Optimal. Leaf size=143 \[ -\frac {22053 \sqrt {3+2 x+5 x^2}}{31250}+\frac {36073 x \sqrt {3+2 x+5 x^2}}{1875}-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}-\frac {1719097 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{31250 \sqrt {5}} \]
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Rubi [A]
time = 0.12, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1675, 654, 633,
221} \begin {gather*} -\frac {207427 \sqrt {5 x^2+2 x+3} x^2}{37500}+\frac {36073 \sqrt {5 x^2+2 x+3} x}{1875}-\frac {22053 \sqrt {5 x^2+2 x+3}}{31250}+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5+\frac {5131}{750} \sqrt {5 x^2+2 x+3} x^4-\frac {33259 \sqrt {5 x^2+2 x+3} x^3}{2500}-\frac {1719097 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{31250 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
Rule 1675
Rubi steps
\begin {align*} \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx &=\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}+\frac {1}{30} \int \frac {60+630 x+1350 x^2-2820 x^3-6135 x^4+5131 x^5}{\sqrt {3+2 x+5 x^2}} \, dx\\ &=\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}+\frac {1}{750} \int \frac {1500+15750 x+33750 x^2-132072 x^3-199554 x^4}{\sqrt {3+2 x+5 x^2}} \, dx\\ &=-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {30000+315000 x+2470986 x^2-1244562 x^3}{\sqrt {3+2 x+5 x^2}} \, dx}{15000}\\ &=-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {450000+12192372 x+43287600 x^2}{\sqrt {3+2 x+5 x^2}} \, dx}{225000}\\ &=\frac {36073 x \sqrt {3+2 x+5 x^2}}{1875}-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {-125362800-7939080 x}{\sqrt {3+2 x+5 x^2}} \, dx}{2250000}\\ &=-\frac {22053 \sqrt {3+2 x+5 x^2}}{31250}+\frac {36073 x \sqrt {3+2 x+5 x^2}}{1875}-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}-\frac {1719097 \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx}{31250}\\ &=-\frac {22053 \sqrt {3+2 x+5 x^2}}{31250}+\frac {36073 x \sqrt {3+2 x+5 x^2}}{1875}-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}-\frac {1719097 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )}{62500 \sqrt {70}}\\ &=-\frac {22053 \sqrt {3+2 x+5 x^2}}{31250}+\frac {36073 x \sqrt {3+2 x+5 x^2}}{1875}-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}-\frac {1719097 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{31250 \sqrt {5}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 79, normalized size = 0.55 \begin {gather*} \frac {\sqrt {3+2 x+5 x^2} \left (-132318+3607300 x-1037135 x^2-2494425 x^3+1282750 x^4+306250 x^5\right )}{187500}+\frac {1719097 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{31250 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 113, normalized size = 0.79
method | result | size |
risch | \(\frac {\left (306250 x^{5}+1282750 x^{4}-2494425 x^{3}-1037135 x^{2}+3607300 x -132318\right ) \sqrt {5 x^{2}+2 x +3}}{187500}-\frac {1719097 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{156250}\) | \(55\) |
trager | \(\left (\frac {49}{30} x^{5}+\frac {5131}{750} x^{4}-\frac {33259}{2500} x^{3}-\frac {207427}{37500} x^{2}+\frac {36073}{1875} x -\frac {22053}{31250}\right ) \sqrt {5 x^{2}+2 x +3}-\frac {1719097 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (5 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +\RootOf \left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{156250}\) | \(79\) |
default | \(\frac {5131 x^{4} \sqrt {5 x^{2}+2 x +3}}{750}+\frac {49 x^{5} \sqrt {5 x^{2}+2 x +3}}{30}+\frac {36073 x \sqrt {5 x^{2}+2 x +3}}{1875}-\frac {207427 x^{2} \sqrt {5 x^{2}+2 x +3}}{37500}-\frac {33259 x^{3} \sqrt {5 x^{2}+2 x +3}}{2500}-\frac {22053 \sqrt {5 x^{2}+2 x +3}}{31250}-\frac {1719097 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{156250}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 114, normalized size = 0.80 \begin {gather*} \frac {49}{30} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{5} + \frac {5131}{750} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{4} - \frac {33259}{2500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{3} - \frac {207427}{37500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{2} + \frac {36073}{1875} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {1719097}{156250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {22053}{31250} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 77, normalized size = 0.54 \begin {gather*} \frac {1}{187500} \, {\left (306250 \, x^{5} + 1282750 \, x^{4} - 2494425 \, x^{3} - 1037135 \, x^{2} + 3607300 \, x - 132318\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1719097}{312500} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \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 {\left (x^{2} + 5 x + 2\right ) \left (7 x^{2} - 4 x - 1\right )^{2}}{\sqrt {5 x^{2} + 2 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.56, size = 72, normalized size = 0.50 \begin {gather*} \frac {1}{187500} \, {\left (5 \, {\left ({\left (5 \, {\left (70 \, {\left (175 \, x + 733\right )} x - 99777\right )} x - 207427\right )} x + 721460\right )} x - 132318\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1719097}{156250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \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 = -0.01 \begin {gather*} \int \frac {\left (x^2+5\,x+2\right )\,{\left (-7\,x^2+4\,x+1\right )}^2}{\sqrt {5\,x^2+2\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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