Optimal. Leaf size=652 \[ \frac {\sqrt {\frac {23}{11}} \left (1-i \sqrt {23}-4 x\right ) \sqrt {-1+i \sqrt {23}+4 x} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {\frac {\left (11 i-\sqrt {23}\right ) \left (2+3 x+5 x^2\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}} \left (1-\frac {\sqrt {-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{1-i \sqrt {23}-4 x}\right ) \sqrt {\frac {11-\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )}-\frac {11 \left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}}{\left (1-\frac {\sqrt {-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{1-i \sqrt {23}-4 x}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {6-\left (1-i \sqrt {23}\right ) x}}{\sqrt {-1+i \sqrt {23}+4 x}}\right )|\frac {1}{88} \left (44-\frac {41 \left (i+\sqrt {23}\right )}{\sqrt {11+i \sqrt {23}}}\right )\right )}{\left (23+i \sqrt {23}\right ) \sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2} \sqrt {11-\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )}-\frac {11 \left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}}} \]
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Rubi [A]
time = 0.50, antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1006, 949,
1117} \begin {gather*} \frac {\sqrt {\frac {23}{11}} \left (-4 x-i \sqrt {23}+1\right ) \sqrt {4 x+i \sqrt {23}-1} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {\frac {\left (-\sqrt {23}+11 i\right ) \left (5 x^2+3 x+2\right )}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )^2}} \left (1-\frac {\sqrt {-\frac {-\sqrt {23}+3 i}{\sqrt {23}+7 i}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{-4 x-i \sqrt {23}+1}\right ) \sqrt {\frac {-\frac {11 \left (-\sqrt {23}+3 i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )^2}-\frac {41 \left (\sqrt {23}+i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )}+11}{\left (1-\frac {\sqrt {-\frac {-\sqrt {23}+3 i}{\sqrt {23}+7 i}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{-4 x-i \sqrt {23}+1}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {6-\left (1-i \sqrt {23}\right ) x}}{\sqrt {4 x+i \sqrt {23}-1}}\right )|\frac {1}{88} \left (44-\frac {41 \left (i+\sqrt {23}\right )}{\sqrt {11+i \sqrt {23}}}\right )\right )}{\left (23+i \sqrt {23}\right ) \sqrt [4]{-\frac {-\sqrt {23}+3 i}{\sqrt {23}+7 i}} \sqrt {2 x^2-x+3} \sqrt {5 x^2+3 x+2} \sqrt {-\frac {11 \left (-\sqrt {23}+3 i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )^2}-\frac {41 \left (\sqrt {23}+i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )}+11}} \end {gather*}
Antiderivative was successfully verified.
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Rule 949
Rule 1006
Rule 1117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx &=\frac {\left (\sqrt {-1+i \sqrt {23}+4 x} \sqrt {6+\left (-1+i \sqrt {23}\right ) x}\right ) \int \frac {1}{\sqrt {-1+i \sqrt {23}+4 x} \sqrt {6+\left (-1+i \sqrt {23}\right ) x} \sqrt {2+3 x+5 x^2}} \, dx}{\sqrt {3-x+2 x^2}}\\ &=-\frac {\left (2 \left (-1+i \sqrt {23}+4 x\right )^{3/2} \sqrt {6+\left (-1+i \sqrt {23}\right ) x} \sqrt {\frac {\left (24-\left (-1+i \sqrt {23}\right )^2\right )^2 \left (2+3 x+5 x^2\right )}{\left (180-18 \left (-1+i \sqrt {23}\right )+2 \left (-1+i \sqrt {23}\right )^2\right ) \left (-1+i \sqrt {23}+4 x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {\left (-72+76 \left (-1+i \sqrt {23}\right )-3 \left (-1+i \sqrt {23}\right )^2\right ) x^2}{180-18 \left (-1+i \sqrt {23}\right )+2 \left (-1+i \sqrt {23}\right )^2}+\frac {\left (32-12 \left (-1+i \sqrt {23}\right )+5 \left (-1+i \sqrt {23}\right )^2\right ) x^4}{180-18 \left (-1+i \sqrt {23}\right )+2 \left (-1+i \sqrt {23}\right )^2}}} \, dx,x,\frac {\sqrt {6+\left (-1+i \sqrt {23}\right ) x}}{\sqrt {-1+i \sqrt {23}+4 x}}\right )}{\left (24-\left (-1+i \sqrt {23}\right )^2\right ) \sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}}\\ &=-\frac {\sqrt {\frac {23}{11}} \left (-1+i \sqrt {23}+4 x\right )^{3/2} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {\frac {\left (11 i-\sqrt {23}\right ) \left (2+3 x+5 x^2\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}} \left (1-\frac {\sqrt {-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{1-i \sqrt {23}-4 x}\right ) \sqrt {\frac {11-\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )}-\frac {11 \left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}}{\left (1-\frac {\sqrt {-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{1-i \sqrt {23}-4 x}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {6-\left (1-i \sqrt {23}\right ) x}}{\sqrt {-1+i \sqrt {23}+4 x}}\right )|\frac {1}{88} \left (44-\frac {41 \left (i+\sqrt {23}\right )}{\sqrt {11+i \sqrt {23}}}\right )\right )}{\left (23+i \sqrt {23}\right ) \sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2} \sqrt {11-\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )}-\frac {11 \left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}}}\\ \end {align*}
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Mathematica [A]
time = 2.25, size = 390, normalized size = 0.60 \begin {gather*} \frac {\left (1+i \sqrt {23}-4 x\right ) \left (3 i+\sqrt {31}+10 i x\right ) \sqrt {\frac {6 i-2 \sqrt {31}+20 i x}{\left (11 i+5 \sqrt {23}-2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}} \sqrt {\frac {63-3 i \sqrt {23}-i \sqrt {31}-\sqrt {713}+\left (-22-10 i \sqrt {23}+4 i \sqrt {31}\right ) x}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}} F\left (\sin ^{-1}\left (\sqrt {2} \sqrt {-\frac {-63+3 i \sqrt {23}+i \sqrt {31}+\sqrt {713}+2 \left (11+5 i \sqrt {23}-2 i \sqrt {31}\right ) x}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}}\right )|\frac {1}{484} \left (1197+41 \sqrt {713}\right )\right )}{\left (-11 i+5 \sqrt {23}-2 \sqrt {31}\right ) \sqrt {\frac {3 i+\sqrt {31}+10 i x}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}} \sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.39, size = 420, normalized size = 0.64
method | result | size |
elliptic | \(-\frac {i \sqrt {\left (2 x^{2}-x +3\right ) \left (5 x^{2}+3 x +2\right )}\, \left (\frac {11}{20}-\frac {i \sqrt {23}}{4}-\frac {i \sqrt {31}}{10}\right ) \sqrt {\frac {\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}-\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}\, \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )^{2} \sqrt {\frac {i \sqrt {23}\, \left (x +\frac {3}{10}+\frac {i \sqrt {31}}{10}\right )}{\left (-\frac {11}{20}-\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}\, \sqrt {\frac {i \sqrt {23}\, \left (x +\frac {3}{10}-\frac {i \sqrt {31}}{10}\right )}{\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}\, \sqrt {23}\, \sqrt {10}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}-\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}, \sqrt {\frac {\left (\frac {11}{20}+\frac {i \sqrt {23}}{4}+\frac {i \sqrt {31}}{10}\right ) \left (\frac {11}{20}-\frac {i \sqrt {23}}{4}-\frac {i \sqrt {31}}{10}\right )}{\left (\frac {11}{20}-\frac {i \sqrt {23}}{4}+\frac {i \sqrt {31}}{10}\right ) \left (\frac {11}{20}+\frac {i \sqrt {23}}{4}-\frac {i \sqrt {31}}{10}\right )}}\right )}{115 \sqrt {2 x^{2}-x +3}\, \sqrt {5 x^{2}+3 x +2}\, \left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}-\frac {i \sqrt {23}}{4}\right ) \sqrt {\left (x -\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) \left (x +\frac {3}{10}+\frac {i \sqrt {31}}{10}\right ) \left (x +\frac {3}{10}-\frac {i \sqrt {31}}{10}\right )}}\) | \(395\) |
default | \(\frac {4 i \sqrt {5 x^{2}+3 x +2}\, \sqrt {2 x^{2}-x +3}\, \left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \sqrt {-\frac {\left (2 i \sqrt {31}-5 i \sqrt {23}-11\right ) \left (-1+4 x +i \sqrt {23}\right )}{\left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \left (i \sqrt {23}-4 x +1\right )}}\, \left (i \sqrt {23}-4 x +1\right )^{2} \sqrt {\frac {i \sqrt {23}\, \left (i \sqrt {31}+10 x +3\right )}{\left (2 i \sqrt {31}-5 i \sqrt {23}+11\right ) \left (i \sqrt {23}-4 x +1\right )}}\, \sqrt {\frac {i \sqrt {23}\, \left (i \sqrt {31}-10 x -3\right )}{\left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \left (i \sqrt {23}-4 x +1\right )}}\, \sqrt {23}\, \sqrt {10}\, \EllipticF \left (\sqrt {-\frac {\left (2 i \sqrt {31}-5 i \sqrt {23}-11\right ) \left (-1+4 x +i \sqrt {23}\right )}{\left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \left (i \sqrt {23}-4 x +1\right )}}, \sqrt {\frac {\left (2 i \sqrt {31}+5 i \sqrt {23}+11\right ) \left (2 i \sqrt {31}+5 i \sqrt {23}-11\right )}{\left (2 i \sqrt {31}-5 i \sqrt {23}+11\right ) \left (2 i \sqrt {31}-5 i \sqrt {23}-11\right )}}\right )}{23 \sqrt {10 x^{4}+x^{3}+16 x^{2}+7 x +6}\, \left (2 i \sqrt {31}-5 i \sqrt {23}-11\right ) \sqrt {\left (-1+4 x +i \sqrt {23}\right ) \left (i \sqrt {23}-4 x +1\right ) \left (i \sqrt {31}+10 x +3\right ) \left (i \sqrt {31}-10 x -3\right )}}\) | \(420\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {2 x^{2} - x + 3} \sqrt {5 x^{2} + 3 x + 2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {2\,x^2-x+3}\,\sqrt {5\,x^2+3\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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