3.392 \(\int \frac {(1+4 x-7 x^2)^3 (2+5 x+x^2)}{(3+2 x+5 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {2583293 \sqrt {5 x^2+2 x+3} x^2}{187500}-\frac {3192602 \sqrt {5 x^2+2 x+3} x}{46875}+\frac {15715799 \sqrt {5 x^2+2 x+3}}{156250}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}-\frac {343}{150} \sqrt {5 x^2+2 x+3} x^5-\frac {25921 \sqrt {5 x^2+2 x+3} x^4}{3750}+\frac {393659 \sqrt {5 x^2+2 x+3} x^3}{12500}+\frac {50047657 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{156250 \sqrt {5}} \]

[Out]

50047657/781250*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)+16/546875*(6122807-5338217*x)/(5*x^2+2*x+3)^(1/2)+15715
799/156250*(5*x^2+2*x+3)^(1/2)-3192602/46875*x*(5*x^2+2*x+3)^(1/2)-2583293/187500*x^2*(5*x^2+2*x+3)^(1/2)+3936
59/12500*x^3*(5*x^2+2*x+3)^(1/2)-25921/3750*x^4*(5*x^2+2*x+3)^(1/2)-343/150*x^5*(5*x^2+2*x+3)^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1660, 1661, 640, 619, 215} \[ -\frac {343}{150} \sqrt {5 x^2+2 x+3} x^5-\frac {25921 \sqrt {5 x^2+2 x+3} x^4}{3750}+\frac {393659 \sqrt {5 x^2+2 x+3} x^3}{12500}-\frac {2583293 \sqrt {5 x^2+2 x+3} x^2}{187500}-\frac {3192602 \sqrt {5 x^2+2 x+3} x}{46875}+\frac {15715799 \sqrt {5 x^2+2 x+3}}{156250}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}+\frac {50047657 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{156250 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2))/(3 + 2*x + 5*x^2)^(3/2),x]

[Out]

(16*(6122807 - 5338217*x))/(546875*Sqrt[3 + 2*x + 5*x^2]) + (15715799*Sqrt[3 + 2*x + 5*x^2])/156250 - (3192602
*x*Sqrt[3 + 2*x + 5*x^2])/46875 - (2583293*x^2*Sqrt[3 + 2*x + 5*x^2])/187500 + (393659*x^3*Sqrt[3 + 2*x + 5*x^
2])/12500 - (25921*x^4*Sqrt[3 + 2*x + 5*x^2])/3750 - (343*x^5*Sqrt[3 + 2*x + 5*x^2])/150 + (50047657*ArcSinh[(
1 + 5*x)/Sqrt[14]])/(156250*Sqrt[5])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}+\frac {1}{28} \int \frac {\frac {473724104}{78125}+\frac {94462228 x}{15625}-\frac {40822404 x^2}{3125}-\frac {1210328 x^3}{625}+\frac {1866704 x^4}{125}-\frac {138572 x^5}{25}-\frac {9604 x^6}{5}}{\sqrt {3+2 x+5 x^2}} \, dx\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {1}{840} \int \frac {\frac {2842344624}{15625}+\frac {566773368 x}{3125}-\frac {244934424 x^2}{625}-\frac {7261968 x^3}{125}+\frac {11920524 x^4}{25}-\frac {725788 x^5}{5}}{\sqrt {3+2 x+5 x^2}} \, dx\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {\frac {2842344624}{625}+\frac {566773368 x}{125}-\frac {244934424 x^2}{25}+\frac {1447488 x^3}{5}+\frac {66134712 x^4}{5}}{\sqrt {3+2 x+5 x^2}} \, dx}{21000}\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {\frac {11369378496}{125}+\frac {2267093472 x}{25}-\frac {1574950104 x^2}{5}-\frac {433993224 x^3}{5}}{\sqrt {3+2 x+5 x^2}} \, dx}{420000}\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}-\frac {2583293 x^2 \sqrt {3+2 x+5 x^2}}{187500}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {\frac {34108135488}{25}[phone] x-4290857088 x^2}{\sqrt {3+2 x+5 x^2}} \, dx}{6300000}\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}-\frac {3192602 x \sqrt {3+2 x+5 x^2}}{46875}-\frac {2583293 x^2 \sqrt {3+2 x+5 x^2}}{187500}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {\int \frac {\frac {132579127296}{5}[phone] x}{\sqrt {3+2 x+5 x^2}} \, dx}{63000000}\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}+\frac {15715799 \sqrt {3+2 x+5 x^2}}{156250}-\frac {3192602 x \sqrt {3+2 x+5 x^2}}{46875}-\frac {2583293 x^2 \sqrt {3+2 x+5 x^2}}{187500}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {50047657 \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx}{156250}\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}+\frac {15715799 \sqrt {3+2 x+5 x^2}}{156250}-\frac {3192602 x \sqrt {3+2 x+5 x^2}}{46875}-\frac {2583293 x^2 \sqrt {3+2 x+5 x^2}}{187500}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {50047657 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )}{312500 \sqrt {70}}\\ &=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}+\frac {15715799 \sqrt {3+2 x+5 x^2}}{156250}-\frac {3192602 x \sqrt {3+2 x+5 x^2}}{46875}-\frac {2583293 x^2 \sqrt {3+2 x+5 x^2}}{187500}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {50047657 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{156250 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 75, normalized size = 0.45 \[ \frac {2102001594 \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )-\frac {5 \left (75031250 x^7+256821250 x^6-897612625 x^5+174819575 x^4+1795638985 x^3-2135143465 x^2+1045703388 x-3155769618\right )}{\sqrt {5 x^2+2 x+3}}}{32812500} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2))/(3 + 2*x + 5*x^2)^(3/2),x]

[Out]

((-5*(-3155769618 + 1045703388*x - 2135143465*x^2 + 1795638985*x^3 + 174819575*x^4 - 897612625*x^5 + 256821250
*x^6 + 75031250*x^7))/Sqrt[3 + 2*x + 5*x^2] + 2102001594*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/32812500

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fricas [A]  time = 0.83, size = 112, normalized size = 0.67 \[ \frac {1051000797 \, \sqrt {5} {\left (5 \, x^{2} + 2 \, x + 3\right )} \log \left (-\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) - 5 \, {\left (75031250 \, x^{7} + 256821250 \, x^{6} - 897612625 \, x^{5} + 174819575 \, x^{4} + 1795638985 \, x^{3} - 2135143465 \, x^{2} + 1045703388 \, x - 3155769618\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{32812500 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(3/2),x, algorithm="fricas")

[Out]

1/32812500*(1051000797*sqrt(5)*(5*x^2 + 2*x + 3)*log(-sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x
- 8) - 5*(75031250*x^7 + 256821250*x^6 - 897612625*x^5 + 174819575*x^4 + 1795638985*x^3 - 2135143465*x^2 + 104
5703388*x - 3155769618)*sqrt(5*x^2 + 2*x + 3))/(5*x^2 + 2*x + 3)

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giac [A]  time = 0.26, size = 81, normalized size = 0.49 \[ -\frac {50047657}{781250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) - \frac {{\left (35 \, {\left ({\left (5 \, {\left (35 \, {\left (70 \, {\left (175 \, x + 599\right )} x - 146549\right )} x + 998969\right )} x + 51303971\right )} x - 61004099\right )} x + 1045703388\right )} x - 3155769618}{6562500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(3/2),x, algorithm="giac")

[Out]

-50047657/781250*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1) - 1/6562500*((35*((5*(35*(70*(1
75*x + 599)*x - 146549)*x + 998969)*x + 51303971)*x - 61004099)*x + 1045703388)*x - 3155769618)/sqrt(5*x^2 + 2
*x + 3)

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maple [A]  time = 0.03, size = 166, normalized size = 1.00 \[ -\frac {343 x^{7}}{30 \sqrt {5 x^{2}+2 x +3}}-\frac {29351 x^{6}}{750 \sqrt {5 x^{2}+2 x +3}}+\frac {1025843 x^{5}}{7500 \sqrt {5 x^{2}+2 x +3}}-\frac {998969 x^{4}}{37500 \sqrt {5 x^{2}+2 x +3}}-\frac {51303971 x^{3}}{187500 \sqrt {5 x^{2}+2 x +3}}+\frac {61004099 x^{2}}{187500 \sqrt {5 x^{2}+2 x +3}}-\frac {50047657 x}{156250 \sqrt {5 x^{2}+2 x +3}}+\frac {50047657 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250}+\frac {\frac {176049701 x}{1093750}+\frac {176049701}{5468750}}{\sqrt {5 x^{2}+2 x +3}}+\frac {175268451}{390625 \sqrt {5 x^{2}+2 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-7*x^2+4*x+1)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(3/2),x)

[Out]

50047657/781250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))[phone]/10937500*(10*x+2)/(5*x^2+2*x+3)^(1/2)-998969/
37500*x^4/(5*x^2+2*x+3)^(1/2)-51303971/187500*x^3/(5*x^2+2*x+3)^(1/2)[phone]/187500*x^2/(5*x^2+2*x+3)^(1/2)-
50047657/156250*x/(5*x^2+2*x+3)^(1/2)-343/30*x^7/(5*x^2+2*x+3)^(1/2)-29351/750*x^6/(5*x^2+2*x+3)^(1/2)+1025843
/7500*x^5/(5*x^2+2*x+3)^(1/2)[phone]/390625/(5*x^2+2*x+3)^(1/2)

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maxima [A]  time = 0.99, size = 148, normalized size = 0.89 \[ -\frac {343 \, x^{7}}{30 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} - \frac {29351 \, x^{6}}{750 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {1025843 \, x^{5}}{7500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} - \frac {998969 \, x^{4}}{37500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} - \frac {51303971 \, x^{3}}{187500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {61004099 \, x^{2}}{187500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {50047657}{781250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {87141949 \, x}{546875 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {525961603}{1093750 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(3/2),x, algorithm="maxima")

[Out]

-343/30*x^7/sqrt(5*x^2 + 2*x + 3) - 29351/750*x^6/sqrt(5*x^2 + 2*x + 3) + 1025843/7500*x^5/sqrt(5*x^2 + 2*x +
3) - 998969/37500*x^4/sqrt(5*x^2 + 2*x + 3) - 51303971/187500*x^3/sqrt(5*x^2 + 2*x + 3) + 61004099/187500*x^2/
sqrt(5*x^2 + 2*x + 3) + 50047657/781250*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 1)) - 87141949/546875*x/sqrt(5*x^
2 + 2*x + 3) + 525961603/1093750/sqrt(5*x^2 + 2*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (x^2+5\,x+2\right )\,{\left (-7\,x^2+4\,x+1\right )}^3}{{\left (5\,x^2+2\,x+3\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x + x^2 + 2)*(4*x - 7*x^2 + 1)^3)/(2*x + 5*x^2 + 3)^(3/2),x)

[Out]

int(((5*x + x^2 + 2)*(4*x - 7*x^2 + 1)^3)/(2*x + 5*x^2 + 3)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {29 x}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \left (- \frac {115 x^{2}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \frac {61 x^{3}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {871 x^{4}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \left (- \frac {127 x^{5}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \left (- \frac {2065 x^{6}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \frac {1127 x^{7}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {343 x^{8}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \left (- \frac {2}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x**2+4*x+1)**3*(x**2+5*x+2)/(5*x**2+2*x+3)**(3/2),x)

[Out]

-Integral(-29*x/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x) -
Integral(-115*x**2/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x)
- Integral(61*x**3/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x
) - Integral(871*x**4/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)),
x) - Integral(-127*x**5/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3
)), x) - Integral(-2065*x**6/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x
+ 3)), x) - Integral(1127*x**7/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 +
2*x + 3)), x) - Integral(343*x**8/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2
+ 2*x + 3)), x) - Integral(-2/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*
x + 3)), x)

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