### 3.386 $$\int \frac{(1+4 x-7 x^2)^3 (2+5 x+x^2)}{\sqrt{3+2 x+5 x^2}} \, dx$$

Optimal. Leaf size=185 $-\frac{343}{40} \sqrt{5 x^2+2 x+3} x^7-\frac{1141}{40} \sqrt{5 x^2+2 x+3} x^6+\frac{26159}{300} \sqrt{5 x^2+2 x+3} x^5-\frac{47807 \sqrt{5 x^2+2 x+3} x^4}{3750}-\frac{5160533 \sqrt{5 x^2+2 x+3} x^3}{50000}+\frac{40722851 \sqrt{5 x^2+2 x+3} x^2}{750000}+\frac{5793077 \sqrt{5 x^2+2 x+3} x}{75000}-\frac{16515809 \sqrt{5 x^2+2 x+3}}{156250}-\frac{77513689 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{625000 \sqrt{5}}$

[Out]

(-16515809*Sqrt[3 + 2*x + 5*x^2])/156250 + (5793077*x*Sqrt[3 + 2*x + 5*x^2])/75000 + (40722851*x^2*Sqrt[3 + 2*
x + 5*x^2])/750000 - (5160533*x^3*Sqrt[3 + 2*x + 5*x^2])/50000 - (47807*x^4*Sqrt[3 + 2*x + 5*x^2])/3750 + (261
59*x^5*Sqrt[3 + 2*x + 5*x^2])/300 - (1141*x^6*Sqrt[3 + 2*x + 5*x^2])/40 - (343*x^7*Sqrt[3 + 2*x + 5*x^2])/40 -
(77513689*ArcSinh[(1 + 5*x)/Sqrt[14]])/(625000*Sqrt[5])

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Rubi [A]  time = 0.312332, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.114, Rules used = {1661, 640, 619, 215} $-\frac{343}{40} \sqrt{5 x^2+2 x+3} x^7-\frac{1141}{40} \sqrt{5 x^2+2 x+3} x^6+\frac{26159}{300} \sqrt{5 x^2+2 x+3} x^5-\frac{47807 \sqrt{5 x^2+2 x+3} x^4}{3750}-\frac{5160533 \sqrt{5 x^2+2 x+3} x^3}{50000}+\frac{40722851 \sqrt{5 x^2+2 x+3} x^2}{750000}+\frac{5793077 \sqrt{5 x^2+2 x+3} x}{75000}-\frac{16515809 \sqrt{5 x^2+2 x+3}}{156250}-\frac{77513689 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{625000 \sqrt{5}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-16515809*Sqrt[3 + 2*x + 5*x^2])/156250 + (5793077*x*Sqrt[3 + 2*x + 5*x^2])/75000 + (40722851*x^2*Sqrt[3 + 2*
x + 5*x^2])/750000 - (5160533*x^3*Sqrt[3 + 2*x + 5*x^2])/50000 - (47807*x^4*Sqrt[3 + 2*x + 5*x^2])/3750 + (261
59*x^5*Sqrt[3 + 2*x + 5*x^2])/300 - (1141*x^6*Sqrt[3 + 2*x + 5*x^2])/40 - (343*x^7*Sqrt[3 + 2*x + 5*x^2])/40 -
(77513689*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 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 )^3 \left (2+5 x+x^2\right )}{\sqrt{3+2 x+5 x^2}} \, dx &=-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{1}{40} \int \frac{80+1160 x+4600 x^2-2440 x^3-34840 x^4+5080 x^5+89803 x^6-39935 x^7}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{2800+40600 x+161000 x^2-85400 x^3-1219400 x^4+896630 x^5+3662260 x^6}{\sqrt{3+2 x+5 x^2}} \, dx}{1400}\\ &=\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{84000+1218000 x+4830000 x^2-2562000 x^3-91515900 x^4-13385960 x^5}{\sqrt{3+2 x+5 x^2}} \, dx}{42000}\\ &=-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{2100000+30450000 x+120750000 x^2+96581520 x^3-2167423860 x^4}{\sqrt{3+2 x+5 x^2}} \, dx}{1050000}\\ &=-\frac{5160533 x^3 \sqrt{3+2 x+5 x^2}}{50000}-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{42000000+609000000 x+21921814740 x^2+17103597420 x^3}{\sqrt{3+2 x+5 x^2}} \, dx}{21000000}\\ &=\frac{40722851 x^2 \sqrt{3+2 x+5 x^2}}{750000}-\frac{5160533 x^3 \sqrt{3+2 x+5 x^2}}{50000}-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{630000000-93486584520 x+243309234000 x^2}{\sqrt{3+2 x+5 x^2}} \, dx}{315000000}\\ &=\frac{5793077 x \sqrt{3+2 x+5 x^2}}{75000}+\frac{40722851 x^2 \sqrt{3+2 x+5 x^2}}{750000}-\frac{5160533 x^3 \sqrt{3+2 x+5 x^2}}{50000}-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{-723627702000-1664793547200 x}{\sqrt{3+2 x+5 x^2}} \, dx}{3150000000}\\ &=-\frac{16515809 \sqrt{3+2 x+5 x^2}}{156250}+\frac{5793077 x \sqrt{3+2 x+5 x^2}}{75000}+\frac{40722851 x^2 \sqrt{3+2 x+5 x^2}}{750000}-\frac{5160533 x^3 \sqrt{3+2 x+5 x^2}}{50000}-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}-\frac{77513689 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{625000}\\ &=-\frac{16515809 \sqrt{3+2 x+5 x^2}}{156250}+\frac{5793077 x \sqrt{3+2 x+5 x^2}}{75000}+\frac{40722851 x^2 \sqrt{3+2 x+5 x^2}}{750000}-\frac{5160533 x^3 \sqrt{3+2 x+5 x^2}}{50000}-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}-\frac{77513689 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{1250000 \sqrt{70}}\\ &=-\frac{16515809 \sqrt{3+2 x+5 x^2}}{156250}+\frac{5793077 x \sqrt{3+2 x+5 x^2}}{75000}+\frac{40722851 x^2 \sqrt{3+2 x+5 x^2}}{750000}-\frac{5160533 x^3 \sqrt{3+2 x+5 x^2}}{50000}-\frac{47807 x^4 \sqrt{3+2 x+5 x^2}}{3750}+\frac{26159}{300} x^5 \sqrt{3+2 x+5 x^2}-\frac{1141}{40} x^6 \sqrt{3+2 x+5 x^2}-\frac{343}{40} x^7 \sqrt{3+2 x+5 x^2}-\frac{77513689 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{625000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.261152, size = 75, normalized size = 0.41 $\frac{-5 \sqrt{5 x^2+2 x+3} \left (32156250 x^7+106968750 x^6-326987500 x^5+47807000 x^4+387039975 x^3-203614255 x^2-289653850 x+396379416\right )-465082134 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{18750000}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*Sqrt[3 + 2*x + 5*x^2]*(396379416 - 289653850*x - 203614255*x^2 + 387039975*x^3 + 47807000*x^4 - 326987500*
x^5 + 106968750*x^6 + 32156250*x^7) - 465082134*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/18750000

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Maple [A]  time = 0.067, size = 147, normalized size = 0.8 \begin{align*} -{\frac{343\,{x}^{7}}{40}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{1141\,{x}^{6}}{40}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{26159\,{x}^{5}}{300}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{47807\,{x}^{4}}{3750}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{77513689\,\sqrt{5}}{3125000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }-{\frac{5160533\,{x}^{3}}{50000}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{5793077\,x}{75000}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{40722851\,{x}^{2}}{750000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{16515809}{156250}\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)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(1/2),x)

[Out]

-343/40*x^7*(5*x^2+2*x+3)^(1/2)-1141/40*x^6*(5*x^2+2*x+3)^(1/2)+26159/300*x^5*(5*x^2+2*x+3)^(1/2)-47807/3750*x
^4*(5*x^2+2*x+3)^(1/2)-77513689/3125000*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))-5160533/50000*x^3*(5*x^2+2*x+3)
^(1/2)+5793077/75000*x*(5*x^2+2*x+3)^(1/2)+40722851/750000*x^2*(5*x^2+2*x+3)^(1/2)-16515809/156250*(5*x^2+2*x+
3)^(1/2)

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Maxima [A]  time = 1.53147, size = 200, normalized size = 1.08 \begin{align*} -\frac{343}{40} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{7} - \frac{1141}{40} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{6} + \frac{26159}{300} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{5} - \frac{47807}{3750} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{4} - \frac{5160533}{50000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{3} + \frac{40722851}{750000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{2} + \frac{5793077}{75000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{77513689}{3125000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{16515809}{156250} \, \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]

-343/40*sqrt(5*x^2 + 2*x + 3)*x^7 - 1141/40*sqrt(5*x^2 + 2*x + 3)*x^6 + 26159/300*sqrt(5*x^2 + 2*x + 3)*x^5 -
47807/3750*sqrt(5*x^2 + 2*x + 3)*x^4 - 5160533/50000*sqrt(5*x^2 + 2*x + 3)*x^3 + 40722851/750000*sqrt(5*x^2 +
2*x + 3)*x^2 + 5793077/75000*sqrt(5*x^2 + 2*x + 3)*x - 77513689/3125000*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 1
)) - 16515809/156250*sqrt(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.40984, size = 332, normalized size = 1.79 \begin{align*} -\frac{1}{3750000} \,{\left (32156250 \, x^{7} + 106968750 \, x^{6} - 326987500 \, x^{5} + 47807000 \, x^{4} + 387039975 \, x^{3} - 203614255 \, x^{2} - 289653850 \, x + 396379416\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{77513689}{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)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")

[Out]

-1/3750000*(32156250*x^7 + 106968750*x^6 - 326987500*x^5 + 47807000*x^4 + 387039975*x^3 - 203614255*x^2 - 2896
53850*x + 396379416)*sqrt(5*x^2 + 2*x + 3) + 77513689/6250000*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 - \frac{29 x}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{115 x^{2}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{61 x^{3}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{871 x^{4}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{127 x^{5}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{2065 x^{6}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{1127 x^{7}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{343 x^{8}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{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 = 1.16367, size = 111, normalized size = 0.6 \begin{align*} -\frac{1}{3750000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (175 \,{\left (15 \,{\left (49 \, x + 163\right )} x - 7474\right )} x + 191228\right )} x + 15481599\right )} x - 40722851\right )} x - 57930770\right )} x + 396379416\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{77513689}{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)^3*(x^2+5*x+2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")

[Out]

-1/3750000*(5*((5*(10*(175*(15*(49*x + 163)*x - 7474)*x + 191228)*x + 15481599)*x - 40722851)*x - 57930770)*x
+ 396379416)*sqrt(5*x^2 + 2*x + 3) + 77513689/3125000*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))
- 1)