrt{2}}}\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}-\frac{\log \left (1+\sqrt{2}+x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+x}\right )}{2 \sqrt{1+\sqrt{2}}}+\frac{\log \left (1+\sqrt{2}+x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+x}\right )}{2 \sqrt{1+\sqrt{2}}}-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{2}\right )-x^2} \, dx,x,-\sqrt{2 \left (1+\sqrt{2}\right )}+2 \sqrt{1+x}\right )-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{2}\right )-x^2} \, dx,x,\sqrt{2 \left (1+\sqrt{2}\right )}+2 \sqrt{1+x}\right )\\ &=-2 \sqrt{1+x}-\frac{2}{3} (1+x)^{3/2}+\frac{2}{5} (1+x)^{5/2}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{1+x}}{\sqrt{2 \left (-1+\sqrt{2}\right )}}\right )}{\sqrt{-1+\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}+2 \sqrt{1+x}}{\sqrt{2 \left (-1+\sqrt{2}\right )}}\right )}{\sqrt{-1+\sqrt{2}}}-\frac{\log \left (1+\sqrt{2}+x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+x}\right )}{2 \sqrt{1+\sqrt{2}}}+\frac{\log \left (1+\sqrt{2}+x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+x}\right )}{2 \sqrt{1+\sqrt{2}}}\\ \end{align*}
Mathematica [A] time = 0.103947, size = 68, normalized size = 0.85 \[ \frac{2}{15} \sqrt{x+1} \left (3 x^2+x-17\right )+(1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1-i}}\right )+(1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1+i}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 443, normalized size = 5.5 \begin{align*}{\frac{2}{5} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+x}-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}}{2}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+2\,{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}}{2}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+2\,{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{3} + 1\right )} \sqrt{x + 1}}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60087, size = 986, normalized size = 12.32 \begin{align*} -\frac{1}{8} \cdot 8^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4}{\left (\sqrt{2} - 2\right )} \log \left (2 \cdot 8^{\frac{1}{4}} \sqrt{x + 1} \sqrt{2 \, \sqrt{2} + 4} + 4 \, x + 4 \, \sqrt{2} + 4\right ) + \frac{1}{8} \cdot 8^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4}{\left (\sqrt{2} - 2\right )} \log \left (-2 \cdot 8^{\frac{1}{4}} \sqrt{x + 1} \sqrt{2 \, \sqrt{2} + 4} + 4 \, x + 4 \, \sqrt{2} + 4\right ) - \frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, \sqrt{2} + 4} \arctan \left (\frac{1}{16} \cdot 8^{\frac{3}{4}} \sqrt{2} \sqrt{2 \cdot 8^{\frac{1}{4}} \sqrt{x + 1} \sqrt{2 \, \sqrt{2} + 4} + 4 \, x + 4 \, \sqrt{2} + 4} \sqrt{2 \, \sqrt{2} + 4} - \frac{1}{8} \cdot 8^{\frac{3}{4}} \sqrt{2} \sqrt{x + 1} \sqrt{2 \, \sqrt{2} + 4} - \sqrt{2} - 1\right ) - \frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, \sqrt{2} + 4} \arctan \left (\frac{1}{16} \cdot 8^{\frac{3}{4}} \sqrt{2} \sqrt{-2 \cdot 8^{\frac{1}{4}} \sqrt{x + 1} \sqrt{2 \, \sqrt{2} + 4} + 4 \, x + 4 \, \sqrt{2} + 4} \sqrt{2 \, \sqrt{2} + 4} - \frac{1}{8} \cdot 8^{\frac{3}{4}} \sqrt{2} \sqrt{x + 1} \sqrt{2 \, \sqrt{2} + 4} + \sqrt{2} + 1\right ) + \frac{2}{15} \,{\left (3 \, x^{2} + x - 17\right )} \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.36695, size = 56, normalized size = 0.7 \begin{align*} \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{5} - \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{x + 1} + 4 \operatorname{RootSum}{\left (512 t^{4} + 32 t^{2} + 1, \left ( t \mapsto t \log{\left (- 128 t^{3} + \sqrt{x + 1} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{3} + 1\right )} \sqrt{x + 1}}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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