Optimal. Leaf size=325 \[ \frac {18}{91} \sqrt {x+1} \sqrt {x^2-x+1} x^2+\frac {54 \sqrt {x+1} \sqrt {x^2-x+1}}{91 \left (x+\sqrt {3}+1\right )}+\frac {2}{13} \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right ) x^2+\frac {18 \sqrt {2} 3^{3/4} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {809, 279, 303, 218, 1877} \[ \frac {2}{13} \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right ) x^2+\frac {18}{91} \sqrt {x+1} \sqrt {x^2-x+1} x^2+\frac {54 \sqrt {x+1} \sqrt {x^2-x+1}}{91 \left (x+\sqrt {3}+1\right )}+\frac {18 \sqrt {2} 3^{3/4} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
Antiderivative was successfully verified.
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Rule 218
Rule 279
Rule 303
Rule 809
Rule 1877
Rubi steps
\begin {align*} \int x (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 \left (1+x^3\right )^{3/2} \, dx}{\sqrt {1+x^3}}\\ &=\frac {2}{13} x^2 \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 \sqrt {1+x^3} \, dx}{13 \sqrt {1+x^3}}\\ &=\frac {18}{91} x^2 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{13} x^2 \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}{\sqrt {1+x^3}} \, dx}{91 \sqrt {1+x^3}}\\ &=\frac {18}{91} x^2 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{13} x^2 \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 {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx}{91 \sqrt {1+x^3}}+\frac {\left (27 \sqrt {2 \left (2-\sqrt {3}\right )} \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{91 \sqrt {1+x^3}}\\ &=\frac {18}{91} x^2 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {54 \sqrt {1+x} \sqrt {1-x+x^2}}{91 \left (1+\sqrt {3}+x\right )}+\frac {2}{13} x^2 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {27 \sqrt [4]{3} \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}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {18 \sqrt {2} 3^{3/4} (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 )}{91 \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 = 0.49, size = 244, normalized size = 0.75 \[ \frac {\sqrt {x+1} \left (4 x^2 \left (x^2-x+1\right ) \left (7 x^3+16\right )-\frac {27 \sqrt {2} \sqrt {\frac {2 i x+\sqrt {3}-i}{\sqrt {3}-3 i}} \left (\left (\sqrt {3}-3 i\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {i (x+1)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )-\left (\sqrt {3}-i\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {i (x+1)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )\right )}{\sqrt {-\frac {i (x+1)}{-2 i x+\sqrt {3}+i}}}\right )}{182 \sqrt {x^2-x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{4} + x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}, x\right ) \]
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 \[ \int {\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 366, normalized size = 1.13 \[ \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (14 x^{8}+46 x^{5}+32 x^{2}-162 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticE \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )+27 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )+81 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )\right )}{91 x^{3}+91} \]
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 \[ \int {\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}} x\,{d x} \]
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 \[ \int x\,{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2} \,d x \]
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 \[ \int x \left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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