Optimal. Leaf size=287 \[ -\frac {\sqrt {x^2-x+1} \sqrt {x+1}}{x}+\frac {3 \sqrt {x^2-x+1} \sqrt {x+1}}{x+\sqrt {3}+1}+\frac {\sqrt {2} 3^{3/4} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {915, 277, 303, 218, 1877} \[ -\frac {\sqrt {x^2-x+1} \sqrt {x+1}}{x}+\frac {3 \sqrt {x^2-x+1} \sqrt {x+1}}{x+\sqrt {3}+1}+\frac {\sqrt {2} 3^{3/4} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )} \]
Antiderivative was successfully verified.
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Rule 218
Rule 277
Rule 303
Rule 915
Rule 1877
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^2} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\sqrt {1+x^3}}{x^2} \, dx}{\sqrt {1+x^3}}\\ &=-\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x}+\frac {\left (3 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {x}{\sqrt {1+x^3}} \, dx}{2 \sqrt {1+x^3}}\\ &=-\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x}+\frac {\left (3 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx}{2 \sqrt {1+x^3}}+\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{\sqrt {1+x^3}}\\ &=-\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x}+\frac {3 \sqrt {1+x} \sqrt {1-x+x^2}}{1+\sqrt {3}+x}-\frac {3 \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 )}{2 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {\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 )}{\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 = 0.40, size = 349, normalized size = 1.22 \[ -\frac {\sqrt {x+1} \sqrt {x^2-x+1}}{x}+\frac {3 \sqrt {1+\frac {2 i (x+1)}{\sqrt {3}-3 i}} \sqrt {1-\frac {2 i (x+1)}{\sqrt {3}+3 i}} \left (\frac {\left (\sqrt {3}-i\right ) \sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x+1} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {i (x+1)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i (x+1)}{\sqrt {3}+3 i}}}-\frac {\left (\sqrt {3}-3 i\right ) \sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x+1} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {i (x+1)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i (x+1)}{\sqrt {3}+3 i}}}\right )}{2 \sqrt {2} \sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {(x+1)^2-3 (x+1)+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 363, normalized size = 1.26 \[ \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (-2 x^{3}-18 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x \EllipticE \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )+3 i \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, \sqrt {3}\, x \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )+9 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{i \sqrt {3}+3}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-2\right )}{2 \left (x^{3}+1\right ) x} \]
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 \[ \int \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {x+1}\,\sqrt {x^2-x+1}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {x + 1} \sqrt {x^{2} - x + 1}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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