3.374 \(\int (1+4 x-7 x^2)^3 (2+5 x+x^2) \sqrt{3+2 x+5 x^2} \, dx\)

Optimal. Leaf size=208 \[ -\frac{343}{50} \left (5 x^2+2 x+3\right )^{3/2} x^7-\frac{50519 \left (5 x^2+2 x+3\right )^{3/2} x^6}{2250}+\frac{190939 \left (5 x^2+2 x+3\right )^{3/2} x^5}{3000}-\frac{888751 \left (5 x^2+2 x+3\right )^{3/2} x^4}{105000}-\frac{90960857 \left (5 x^2+2 x+3\right )^{3/2} x^3}{1575000}+\frac{98060877 \left (5 x^2+2 x+3\right )^{3/2} x^2}{4375000}+\frac{1045360143 \left (5 x^2+2 x+3\right )^{3/2} x}{43750000}-\frac{1968340667 \left (5 x^2+2 x+3\right )^{3/2}}{131250000}-\frac{77159983 (5 x+1) \sqrt{5 x^2+2 x+3}}{31250000}-\frac{540119881 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{15625000 \sqrt{5}} \]

[Out]

(-77159983*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/31250000 - (1968340667*(3 + 2*x + 5*x^2)^(3/2))/131250000 + (10453
60143*x*(3 + 2*x + 5*x^2)^(3/2))/43750000 + (98060877*x^2*(3 + 2*x + 5*x^2)^(3/2))/4375000 - (90960857*x^3*(3
+ 2*x + 5*x^2)^(3/2))/1575000 - (888751*x^4*(3 + 2*x + 5*x^2)^(3/2))/105000 + (190939*x^5*(3 + 2*x + 5*x^2)^(3
/2))/3000 - (50519*x^6*(3 + 2*x + 5*x^2)^(3/2))/2250 - (343*x^7*(3 + 2*x + 5*x^2)^(3/2))/50 - (540119881*ArcSi
nh[(1 + 5*x)/Sqrt[14]])/(15625000*Sqrt[5])

________________________________________________________________________________________

Rubi [A]  time = 0.352348, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1661, 640, 612, 619, 215} \[ -\frac{343}{50} \left (5 x^2+2 x+3\right )^{3/2} x^7-\frac{50519 \left (5 x^2+2 x+3\right )^{3/2} x^6}{2250}+\frac{190939 \left (5 x^2+2 x+3\right )^{3/2} x^5}{3000}-\frac{888751 \left (5 x^2+2 x+3\right )^{3/2} x^4}{105000}-\frac{90960857 \left (5 x^2+2 x+3\right )^{3/2} x^3}{1575000}+\frac{98060877 \left (5 x^2+2 x+3\right )^{3/2} x^2}{4375000}+\frac{1045360143 \left (5 x^2+2 x+3\right )^{3/2} x}{43750000}-\frac{1968340667 \left (5 x^2+2 x+3\right )^{3/2}}{131250000}-\frac{77159983 (5 x+1) \sqrt{5 x^2+2 x+3}}{31250000}-\frac{540119881 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{15625000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-77159983*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/31250000 - (1968340667*(3 + 2*x + 5*x^2)^(3/2))/131250000 + (10453
60143*x*(3 + 2*x + 5*x^2)^(3/2))/43750000 + (98060877*x^2*(3 + 2*x + 5*x^2)^(3/2))/4375000 - (90960857*x^3*(3
+ 2*x + 5*x^2)^(3/2))/1575000 - (888751*x^4*(3 + 2*x + 5*x^2)^(3/2))/105000 + (190939*x^5*(3 + 2*x + 5*x^2)^(3
/2))/3000 - (50519*x^6*(3 + 2*x + 5*x^2)^(3/2))/2250 - (343*x^7*(3 + 2*x + 5*x^2)^(3/2))/50 - (540119881*ArcSi
nh[(1 + 5*x)/Sqrt[14]])/(15625000*Sqrt[5])

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]

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 612

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

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 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]

Rubi steps

\begin{align*} \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt{3+2 x+5 x^2} \, dx &=-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{1}{50} \int \sqrt{3+2 x+5 x^2} \left (100+1450 x+5750 x^2-3050 x^3-43550 x^4+6350 x^5+110453 x^6-50519 x^7\right ) \, dx\\ &=-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (4500+65250 x+258750 x^2-137250 x^3-1959750 x^4+1195092 x^5+5728170 x^6\right ) \, dx}{2250}\\ &=\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (180000+2610000 x+10350000 x^2-5490000 x^3-164312550 x^4-26662530 x^5\right ) \, dx}{90000}\\ &=-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (6300000+91350000 x+362250000 x^2+127800360 x^3-5457651420 x^4\right ) \, dx}{3150000}\\ &=-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (189000000+2740500000 x+59986362780 x^2+52952873580 x^3\right ) \, dx}{94500000}\\ &=\frac{98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (4725000000-249204741480 x+1128988954440 x^2\right ) \, dx}{2362500000}\\ &=\frac{1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac{98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int (-3292466863320-10629039601800 x) \sqrt{3+2 x+5 x^2} \, dx}{47250000000}\\ &=-\frac{1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac{1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac{98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac{77159983 \int \sqrt{3+2 x+5 x^2} \, dx}{3125000}\\ &=-\frac{77159983 (1+5 x) \sqrt{3+2 x+5 x^2}}{31250000}-\frac{1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac{1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac{98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac{540119881 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{15625000}\\ &=-\frac{77159983 (1+5 x) \sqrt{3+2 x+5 x^2}}{31250000}-\frac{1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac{1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac{98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac{\left (77159983 \sqrt{\frac{7}{10}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{31250000}\\ &=-\frac{77159983 (1+5 x) \sqrt{3+2 x+5 x^2}}{31250000}-\frac{1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac{1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac{98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac{90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac{888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac{190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac{50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac{343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac{540119881 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{15625000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.315134, size = 85, normalized size = 0.41 \[ \frac{-5 \sqrt{5 x^2+2 x+3} \left (67528125000 x^9+248031875000 x^8-497593468750 x^7-34674656250 x^6+225922362500 x^5+56757413000 x^4+17642392275 x^3-78839046795 x^2-57768004650 x+93436408944\right )-68055105006 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{9843750000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[3 + 2*x + 5*x^2]*(93436408944 - 57768004650*x - 78839046795*x^2 + 17642392275*x^3 + 56757413000*x^4 +
225922362500*x^5 - 34674656250*x^6 - 497593468750*x^7 + 248031875000*x^8 + 67528125000*x^9) - 68055105006*Sqr
t[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/9843750000

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Maple [A]  time = 0.075, size = 166, normalized size = 0.8 \begin{align*}{\frac{190939\,{x}^{5}}{3000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{888751\,{x}^{4}}{105000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{90960857\,{x}^{3}}{1575000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{98060877\,{x}^{2}}{4375000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1045360143\,x}{43750000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{771599830\,x+154319966}{62500000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{540119881\,\sqrt{5}}{78125000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }-{\frac{1968340667}{131250000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{343\,{x}^{7}}{50} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{50519\,{x}^{6}}{2250} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}} \end{align*}

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)^(1/2),x)

[Out]

190939/3000*x^5*(5*x^2+2*x+3)^(3/2)-888751/105000*x^4*(5*x^2+2*x+3)^(3/2)-90960857/1575000*x^3*(5*x^2+2*x+3)^(
3/2)[phone]/4375000*x^2*(5*x^2+2*x+3)^(3/2)[phone]/43750000*x*(5*x^2+2*x+3)^(3/2)-77159983/62500000*(10*
x+2)*(5*x^2+2*x+3)^(1/2)-540119881/78125000*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))-1968340667/131250000*(5*x^2
+2*x+3)^(3/2)-343/50*x^7*(5*x^2+2*x+3)^(3/2)-50519/2250*x^6*(5*x^2+2*x+3)^(3/2)

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Maxima [A]  time = 1.49426, size = 239, normalized size = 1.15 \begin{align*} -\frac{343}{50} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{7} - \frac{50519}{2250} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{6} + \frac{190939}{3000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{5} - \frac{888751}{105000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{90960857}{1575000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{98060877}{4375000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{1045360143}{43750000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x - \frac{1968340667}{131250000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{77159983}{6250000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{540119881}{78125000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{77159983}{31250000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} \end{align*}

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)^(1/2),x, algorithm="maxima")

[Out]

-343/50*(5*x^2 + 2*x + 3)^(3/2)*x^7 - 50519/2250*(5*x^2 + 2*x + 3)^(3/2)*x^6 + 190939/3000*(5*x^2 + 2*x + 3)^(
3/2)*x^5 - 888751/105000*(5*x^2 + 2*x + 3)^(3/2)*x^4 - 90960857/1575000*(5*x^2 + 2*x + 3)^(3/2)*x^3 + 98060877
/4375000*(5*x^2 + 2*x + 3)^(3/2)*x^2 + 1045360143/43750000*(5*x^2 + 2*x + 3)^(3/2)*x - 1968340667/131250000*(5
*x^2 + 2*x + 3)^(3/2) - 77159983/6250000*sqrt(5*x^2 + 2*x + 3)*x - 540119881/78125000*sqrt(5)*arcsinh(1/14*sqr
t(14)*(5*x + 1)) - 77159983/31250000*sqrt(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.13303, size = 417, normalized size = 2. \begin{align*} -\frac{1}{1968750000} \,{\left (67528125000 \, x^{9} + 248031875000 \, x^{8} - 497593468750 \, x^{7} - 34674656250 \, x^{6} + 225922362500 \, x^{5} + 56757413000 \, x^{4} + 17642392275 \, x^{3} - 78839046795 \, x^{2} - 57768004650 \, x + 93436408944\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{540119881}{156250000} \, \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{align*}

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)^(1/2),x, algorithm="fricas")

[Out]

-1/1968750000*(67528125000*x^9 + 248031875000*x^8 - 497593468750*x^7 - 34674656250*x^6 + 225922362500*x^5 + 56
757413000*x^4 + 17642392275*x^3 - 78839046795*x^2 - 57768004650*x + 93436408944)*sqrt(5*x^2 + 2*x + 3) + 54011
9881/156250000*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 29 x \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 115 x^{2} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 61 x^{3} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 871 x^{4} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 127 x^{5} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 2065 x^{6} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 1127 x^{7} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 343 x^{8} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 2 \sqrt{5 x^{2} + 2 x + 3}\, dx \end{align*}

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)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.28566, size = 124, normalized size = 0.6 \begin{align*} -\frac{1}{1968750000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (25 \,{\left (5 \,{\left (49 \,{\left (140 \,{\left (315 \, x + 1157\right )} x - 324959\right )} x - 1109589\right )} x + 36147578\right )} x + 227029652\right )} x + 705695691\right )} x - 15767809359\right )} x - 11553600930\right )} x + 93436408944\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{540119881}{78125000} \, \sqrt{5} \log \left (-\sqrt{5}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \end{align*}

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)^(1/2),x, algorithm="giac")

[Out]

-1/1968750000*(5*((5*(10*(25*(5*(49*(140*(315*x + 1157)*x - 324959)*x - 1109589)*x + 36147578)*x + 227029652)*
x + 705695691)*x - 15767809359)*x - 11553600930)*x + 93436408944)*sqrt(5*x^2 + 2*x + 3) + 540119881/78125000*s
qrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)