### 3.382 $$\int (1+4 x-7 x^2) (2+5 x+x^2) (3+2 x+5 x^2)^{3/2} \, dx$$

Optimal. Leaf size=147 $-\frac{7}{40} \left (5 x^2+2 x+3\right )^{5/2} x^3-\frac{1163 \left (5 x^2+2 x+3\right )^{5/2} x^2}{1400}+\frac{2809 \left (5 x^2+2 x+3\right )^{5/2} x}{5250}+\frac{149509 \left (5 x^2+2 x+3\right )^{5/2}}{262500}-\frac{18397 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{150000}-\frac{128779 (5 x+1) \sqrt{5 x^2+2 x+3}}{250000}-\frac{901453 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{125000 \sqrt{5}}$

[Out]

(-128779*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/250000 - (18397*(1 + 5*x)*(3 + 2*x + 5*x^2)^(3/2))/150000 + (149509*
(3 + 2*x + 5*x^2)^(5/2))/262500 + (2809*x*(3 + 2*x + 5*x^2)^(5/2))/5250 - (1163*x^2*(3 + 2*x + 5*x^2)^(5/2))/1
400 - (7*x^3*(3 + 2*x + 5*x^2)^(5/2))/40 - (901453*ArcSinh[(1 + 5*x)/Sqrt[14]])/(125000*Sqrt[5])

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Rubi [A]  time = 0.129781, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.152, Rules used = {1661, 640, 612, 619, 215} $-\frac{7}{40} \left (5 x^2+2 x+3\right )^{5/2} x^3-\frac{1163 \left (5 x^2+2 x+3\right )^{5/2} x^2}{1400}+\frac{2809 \left (5 x^2+2 x+3\right )^{5/2} x}{5250}+\frac{149509 \left (5 x^2+2 x+3\right )^{5/2}}{262500}-\frac{18397 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{150000}-\frac{128779 (5 x+1) \sqrt{5 x^2+2 x+3}}{250000}-\frac{901453 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{125000 \sqrt{5}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-128779*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/250000 - (18397*(1 + 5*x)*(3 + 2*x + 5*x^2)^(3/2))/150000 + (149509*
(3 + 2*x + 5*x^2)^(5/2))/262500 + (2809*x*(3 + 2*x + 5*x^2)^(5/2))/5250 - (1163*x^2*(3 + 2*x + 5*x^2)^(5/2))/1
400 - (7*x^3*(3 + 2*x + 5*x^2)^(5/2))/40 - (901453*ArcSinh[(1 + 5*x)/Sqrt[14]])/(125000*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 ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx &=-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}+\frac{1}{40} \int \left (3+2 x+5 x^2\right )^{3/2} \left (80+520 x+343 x^2-1163 x^3\right ) \, dx\\ &=-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (2800+25178 x+22472 x^2\right ) \, dx}{1400}\\ &=\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int (16584+598036 x) \left (3+2 x+5 x^2\right )^{3/2} \, dx}{42000}\\ &=\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{18397 \int \left (3+2 x+5 x^2\right )^{3/2} \, dx}{7500}\\ &=-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{128779 \int \sqrt{3+2 x+5 x^2} \, dx}{25000}\\ &=-\frac{128779 (1+5 x) \sqrt{3+2 x+5 x^2}}{250000}-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{901453 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{125000}\\ &=-\frac{128779 (1+5 x) \sqrt{3+2 x+5 x^2}}{250000}-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{\left (128779 \sqrt{\frac{7}{10}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{250000}\\ &=-\frac{128779 (1+5 x) \sqrt{3+2 x+5 x^2}}{250000}-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{901453 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{125000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.148884, size = 75, normalized size = 0.51 $\frac{-5 \sqrt{5 x^2+2 x+3} \left (22968750 x^7+127406250 x^6+48237500 x^5+28373000 x^4-78608475 x^3-86464445 x^2-36695150 x-22275576\right )-37861026 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{26250000}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*Sqrt[3 + 2*x + 5*x^2]*(-22275576 - 36695150*x - 86464445*x^2 - 78608475*x^3 + 28373000*x^4 + 48237500*x^5
+ 127406250*x^6 + 22968750*x^7) - 37861026*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/26250000

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Maple [A]  time = 0.056, size = 117, normalized size = 0.8 \begin{align*} -{\frac{7\,{x}^{3}}{40} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{1163\,{x}^{2}}{1400} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{1287790\,x+257558}{500000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{901453\,\sqrt{5}}{625000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{2809\,x}{5250} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{183970\,x+36794}{300000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{149509}{262500} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/40*x^3*(5*x^2+2*x+3)^(5/2)-1163/1400*x^2*(5*x^2+2*x+3)^(5/2)-128779/500000*(10*x+2)*(5*x^2+2*x+3)^(1/2)-901
453/625000*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))+2809/5250*x*(5*x^2+2*x+3)^(5/2)-18397/300000*(10*x+2)*(5*x^2
+2*x+3)^(3/2)+149509/262500*(5*x^2+2*x+3)^(5/2)

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Maxima [A]  time = 1.47904, size = 186, normalized size = 1.27 \begin{align*} -\frac{7}{40} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{3} - \frac{1163}{1400} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{2809}{5250} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x + \frac{149509}{262500} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} - \frac{18397}{30000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x - \frac{18397}{150000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{128779}{50000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{901453}{625000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{128779}{250000} \, \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)*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="maxima")

[Out]

-7/40*(5*x^2 + 2*x + 3)^(5/2)*x^3 - 1163/1400*(5*x^2 + 2*x + 3)^(5/2)*x^2 + 2809/5250*(5*x^2 + 2*x + 3)^(5/2)*
x + 149509/262500*(5*x^2 + 2*x + 3)^(5/2) - 18397/30000*(5*x^2 + 2*x + 3)^(3/2)*x - 18397/150000*(5*x^2 + 2*x
+ 3)^(3/2) - 128779/50000*sqrt(5*x^2 + 2*x + 3)*x - 901453/625000*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 1)) - 1
28779/250000*sqrt(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.26446, size = 323, normalized size = 2.2 \begin{align*} -\frac{1}{5250000} \,{\left (22968750 \, x^{7} + 127406250 \, x^{6} + 48237500 \, x^{5} + 28373000 \, x^{4} - 78608475 \, x^{3} - 86464445 \, x^{2} - 36695150 \, x - 22275576\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{901453}{1250000} \, \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)*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="fricas")

[Out]

-1/5250000*(22968750*x^7 + 127406250*x^6 + 48237500*x^5 + 28373000*x^4 - 78608475*x^3 - 86464445*x^2 - 3669515
0*x - 22275576)*sqrt(5*x^2 + 2*x + 3) + 901453/1250000*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 2
5*x^2 - 10*x - 8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 43 x \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 57 x^{2} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 14 x^{3} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 48 x^{4} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 169 x^{5} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 35 x^{6} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 6 \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)*(x**2+5*x+2)*(5*x**2+2*x+3)**(3/2),x)

[Out]

-Integral(-43*x*sqrt(5*x**2 + 2*x + 3), x) - Integral(-57*x**2*sqrt(5*x**2 + 2*x + 3), x) - Integral(14*x**3*s
qrt(5*x**2 + 2*x + 3), x) - Integral(48*x**4*sqrt(5*x**2 + 2*x + 3), x) - Integral(169*x**5*sqrt(5*x**2 + 2*x
+ 3), x) - Integral(35*x**6*sqrt(5*x**2 + 2*x + 3), x) - Integral(-6*sqrt(5*x**2 + 2*x + 3), x)

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Giac [A]  time = 1.30397, size = 111, normalized size = 0.76 \begin{align*} -\frac{1}{5250000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (25 \,{\left (15 \,{\left (245 \, x + 1359\right )} x + 7718\right )} x + 113492\right )} x - 3144339\right )} x - 17292889\right )} x - 7339030\right )} x - 22275576\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{901453}{625000} \, \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)*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="giac")

[Out]

-1/5250000*(5*((5*(10*(25*(15*(245*x + 1359)*x + 7718)*x + 113492)*x - 3144339)*x - 17292889)*x - 7339030)*x -
22275576)*sqrt(5*x^2 + 2*x + 3) + 901453/625000*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)