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

Optimal. Leaf size=189 \[ \frac{2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4+\frac{29}{330} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^3+\frac{133 \left (3 x^2-x+2\right )^{7/2} (2 x+1)^2}{1485}-\frac{(26353-21350 x) \left (3 x^2-x+2\right )^{7/2}}{498960}+\frac{5089 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{155520}+\frac{117047 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{1492992}+\frac{2692081 (1-6 x) \sqrt{3 x^2-x+2}}{11943936}+\frac{61917863 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{23887872 \sqrt{3}} \]

[Out]

(2692081*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/11943936 + (117047*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/1492992 + (5089*(1
- 6*x)*(2 - x + 3*x^2)^(5/2))/155520 - ((26353 - 21350*x)*(2 - x + 3*x^2)^(7/2))/498960 + (133*(1 + 2*x)^2*(2
- x + 3*x^2)^(7/2))/1485 + (29*(1 + 2*x)^3*(2 - x + 3*x^2)^(7/2))/330 + (2*(1 + 2*x)^4*(2 - x + 3*x^2)^(7/2))
/33 + (61917863*ArcSinh[(1 - 6*x)/Sqrt[23]])/(23887872*Sqrt[3])

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Rubi [A]  time = 0.155952, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2}{33} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^4+\frac{29}{330} \left (3 x^2-x+2\right )^{7/2} (2 x+1)^3+\frac{133 \left (3 x^2-x+2\right )^{7/2} (2 x+1)^2}{1485}-\frac{(26353-21350 x) \left (3 x^2-x+2\right )^{7/2}}{498960}+\frac{5089 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{155520}+\frac{117047 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{1492992}+\frac{2692081 (1-6 x) \sqrt{3 x^2-x+2}}{11943936}+\frac{61917863 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{23887872 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2692081*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/11943936 + (117047*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/1492992 + (5089*(1
- 6*x)*(2 - x + 3*x^2)^(5/2))/155520 - ((26353 - 21350*x)*(2 - x + 3*x^2)^(7/2))/498960 + (133*(1 + 2*x)^2*(2
- x + 3*x^2)^(7/2))/1485 + (29*(1 + 2*x)^3*(2 - x + 3*x^2)^(7/2))/330 + (2*(1 + 2*x)^4*(2 - x + 3*x^2)^(7/2))
/33 + (61917863*ArcSinh[(1 - 6*x)/Sqrt[23]])/(23887872*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)^3 \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{132} \int (1+2 x)^3 (32+348 x) \left (2-x+3 x^2\right )^{5/2} \, dx\\ &=\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}+\frac{\int (1+2 x)^2 (-1998+9576 x) \left (2-x+3 x^2\right )^{5/2} \, dx}{3960}\\ &=\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}+\frac{\int (1+2 x) (-97038+54900 x) \left (2-x+3 x^2\right )^{5/2} \, dx}{106920}\\ &=-\frac{(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}-\frac{5089 \int \left (2-x+3 x^2\right )^{5/2} \, dx}{4320}\\ &=\frac{5089 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{155520}-\frac{(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}-\frac{117047 \int \left (2-x+3 x^2\right )^{3/2} \, dx}{62208}\\ &=\frac{117047 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{1492992}+\frac{5089 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{155520}-\frac{(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}-\frac{2692081 \int \sqrt{2-x+3 x^2} \, dx}{995328}\\ &=\frac{2692081 (1-6 x) \sqrt{2-x+3 x^2}}{11943936}+\frac{117047 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{1492992}+\frac{5089 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{155520}-\frac{(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}-\frac{61917863 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{23887872}\\ &=\frac{2692081 (1-6 x) \sqrt{2-x+3 x^2}}{11943936}+\frac{117047 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{1492992}+\frac{5089 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{155520}-\frac{(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}-\frac{\left (2692081 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{23887872}\\ &=\frac{2692081 (1-6 x) \sqrt{2-x+3 x^2}}{11943936}+\frac{117047 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{1492992}+\frac{5089 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{155520}-\frac{(26353-21350 x) \left (2-x+3 x^2\right )^{7/2}}{498960}+\frac{133 (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}}{1485}+\frac{29}{330} (1+2 x)^3 \left (2-x+3 x^2\right )^{7/2}+\frac{2}{33} (1+2 x)^4 \left (2-x+3 x^2\right )^{7/2}+\frac{61917863 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{23887872 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0599622, size = 90, normalized size = 0.48 \[ \frac{6 \sqrt{3 x^2-x+2} \left (120394874880 x^{10}[phone] x^9+308846297088 x^8+419978151936 x^7+415908006912 x^6+347247744768 x^5+263636134272 x^4+161269204752 x^3+72088585464 x^2+26646633218 x+9173509857\right )-23838377255 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{27590492160} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(9173509857 + 26646633218*x + 72088585464*x^2 + 161269204752*x^3 + 263636134272*x^4 + 3
47247744768*x^5 + 415908006912*x^6 + 419978151936*x^7 + 308846297088*x^8 + 207681159168*x^9 + 120394874880*x^1
0) - 23838377255*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/27590492160

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Maple [A]  time = 0.059, size = 153, normalized size = 0.8 \begin{align*}{\frac{92423}{498960} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{32\,{x}^{4}}{33} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{436\,{x}^{3}}{165} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{4258\,{x}^{2}}{1485} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{10073\,x}{7128} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{-2692081+16152486\,x}{11943936}\sqrt{3\,{x}^{2}-x+2}}-{\frac{61917863\,\sqrt{3}}{71663616}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{-5089+30534\,x}{155520} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{-117047+702282\,x}{1492992} \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)^3*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x)

[Out]

92423/498960*(3*x^2-x+2)^(7/2)+32/33*x^4*(3*x^2-x+2)^(7/2)+436/165*x^3*(3*x^2-x+2)^(7/2)+4258/1485*x^2*(3*x^2-
x+2)^(7/2)+10073/7128*x*(3*x^2-x+2)^(7/2)-2692081/11943936*(-1+6*x)*(3*x^2-x+2)^(1/2)-61917863/71663616*3^(1/2
)*arcsinh(6/23*23^(1/2)*(x-1/6))-5089/155520*(-1+6*x)*(3*x^2-x+2)^(5/2)-117047/1492992*(-1+6*x)*(3*x^2-x+2)^(3
/2)

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Maxima [A]  time = 1.49075, size = 248, normalized size = 1.31 \begin{align*} \frac{32}{33} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{4} + \frac{436}{165} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{4258}{1485} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{10073}{7128} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x + \frac{92423}{498960} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} - \frac{5089}{25920} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{5089}{155520} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} - \frac{117047}{248832} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{117047}{1492992} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{2692081}{1990656} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{61917863}{71663616} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{2692081}{11943936} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/33*(3*x^2 - x + 2)^(7/2)*x^4 + 436/165*(3*x^2 - x + 2)^(7/2)*x^3 + 4258/1485*(3*x^2 - x + 2)^(7/2)*x^2 + 10
073/7128*(3*x^2 - x + 2)^(7/2)*x + 92423/498960*(3*x^2 - x + 2)^(7/2) - 5089/25920*(3*x^2 - x + 2)^(5/2)*x + 5
089/155520*(3*x^2 - x + 2)^(5/2) - 117047/248832*(3*x^2 - x + 2)^(3/2)*x + 117047/1492992*(3*x^2 - x + 2)^(3/2
) - 2692081/1990656*sqrt(3*x^2 - x + 2)*x - 61917863/71663616*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) + 26920
81/11943936*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.45142, size = 444, normalized size = 2.35 \begin{align*} \frac{1}{4598415360} \,{\left (120394874880 \, x^{10} + 207681159168 \, x^{9} + 308846297088 \, x^{8} + 419978151936 \, x^{7} + 415908006912 \, x^{6} + 347247744768 \, x^{5} + 263636134272 \, x^{4} + 161269204752 \, x^{3} + 72088585464 \, x^{2} + 26646633218 \, x + 9173509857\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{61917863}{143327232} \, \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)^3*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="fricas")

[Out]

1/4598415360*(120394874880*x^10 + 207681159168*x^9 + 308846297088*x^8 + 419978151936*x^7 + 415908006912*x^6 +
347247744768*x^5 + 263636134272*x^4 + 161269204752*x^3 + 72088585464*x^2 + 26646633218*x + 9173509857)*sqrt(3*
x^2 - x + 2) + 61917863/143327232*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 )^{3} \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)**3*(3*x**2-x+2)**(5/2)*(4*x**2+3*x+1),x)

[Out]

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

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Giac [A]  time = 1.27783, size = 132, normalized size = 0.7 \begin{align*} \frac{1}{4598415360} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \,{\left (18 \,{\left (40 \, x + 69\right )} x + 1847\right )} x + 120557\right )} x + 1671441\right )} x + 50238389\right )} x + 228850811\right )} x + 1119925033\right )} x + 3003691061\right )} x + 13323316609\right )} x + 9173509857\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{61917863}{71663616} \, \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)^3*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="giac")

[Out]

1/4598415360*(2*(12*(6*(8*(6*(36*(14*(48*(18*(40*x + 69)*x + 1847)*x + 120557)*x + 1671441)*x + 50238389)*x +
228850811)*x + 1119925033)*x + 3003691061)*x + 13323316609)*x + 9173509857)*sqrt(3*x^2 - x + 2) + 61917863/716
63616*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)