Optimal. Leaf size=175 \[ \frac {9}{10} x \sqrt {x^2-x+1} \sqrt {x+1}-\frac {\sqrt {x^2-x+1} \left (x^3+1\right ) \sqrt {x+1}}{2 x^2}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{10 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {915, 277, 195, 218} \[ -\frac {\sqrt {x^2-x+1} \left (x^3+1\right ) \sqrt {x+1}}{2 x^2}+\frac {9}{10} x \sqrt {x^2-x+1} \sqrt {x+1}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{10 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
Antiderivative was successfully verified.
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Rule 195
Rule 218
Rule 277
Rule 915
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\left (1+x^3\right )^{3/2}}{x^3} \, dx}{\sqrt {1+x^3}}\\ &=-\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {\left (9 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \sqrt {1+x^3} \, dx}{4 \sqrt {1+x^3}}\\ &=\frac {9}{10} x \sqrt {1+x} \sqrt {1-x+x^2}-\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {\left (27 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{20 \sqrt {1+x^3}}\\ &=\frac {9}{10} x \sqrt {1+x} \sqrt {1-x+x^2}-\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {9\ 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 )}{10 \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 = 0.35, size = 192, normalized size = 1.10 \[ \frac {\sqrt {x+1} \left (\frac {2 \left (x^2-x+1\right ) \left (4 x^3-5\right )}{x^2}-\frac {27 i \sqrt {2} \sqrt {\frac {-2 i x+\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {\frac {2 i x+\sqrt {3}-i}{\sqrt {3}-3 i}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {i (x+1)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i (x+1)}{\sqrt {3}+3 i}}}\right )}{20 \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 264, normalized size = 1.51 \[ -\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (-8 x^{6}+2 x^{3}+27 i \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \sqrt {3}\, x^{2} \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-81 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x^{2} \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )+10\right )}{20 \left (x^{3}+1\right ) x^{2}} \]
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 \[ \int \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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