3.208 \(\int (1+2 x)^3 \sqrt{2-x+3 x^2} (1+3 x+4 x^2) \, dx\)

Optimal. Leaf size=143 \[ \frac{2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4+\frac{67}{378} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3+\frac{17}{105} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2-\frac{(26982 x+75295) \left (3 x^2-x+2\right )^{3/2}}{68040}+\frac{5393 (1-6 x) \sqrt{3 x^2-x+2}}{15552}+\frac{124039 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{31104 \sqrt{3}} \]

[Out]

(5393*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/15552 + (17*(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2))/105 + (67*(1 + 2*x)^3*(2 -
x + 3*x^2)^(3/2))/378 + (2*(1 + 2*x)^4*(2 - x + 3*x^2)^(3/2))/21 - ((75295 + 26982*x)*(2 - x + 3*x^2)^(3/2))/
68040 + (124039*ArcSinh[(1 - 6*x)/Sqrt[23]])/(31104*Sqrt[3])

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Rubi [A]  time = 0.138362, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2}{21} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^4+\frac{67}{378} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^3+\frac{17}{105} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2-\frac{(26982 x+75295) \left (3 x^2-x+2\right )^{3/2}}{68040}+\frac{5393 (1-6 x) \sqrt{3 x^2-x+2}}{15552}+\frac{124039 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{31104 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5393*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/15552 + (17*(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2))/105 + (67*(1 + 2*x)^3*(2 -
x + 3*x^2)^(3/2))/378 + (2*(1 + 2*x)^4*(2 - x + 3*x^2)^(3/2))/21 - ((75295 + 26982*x)*(2 - x + 3*x^2)^(3/2))/
68040 + (124039*ArcSinh[(1 - 6*x)/Sqrt[23]])/(31104*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 \sqrt{2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}+\frac{1}{84} \int (1+2 x)^3 (-32+268 x) \sqrt{2-x+3 x^2} \, dx\\ &=\frac{67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}+\frac{\int (1+2 x)^2 (-3390+3672 x) \sqrt{2-x+3 x^2} \, dx}{1512}\\ &=\frac{17}{105} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}+\frac{\int (-74718-53964 x) (1+2 x) \sqrt{2-x+3 x^2} \, dx}{22680}\\ &=\frac{17}{105} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}-\frac{(75295+26982 x) \left (2-x+3 x^2\right )^{3/2}}{68040}-\frac{5393 \int \sqrt{2-x+3 x^2} \, dx}{1296}\\ &=\frac{5393 (1-6 x) \sqrt{2-x+3 x^2}}{15552}+\frac{17}{105} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}-\frac{(75295+26982 x) \left (2-x+3 x^2\right )^{3/2}}{68040}-\frac{124039 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{31104}\\ &=\frac{5393 (1-6 x) \sqrt{2-x+3 x^2}}{15552}+\frac{17}{105} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}-\frac{(75295+26982 x) \left (2-x+3 x^2\right )^{3/2}}{68040}-\frac{\left (5393 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{31104}\\ &=\frac{5393 (1-6 x) \sqrt{2-x+3 x^2}}{15552}+\frac{17}{105} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{67}{378} (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}+\frac{2}{21} (1+2 x)^4 \left (2-x+3 x^2\right )^{3/2}-\frac{(75295+26982 x) \left (2-x+3 x^2\right )^{3/2}}{68040}+\frac{124039 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{31104 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0474035, size = 70, normalized size = 0.49 \[ \frac{6 \sqrt{3 x^2-x+2} \left (2488320 x^6+6462720 x^5+7491456 x^4+5497776 x^3+3280872 x^2+1493894 x-543069\right )-4341365 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{3265920} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(-543069 + 1493894*x + 3280872*x^2 + 5497776*x^3 + 7491456*x^4 + 6462720*x^5 + 2488320*
x^6) - 4341365*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/3265920

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Maple [A]  time = 0.057, size = 115, normalized size = 0.8 \begin{align*}{\frac{32\,{x}^{4}}{21} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{844\,{x}^{3}}{189} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1594\,{x}^{2}}{315} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{7849\,x}{3780} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{-5393+32358\,x}{15552}\sqrt{3\,{x}^{2}-x+2}}-{\frac{124039\,\sqrt{3}}{93312}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{45739}{68040} \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*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x)

[Out]

32/21*x^4*(3*x^2-x+2)^(3/2)+844/189*x^3*(3*x^2-x+2)^(3/2)+1594/315*x^2*(3*x^2-x+2)^(3/2)+7849/3780*x*(3*x^2-x+
2)^(3/2)-5393/15552*(-1+6*x)*(3*x^2-x+2)^(1/2)-124039/93312*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-45739/68040
*(3*x^2-x+2)^(3/2)

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Maxima [A]  time = 1.52799, size = 170, normalized size = 1.19 \begin{align*} \frac{32}{21} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x^{4} + \frac{844}{189} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x^{3} + \frac{1594}{315} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{7849}{3780} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{45739}{68040} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{5393}{2592} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{124039}{93312} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{5393}{15552} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/21*(3*x^2 - x + 2)^(3/2)*x^4 + 844/189*(3*x^2 - x + 2)^(3/2)*x^3 + 1594/315*(3*x^2 - x + 2)^(3/2)*x^2 + 784
9/3780*(3*x^2 - x + 2)^(3/2)*x - 45739/68040*(3*x^2 - x + 2)^(3/2) - 5393/2592*sqrt(3*x^2 - x + 2)*x - 124039/
93312*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) + 5393/15552*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.90773, size = 285, normalized size = 1.99 \begin{align*} \frac{1}{544320} \,{\left (2488320 \, x^{6} + 6462720 \, x^{5} + 7491456 \, x^{4} + 5497776 \, x^{3} + 3280872 \, x^{2} + 1493894 \, x - 543069\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{124039}{186624} \, \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*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x, algorithm="fricas")

[Out]

1/544320*(2488320*x^6 + 6462720*x^5 + 7491456*x^4 + 5497776*x^3 + 3280872*x^2 + 1493894*x - 543069)*sqrt(3*x^2
- x + 2) + 124039/186624*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} \sqrt{3 x^{2} - x + 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*(4*x**2+3*x+1)*(3*x**2-x+2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.34908, size = 105, normalized size = 0.73 \begin{align*} \frac{1}{544320} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (72 \, x + 187\right )} x + 6503\right )} x + 38179\right )} x + 136703\right )} x + 746947\right )} x - 543069\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{124039}{93312} \, \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*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x, algorithm="giac")

[Out]

1/544320*(2*(12*(6*(8*(30*(72*x + 187)*x + 6503)*x + 38179)*x + 136703)*x + 746947)*x - 543069)*sqrt(3*x^2 - x
+ 2) + 124039/93312*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)