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

[Out]

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

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Rubi [A]  time = 0.20, 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 = {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 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 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 )^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*}

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Mathematica [A]  time = 0.14, 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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fricas [A]  time = 0.87, size = 77, normalized size = 0.54 \[ \frac {1}{187500} \, {\left (306250 \, x^{5} + 1282750 \, x^{4} - 2494425 \, x^{3} - 1037135 \, x^{2} + 3607300 \, x - 132318\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1719097}{312500} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]

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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giac [A]  time = 0.39, size = 72, normalized size = 0.50 \[ \frac {1}{187500} \, {\left (5 \, {\left ({\left (5 \, {\left (70 \, {\left (175 \, x + 733\right )} x - 99777\right )} x - 207427\right )} x + 721460\right )} x - 132318\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1719097}{156250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]

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)

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maple [A]  time = 0.01, size = 113, normalized size = 0.79 \[ \frac {49 \sqrt {5 x^{2}+2 x +3}\, x^{5}}{30}+\frac {5131 \sqrt {5 x^{2}+2 x +3}\, x^{4}}{750}-\frac {33259 \sqrt {5 x^{2}+2 x +3}\, x^{3}}{2500}-\frac {207427 \sqrt {5 x^{2}+2 x +3}\, x^{2}}{37500}+\frac {36073 \sqrt {5 x^{2}+2 x +3}\, x}{1875}-\frac {1719097 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{156250}-\frac {22053 \sqrt {5 x^{2}+2 x +3}}{31250} \]

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]

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

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maxima [A]  time = 0.97, size = 114, normalized size = 0.80 \[ \frac {49}{30} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{5} + \frac {5131}{750} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{4} - \frac {33259}{2500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{3} - \frac {207427}{37500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{2} + \frac {36073}{1875} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {1719097}{156250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {22053}{31250} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} + 5 x + 2\right ) \left (7 x^{2} - 4 x - 1\right )^{2}}{\sqrt {5 x^{2} + 2 x + 3}}\, dx \]

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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