Optimal. Leaf size=168 \[ \frac {14 x}{27 \sqrt {x+1} \sqrt {x^2-x+1}}+\frac {14 \sqrt {2+\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}+\frac {2 x}{9 \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {713, 199, 218} \[ \frac {14 x}{27 \sqrt {x+1} \sqrt {x^2-x+1}}+\frac {2 x}{9 \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right )}+\frac {14 \sqrt {2+\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 218
Rule 713
Rubi steps
\begin {align*} \int \frac {1}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1+x^3} \int \frac {1}{\left (1+x^3\right )^{5/2}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {2 x}{9 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {\left (7 \sqrt {1+x^3}\right ) \int \frac {1}{\left (1+x^3\right )^{3/2}} \, dx}{9 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {14 x}{27 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2 x}{9 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {\left (7 \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{27 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {14 x}{27 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2 x}{9 \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {14 \sqrt {2+\sqrt {3}} \sqrt {1+x} \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 )}{27 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 178, normalized size = 1.06 \[ \frac {\frac {6 x \left (7 x^3+10\right )}{(x+1)^{3/2} \left (x^2-x+1\right )}+\frac {7 i (x+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (x+1)}} \sqrt {6-\frac {36 i}{\left (\sqrt {3}+3 i\right ) (x+1)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}}}}{81 \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{9} + 3 \, x^{6} + 3 \, x^{3} + 1}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 469, normalized size = 2.79 \[ -\frac {-14 x^{4}+7 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x^{3} \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-21 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x^{3} \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-20 x +7 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-21 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )}{27 \left (x^{2}-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x+1\right )}^{5/2}\,{\left (x^2-x+1\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (x + 1\right )^{\frac {5}{2}} \left (x^{2} - x + 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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