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

Optimal. Leaf size=143 \[ \frac{49}{30} \sqrt{5 x^2+2 x+3} x^5+\frac{5131}{750} \sqrt{5 x^2+2 x+3} x^4-\frac{33259 \sqrt{5 x^2+2 x+3} x^3}{2500}-\frac{207427 \sqrt{5 x^2+2 x+3} x^2}{37500}+\frac{36073 \sqrt{5 x^2+2 x+3} x}{1875}-\frac{22053 \sqrt{5 x^2+2 x+3}}{31250}-\frac{1719097 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{31250 \sqrt{5}} \]

[Out]

(-22053*Sqrt[3 + 2*x + 5*x^2])/31250 + (36073*x*Sqrt[3 + 2*x + 5*x^2])/1875 - (207427*x^2*Sqrt[3 + 2*x + 5*x^2
])/37500 - (33259*x^3*Sqrt[3 + 2*x + 5*x^2])/2500 + (5131*x^4*Sqrt[3 + 2*x + 5*x^2])/750 + (49*x^5*Sqrt[3 + 2*
x + 5*x^2])/30 - (1719097*ArcSinh[(1 + 5*x)/Sqrt[14]])/(31250*Sqrt[5])

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Rubi [A]  time = 0.202813, antiderivative size = 143, normalized size of antiderivative = 1., 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 = {1661, 640, 619, 215} \[ \frac{49}{30} \sqrt{5 x^2+2 x+3} x^5+\frac{5131}{750} \sqrt{5 x^2+2 x+3} x^4-\frac{33259 \sqrt{5 x^2+2 x+3} x^3}{2500}-\frac{207427 \sqrt{5 x^2+2 x+3} x^2}{37500}+\frac{36073 \sqrt{5 x^2+2 x+3} x}{1875}-\frac{22053 \sqrt{5 x^2+2 x+3}}{31250}-\frac{1719097 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{31250 \sqrt{5}} \]

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]

(-22053*Sqrt[3 + 2*x + 5*x^2])/31250 + (36073*x*Sqrt[3 + 2*x + 5*x^2])/1875 - (207427*x^2*Sqrt[3 + 2*x + 5*x^2
])/37500 - (33259*x^3*Sqrt[3 + 2*x + 5*x^2])/2500 + (5131*x^4*Sqrt[3 + 2*x + 5*x^2])/750 + (49*x^5*Sqrt[3 + 2*
x + 5*x^2])/30 - (1719097*ArcSinh[(1 + 5*x)/Sqrt[14]])/(31250*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 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 \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 \operatorname{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*}

Mathematica [A]  time = 0.153379, size = 65, normalized size = 0.45 \[ \frac{5 \sqrt{5 x^2+2 x+3} \left (306250 x^5+1282750 x^4-2494425 x^3-1037135 x^2+3607300 x-132318\right )-10314582 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{937500} \]

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]*(-132318 + 3607300*x - 1037135*x^2 - 2494425*x^3 + 1282750*x^4 + 306250*x^5) - 103145
82*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/937500

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Maple [A]  time = 0.058, size = 113, normalized size = 0.8 \begin{align*}{\frac{49\,{x}^{5}}{30}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{5131\,{x}^{4}}{750}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{1719097\,\sqrt{5}}{156250}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }-{\frac{33259\,{x}^{3}}{2500}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{36073\,x}{1875}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{207427\,{x}^{2}}{37500}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{22053}{31250}\sqrt{5\,{x}^{2}+2\,x+3}} \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]

49/30*x^5*(5*x^2+2*x+3)^(1/2)+5131/750*x^4*(5*x^2+2*x+3)^(1/2)-1719097/156250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x
+1/5))-33259/2500*x^3*(5*x^2+2*x+3)^(1/2)+36073/1875*x*(5*x^2+2*x+3)^(1/2)-207427/37500*x^2*(5*x^2+2*x+3)^(1/2
)-22053/31250*(5*x^2+2*x+3)^(1/2)

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Maxima [A]  time = 1.45245, size = 154, normalized size = 1.08 \begin{align*} \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{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]

49/30*sqrt(5*x^2 + 2*x + 3)*x^5 + 5131/750*sqrt(5*x^2 + 2*x + 3)*x^4 - 33259/2500*sqrt(5*x^2 + 2*x + 3)*x^3 -
207427/37500*sqrt(5*x^2 + 2*x + 3)*x^2 + 36073/1875*sqrt(5*x^2 + 2*x + 3)*x - 1719097/156250*sqrt(5)*arcsinh(1
/14*sqrt(14)*(5*x + 1)) - 22053/31250*sqrt(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.36288, size = 267, normalized size = 1.87 \begin{align*} \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{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]

1/187500*(306250*x^5 + 1282750*x^4 - 2494425*x^3 - 1037135*x^2 + 3607300*x - 132318)*sqrt(5*x^2 + 2*x + 3) + 1
719097/312500*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 \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{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)*(7*x**2 - 4*x - 1)**2/sqrt(5*x**2 + 2*x + 3), x)

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Giac [A]  time = 1.19185, size = 97, normalized size = 0.68 \begin{align*} \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{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]

1/187500*(5*((5*(70*(175*x + 733)*x - 99777)*x - 207427)*x + 721460)*x - 132318)*sqrt(5*x^2 + 2*x + 3) + 17190
97/156250*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)