Optimal. Leaf size=201 \[ \frac {54}{935} \sqrt {x+1} \sqrt {x^2-x+1} x+\frac {18}{187} \sqrt {x+1} \sqrt {x^2-x+1} x^4-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{935 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {2}{17} \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right ) x^4 \]
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Rubi [A] time = 0.08, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {915, 279, 321, 218} \[ \frac {2}{17} \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right ) x^4+\frac {18}{187} \sqrt {x+1} \sqrt {x^2-x+1} x^4+\frac {54}{935} \sqrt {x+1} \sqrt {x^2-x+1} x-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{935 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
Antiderivative was successfully verified.
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Rule 218
Rule 279
Rule 321
Rule 915
Rubi steps
\begin {align*} \int x^3 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int x^3 \left (1+x^3\right )^{3/2} \, dx}{\sqrt {1+x^3}}\\ &=\frac {2}{17} x^4 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (9 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int x^3 \sqrt {1+x^3} \, dx}{17 \sqrt {1+x^3}}\\ &=\frac {18}{187} x^4 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{17} x^4 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (27 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {x^3}{\sqrt {1+x^3}} \, dx}{187 \sqrt {1+x^3}}\\ &=\frac {54}{935} x \sqrt {1+x} \sqrt {1-x+x^2}+\frac {18}{187} x^4 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{17} x^4 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {\left (54 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{935 \sqrt {1+x^3}}\\ &=\frac {54}{935} x \sqrt {1+x} \sqrt {1-x+x^2}+\frac {18}{187} x^4 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{17} x^4 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{935 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}\\ \end {align*}
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Mathematica [C] time = 0.88, size = 235, normalized size = 1.17 \[ \frac {2 \left (x \sqrt {x+1} \left (55 x^8-55 x^7+55 x^6+100 x^5-100 x^4+100 x^3+27 x^2-27 x+27\right )-\frac {9 i \sqrt {6} (x+1) \sqrt {\frac {\left (\sqrt {3}-3 i\right ) x+\sqrt {3}+3 i}{\left (\sqrt {3}-3 i\right ) (x+1)}} \sqrt {\frac {\left (\sqrt {3}+3 i\right ) x+\sqrt {3}-3 i}{\left (\sqrt {3}+3 i\right ) (x+1)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}}}\right )}{935 \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{6} + x^{3}\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 262, normalized size = 1.30 \[ \frac {2 \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (55 x^{10}+155 x^{7}+127 x^{4}+27 x +27 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-81 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )\right )}{935 \left (x^{3}+1\right )} \]
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 \[ \int {\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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