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

Optimal. Leaf size=166 \[ \frac{49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5+\frac{989}{200} \left (5 x^2+2 x+3\right )^{3/2} x^4-\frac{25277 \left (5 x^2+2 x+3\right )^{3/2} x^3}{3000}-\frac{77509 \left (5 x^2+2 x+3\right )^{3/2} x^2}{25000}+\frac{1781669 \left (5 x^2+2 x+3\right )^{3/2} x}{250000}+\frac{198439 \left (5 x^2+2 x+3\right )^{3/2}}{750000}-\frac{2521723 (5 x+1) \sqrt{5 x^2+2 x+3}}{1250000}-\frac{17652061 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{625000 \sqrt{5}} \]

[Out]

(-2521723*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/1250000 + (198439*(3 + 2*x + 5*x^2)^(3/2))/750000 + (1781669*x*(3 +
2*x + 5*x^2)^(3/2))/250000 - (77509*x^2*(3 + 2*x + 5*x^2)^(3/2))/25000 - (25277*x^3*(3 + 2*x + 5*x^2)^(3/2))/
3000 + (989*x^4*(3 + 2*x + 5*x^2)^(3/2))/200 + (49*x^5*(3 + 2*x + 5*x^2)^(3/2))/40 - (17652061*ArcSinh[(1 + 5*
x)/Sqrt[14]])/(625000*Sqrt[5])

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Rubi [A]  time = 0.203479, antiderivative size = 166, normalized size of antiderivative = 1., 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 = {1661, 640, 612, 619, 215} \[ \frac{49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5+\frac{989}{200} \left (5 x^2+2 x+3\right )^{3/2} x^4-\frac{25277 \left (5 x^2+2 x+3\right )^{3/2} x^3}{3000}-\frac{77509 \left (5 x^2+2 x+3\right )^{3/2} x^2}{25000}+\frac{1781669 \left (5 x^2+2 x+3\right )^{3/2} x}{250000}+\frac{198439 \left (5 x^2+2 x+3\right )^{3/2}}{750000}-\frac{2521723 (5 x+1) \sqrt{5 x^2+2 x+3}}{1250000}-\frac{17652061 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{625000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2521723*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/1250000 + (198439*(3 + 2*x + 5*x^2)^(3/2))/750000 + (1781669*x*(3 +
2*x + 5*x^2)^(3/2))/250000 - (77509*x^2*(3 + 2*x + 5*x^2)^(3/2))/25000 - (25277*x^3*(3 + 2*x + 5*x^2)^(3/2))/
3000 + (989*x^4*(3 + 2*x + 5*x^2)^(3/2))/200 + (49*x^5*(3 + 2*x + 5*x^2)^(3/2))/40 - (17652061*ArcSinh[(1 + 5*
x)/Sqrt[14]])/(625000*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 )^2 \left (2+5 x+x^2\right ) \sqrt{3+2 x+5 x^2} \, dx &=\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac{1}{40} \int \sqrt{3+2 x+5 x^2} \left (80+840 x+1800 x^2-3760 x^3-7935 x^4+6923 x^5\right ) \, dx\\ &=\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (2800+29400 x+63000 x^2-214676 x^3-353878 x^4\right ) \, dx}{1400}\\ &=-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (84000+882000 x+5074902 x^2-3255378 x^3\right ) \, dx}{42000}\\ &=-\frac{77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int \sqrt{3+2 x+5 x^2} \left (2100000+41582268 x+149660196 x^2\right ) \, dx}{1050000}\\ &=\frac{1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac{77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac{\int (-406980588+83344380 x) \sqrt{3+2 x+5 x^2} \, dx}{21000000}\\ &=\frac{198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac{1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac{77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac{2521723 \int \sqrt{3+2 x+5 x^2} \, dx}{125000}\\ &=-\frac{2521723 (1+5 x) \sqrt{3+2 x+5 x^2}}{1250000}+\frac{198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac{1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac{77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac{17652061 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{625000}\\ &=-\frac{2521723 (1+5 x) \sqrt{3+2 x+5 x^2}}{1250000}+\frac{198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac{1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac{77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac{\left (2521723 \sqrt{\frac{7}{10}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{1250000}\\ &=-\frac{2521723 (1+5 x) \sqrt{3+2 x+5 x^2}}{1250000}+\frac{198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac{1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac{77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac{25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac{989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac{49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac{17652061 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{625000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.194699, size = 75, normalized size = 0.45 \[ \frac{5 \sqrt{5 x^2+2 x+3} \left (22968750 x^7+101906250 x^6-107112500 x^5-65693000 x^4+15583725 x^3+23531995 x^2+44333650 x-4588584\right )-105912366 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{18750000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[3 + 2*x + 5*x^2]*(-4588584 + 44333650*x + 23531995*x^2 + 15583725*x^3 - 65693000*x^4 - 107112500*x^5 +
101906250*x^6 + 22968750*x^7) - 105912366*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/18750000

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Maple [A]  time = 0.06, size = 132, normalized size = 0.8 \begin{align*}{\frac{49\,{x}^{5}}{40} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{989\,{x}^{4}}{200} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{25277\,{x}^{3}}{3000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{77509\,{x}^{2}}{25000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1781669\,x}{250000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{25217230\,x+5043446}{2500000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{17652061\,\sqrt{5}}{3125000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{198439}{750000} \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)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x)

[Out]

49/40*x^5*(5*x^2+2*x+3)^(3/2)+989/200*x^4*(5*x^2+2*x+3)^(3/2)-25277/3000*x^3*(5*x^2+2*x+3)^(3/2)-77509/25000*x
^2*(5*x^2+2*x+3)^(3/2)+1781669/250000*x*(5*x^2+2*x+3)^(3/2)-2521723/2500000*(10*x+2)*(5*x^2+2*x+3)^(1/2)-17652
061/3125000*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))+198439/750000*(5*x^2+2*x+3)^(3/2)

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Maxima [A]  time = 1.49802, size = 193, normalized size = 1.16 \begin{align*} \frac{49}{40} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{5} + \frac{989}{200} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{25277}{3000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{77509}{25000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{1781669}{250000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x + \frac{198439}{750000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{2521723}{250000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{17652061}{3125000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{2521723}{1250000} \, \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)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")

[Out]

49/40*(5*x^2 + 2*x + 3)^(3/2)*x^5 + 989/200*(5*x^2 + 2*x + 3)^(3/2)*x^4 - 25277/3000*(5*x^2 + 2*x + 3)^(3/2)*x
^3 - 77509/25000*(5*x^2 + 2*x + 3)^(3/2)*x^2 + 1781669/250000*(5*x^2 + 2*x + 3)^(3/2)*x + 198439/750000*(5*x^2
+ 2*x + 3)^(3/2) - 2521723/250000*sqrt(5*x^2 + 2*x + 3)*x - 17652061/3125000*sqrt(5)*arcsinh(1/14*sqrt(14)*(5
*x + 1)) - 2521723/1250000*sqrt(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.10473, size = 324, normalized size = 1.95 \begin{align*} \frac{1}{3750000} \,{\left (22968750 \, x^{7} + 101906250 \, x^{6} - 107112500 \, x^{5} - 65693000 \, x^{4} + 15583725 \, x^{3} + 23531995 \, x^{2} + 44333650 \, x - 4588584\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{17652061}{6250000} \, \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)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")

[Out]

1/3750000*(22968750*x^7 + 101906250*x^6 - 107112500*x^5 - 65693000*x^4 + 15583725*x^3 + 23531995*x^2 + 4433365
0*x - 4588584)*sqrt(5*x^2 + 2*x + 3) + 17652061/6250000*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 \left (x^{2} + 5 x + 2\right ) \sqrt{5 x^{2} + 2 x + 3} \left (7 x^{2} - 4 x - 1\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2 + 5*x + 2)*sqrt(5*x**2 + 2*x + 3)*(7*x**2 - 4*x - 1)**2, x)

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Giac [A]  time = 1.17767, size = 111, normalized size = 0.67 \begin{align*} \frac{1}{3750000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (25 \,{\left (15 \,{\left (245 \, x + 1087\right )} x - 17138\right )} x - 262772\right )} x + 623349\right )} x + 4706399\right )} x + 8866730\right )} x - 4588584\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{17652061}{3125000} \, \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)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="giac")

[Out]

1/3750000*(5*((5*(10*(25*(15*(245*x + 1087)*x - 17138)*x - 262772)*x + 623349)*x + 4706399)*x + 8866730)*x - 4
588584)*sqrt(5*x^2 + 2*x + 3) + 17652061/3125000*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)