### 3.221 $$\int (1+2 x)^2 (2-x+3 x^2)^{5/2} (1+3 x+4 x^2) \, dx$$

Optimal. Leaf size=164 $\frac{1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}+\frac{37}{405} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}+\frac{(3430 x+2731) \left (3 x^2-x+2\right )^{7/2}}{17010}-\frac{293 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{58320}-\frac{6739 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}-\frac{154997 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}-\frac{3564931 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}$

[Out]

(-154997*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/4478976 - (6739*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/559872 - (293*(1 - 6*
x)*(2 - x + 3*x^2)^(5/2))/58320 + (37*(1 + 2*x)^2*(2 - x + 3*x^2)^(7/2))/405 + ((1 + 2*x)^3*(2 - x + 3*x^2)^(7
/2))/15 + ((2731 + 3430*x)*(2 - x + 3*x^2)^(7/2))/17010 - (3564931*ArcSinh[(1 - 6*x)/Sqrt[23]])/(8957952*Sqrt[
3])

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Rubi [A]  time = 0.132842, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {1653, 832, 779, 612, 619, 215} $\frac{1}{15} (2 x+1)^3 \left (3 x^2-x+2\right )^{7/2}+\frac{37}{405} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}+\frac{(3430 x+2731) \left (3 x^2-x+2\right )^{7/2}}{17010}-\frac{293 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{58320}-\frac{6739 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}-\frac{154997 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}-\frac{3564931 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-154997*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/4478976 - (6739*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/559872 - (293*(1 - 6*
x)*(2 - x + 3*x^2)^(5/2))/58320 + (37*(1 + 2*x)^2*(2 - x + 3*x^2)^(7/2))/405 + ((1 + 2*x)^3*(2 - x + 3*x^2)^(7
/2))/15 + ((2731 + 3430*x)*(2 - x + 3*x^2)^(7/2))/17010 - (3564931*ArcSinh[(1 - 6*x)/Sqrt[23]])/(8957952*Sqrt[
3])

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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 (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{120} \int (1+2 x)^2 (52+296 x) \left (2-x+3 x^2\right )^{5/2} \, dx\\ &=\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{\int (1+2 x) (72+7840 x) \left (2-x+3 x^2\right )^{5/2} \, dx}{3240}\\ &=\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{293 \int \left (2-x+3 x^2\right )^{5/2} \, dx}{1620}\\ &=-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{6739 \int \left (2-x+3 x^2\right )^{3/2} \, dx}{23328}\\ &=-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{154997 \int \sqrt{2-x+3 x^2} \, dx}{373248}\\ &=-\frac{154997 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{3564931 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{8957952}\\ &=-\frac{154997 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}+\frac{\left (154997 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{8957952}\\ &=-\frac{154997 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}-\frac{6739 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{293 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{37}{405} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{15} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{(2731+3430 x) \left (2-x+3 x^2\right )^{7/2}}{17010}-\frac{3564931 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0550144, size = 85, normalized size = 0.52 $\frac{6 \sqrt{3 x^2-x+2} \left (2257403904 x^9+2675441664 x^8+4427716608 x^7+5671627776 x^6+4996802304 x^5+4171579776 x^4+3096104976 x^3+1693765752 x^2+692659234 x+387182961\right )+124772585 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{940584960}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(387182961 + 692659234*x + 1693765752*x^2 + 3096104976*x^3 + 4171579776*x^4 + 499680230
4*x^5 + 5671627776*x^6 + 4427716608*x^7 + 2675441664*x^8 + 2257403904*x^9) + 124772585*Sqrt[3]*ArcSinh[(-1 + 6
*x)/Sqrt[23]])/940584960

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Maple [A]  time = 0.056, size = 136, normalized size = 0.8 \begin{align*}{\frac{5419}{17010} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{x}^{3}}{15} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{472\,{x}^{2}}{405} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{235\,x}{243} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{-154997+929982\,x}{4478976}\sqrt{3\,{x}^{2}-x+2}}+{\frac{3564931\,\sqrt{3}}{26873856}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{-293+1758\,x}{58320} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-6739+40434\,x}{559872} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5419/17010*(3*x^2-x+2)^(7/2)+8/15*x^3*(3*x^2-x+2)^(7/2)+472/405*x^2*(3*x^2-x+2)^(7/2)+235/243*x*(3*x^2-x+2)^(7
/2)+154997/4478976*(-1+6*x)*(3*x^2-x+2)^(1/2)+3564931/26873856*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))+293/5832
0*(-1+6*x)*(3*x^2-x+2)^(5/2)+6739/559872*(-1+6*x)*(3*x^2-x+2)^(3/2)

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Maxima [A]  time = 1.48004, size = 225, normalized size = 1.37 \begin{align*} \frac{8}{15} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{472}{405} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{235}{243} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x + \frac{5419}{17010} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} + \frac{293}{9720} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x - \frac{293}{58320} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{6739}{93312} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{6739}{559872} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{154997}{746496} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{3564931}{26873856} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{154997}{4478976} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^2*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="maxima")

[Out]

8/15*(3*x^2 - x + 2)^(7/2)*x^3 + 472/405*(3*x^2 - x + 2)^(7/2)*x^2 + 235/243*(3*x^2 - x + 2)^(7/2)*x + 5419/17
010*(3*x^2 - x + 2)^(7/2) + 293/9720*(3*x^2 - x + 2)^(5/2)*x - 293/58320*(3*x^2 - x + 2)^(5/2) + 6739/93312*(3
*x^2 - x + 2)^(3/2)*x - 6739/559872*(3*x^2 - x + 2)^(3/2) + 154997/746496*sqrt(3*x^2 - x + 2)*x + 3564931/2687
3856*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 154997/4478976*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.32793, size = 390, normalized size = 2.38 \begin{align*} \frac{1}{156764160} \,{\left (2257403904 \, x^{9} + 2675441664 \, x^{8} + 4427716608 \, x^{7} + 5671627776 \, x^{6} + 4996802304 \, x^{5} + 4171579776 \, x^{4} + 3096104976 \, x^{3} + 1693765752 \, x^{2} + 692659234 \, x + 387182961\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{3564931}{53747712} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^2*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="fricas")

[Out]

1/156764160*(2257403904*x^9 + 2675441664*x^8 + 4427716608*x^7 + 5671627776*x^6 + 4996802304*x^5 + 4171579776*x
^4 + 3096104976*x^3 + 1693765752*x^2 + 692659234*x + 387182961)*sqrt(3*x^2 - x + 2) + 3564931/53747712*sqrt(3)
*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x + 1\right )^{2} \left (3 x^{2} - x + 2\right )^{\frac{5}{2}} \left (4 x^{2} + 3 x + 1\right )\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19182, size = 126, normalized size = 0.77 \begin{align*} \frac{1}{156764160} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (24 \,{\left (27 \, x + 32\right )} x + 1271\right )} x + 22793\right )} x + 722917\right )} x + 3621163\right )} x + 21500729\right )} x + 70573573\right )} x + 346329617\right )} x + 387182961\right )} \sqrt{3 \, x^{2} - x + 2} - \frac{3564931}{26873856} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^2*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="giac")

[Out]

1/156764160*(2*(12*(6*(8*(6*(36*(14*(24*(27*x + 32)*x + 1271)*x + 22793)*x + 722917)*x + 3621163)*x + 21500729
)*x + 70573573)*x + 346329617)*x + 387182961)*sqrt(3*x^2 - x + 2) - 3564931/26873856*sqrt(3)*log(-2*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)