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]

-6739/559872*(1-6*x)*(3*x^2-x+2)^(3/2)-293/58320*(1-6*x)*(3*x^2-x+2)^(5/2)+37/405*(1+2*x)^2*(3*x^2-x+2)^(7/2)+
1/15*(1+2*x)^3*(3*x^2-x+2)^(7/2)+1/17010*(2731+3430*x)*(3*x^2-x+2)^(7/2)-3564931/26873856*arcsinh(1/23*(1-6*x)
*23^(1/2))*3^(1/2)-154997/4478976*(1-6*x)*(3*x^2-x+2)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 164, normalized size of antiderivative = 1.00, 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 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 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 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 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 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]))

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*}

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Mathematica [A]  time = 0.06, size = 85, normalized size = 0.52 \[ \frac {6 \sqrt {3 x^2-x+2} \left (2257403904 x^9+2675441664 x^8+4427716608 x^7+5671627776 x^6+4996802304 x^5+4171579776 x^4+3096104976 x^3+1693765752 x^2+692659234 x+387182961\right )[phone] \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{940584960} \]

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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fricas [A]  time = 0.95, size = 98, normalized size = 0.60 \[ \frac {1}{156764160} \, {\left (2257403904 \, x^{9} + 2675441664 \, x^{8} + 4427716608 \, x^{7} + 5671627776 \, x^{6} + 4996802304 \, x^{5} + 4171579776 \, x^{4} + 3096104976 \, x^{3} + 1693765752 \, x^{2} + 692659234 \, x + 387182961\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {3564931}{53747712} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]

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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giac [A]  time = 0.21, size = 93, normalized size = 0.57 \[ \frac {1}{156764160} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (24 \, {\left (27 \, x + 32\right )} x + 1271\right )} x + 22793\right )} x + 722917\right )} x + 3621163\right )} x + 21500729\right )} x + 70573573\right )} x + 346329617\right )} x + 387182961\right )} \sqrt {3 \, x^{2} - x + 2} - \frac {3564931}{26873856} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]

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)

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maple [A]  time = 0.01, size = 136, normalized size = 0.83 \[ \frac {8 \left (3 x^{2}-x +2\right )^{\frac {7}{2}} x^{3}}{15}+\frac {472 \left (3 x^{2}-x +2\right )^{\frac {7}{2}} x^{2}}{405}+\frac {235 \left (3 x^{2}-x +2\right )^{\frac {7}{2}} x}{243}+\frac {3564931 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{26873856}+\frac {293 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{58320}+\frac {6739 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{559872}+\frac {154997 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{4478976}+\frac {5419 \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{17010} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+293/58320*(6*x-1)*(3*x^2-
x+2)^(5/2)+6739/559872*(6*x-1)*(3*x^2-x+2)^(3/2)+3564931/26873856*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))+15499
7/4478976*(6*x-1)*(3*x^2-x+2)^(1/2)+5419/17010*(3*x^2-x+2)^(7/2)

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maxima [A]  time = 0.97, size = 167, normalized size = 1.02 \[ \frac {8}{15} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {472}{405} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {235}{243} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x + \frac {5419}{17010} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} + \frac {293}{9720} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x - \frac {293}{58320} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {6739}{93312} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {6739}{559872} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {154997}{746496} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {3564931}{26873856} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {154997}{4478976} \, \sqrt {3 \, x^{2} - x + 2} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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