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 = 0.29, antiderivative size = 224, normalized size of antiderivative = 2.80, number of steps used = 16, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 204
Rule 618
Rule 628
Rule 634
Rule 708
Rule 825
Rule 1094
Rule 1625
Rule 1629
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*}
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Mathematica [A] time = 0.10, 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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fricas [B] time = 0.52, size = 302, normalized size = 3.78 \[ -\frac {1}{8} \cdot 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} \log \left (2 \cdot 8^{\frac {1}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) + \frac {1}{8} \cdot 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} \log \left (-2 \cdot 8^{\frac {1}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) - \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{16} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \cdot 8^{\frac {1}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} \sqrt {2 \, \sqrt {2} + 4} - \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} - \sqrt {2} - 1\right ) - \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{16} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {-2 \cdot 8^{\frac {1}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} \sqrt {2 \, \sqrt {2} + 4} - \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + \sqrt {2} + 1\right ) + \frac {2}{15} \, {\left (3 \, x^{2} + x - 17\right )} \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.71, size = 171, normalized size = 2.14 \[ \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} + \sqrt {\sqrt {2} + 1} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (-2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) - 2 \, \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 443, normalized size = 5.54 \[ \frac {\left (2+2 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\left (2+2 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2+2 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (x +1+\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (x +1+\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (x +1+\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (x +1+\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (x +1\right )^{\frac {5}{2}}}{5}-\frac {2 \left (x +1\right )^{\frac {3}{2}}}{3}-2 \sqrt {x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{3} + 1\right )} \sqrt {x + 1}}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 255, normalized size = 3.19 \[ \frac {2\,{\left (x+1\right )}^{5/2}}{5}-\frac {2\,{\left (x+1\right )}^{3/2}}{3}-2\,\sqrt {x+1}-\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.15, size = 56, normalized size = 0.70 \[ \frac {2 \left (x + 1\right )^{\frac {5}{2}}}{5} - \frac {2 \left (x + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {x + 1} + 4 \operatorname {RootSum} {\left (512 t^{4} + 32 t^{2} + 1, \left (t \mapsto t \log {\left (- 128 t^{3} + \sqrt {x + 1} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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