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

Optimal. Leaf size=139 \[ \frac{2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}+\frac{1}{648} (122 x+137) \left (3 x^2-x+2\right )^{7/2}-\frac{445 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{15552}-\frac{51175 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{746496}-\frac{1177025 (1-6 x) \sqrt{3 x^2-x+2}}{5971968}-\frac{27071575 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{11943936 \sqrt{3}} \]

[Out]

(-1177025*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/5971968 - (51175*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/746496 - (445*(1 -
6*x)*(2 - x + 3*x^2)^(5/2))/15552 + (2*(1 + 2*x)^2*(2 - x + 3*x^2)^(7/2))/27 + ((137 + 122*x)*(2 - x + 3*x^2)^
(7/2))/648 - (27071575*ArcSinh[(1 - 6*x)/Sqrt[23]])/(11943936*Sqrt[3])

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Rubi [A]  time = 0.0917934, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1653, 779, 612, 619, 215} \[ \frac{2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}+\frac{1}{648} (122 x+137) \left (3 x^2-x+2\right )^{7/2}-\frac{445 (1-6 x) \left (3 x^2-x+2\right )^{5/2}}{15552}-\frac{51175 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{746496}-\frac{1177025 (1-6 x) \sqrt{3 x^2-x+2}}{5971968}-\frac{27071575 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{11943936 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1177025*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/5971968 - (51175*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/746496 - (445*(1 -
6*x)*(2 - x + 3*x^2)^(5/2))/15552 + (2*(1 + 2*x)^2*(2 - x + 3*x^2)^(7/2))/27 + ((137 + 122*x)*(2 - x + 3*x^2)^
(7/2))/648 - (27071575*ArcSinh[(1 - 6*x)/Sqrt[23]])/(11943936*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 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) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{108} \int (1+2 x) (72+244 x) \left (2-x+3 x^2\right )^{5/2} \, dx\\ &=\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}+\frac{445}{432} \int \left (2-x+3 x^2\right )^{5/2} \, dx\\ &=-\frac{445 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{15552}+\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}+\frac{51175 \int \left (2-x+3 x^2\right )^{3/2} \, dx}{31104}\\ &=-\frac{51175 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{746496}-\frac{445 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{15552}+\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}+\frac{1177025 \int \sqrt{2-x+3 x^2} \, dx}{497664}\\ &=-\frac{1177025 (1-6 x) \sqrt{2-x+3 x^2}}{5971968}-\frac{51175 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{746496}-\frac{445 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{15552}+\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}+\frac{27071575 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{11943936}\\ &=-\frac{1177025 (1-6 x) \sqrt{2-x+3 x^2}}{5971968}-\frac{51175 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{746496}-\frac{445 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{15552}+\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}+\frac{\left (1177025 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{11943936}\\ &=-\frac{1177025 (1-6 x) \sqrt{2-x+3 x^2}}{5971968}-\frac{51175 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{746496}-\frac{445 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{15552}+\frac{2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac{1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}-\frac{27071575 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{11943936 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0447259, size = 80, normalized size = 0.58 \[ \frac{6 \sqrt{3 x^2-x+2} \left (47775744 x^8+30357504 x^7+79377408 x^6+80034048 x^5+66969216 x^4+58946544 x^3+41031048 x^2+19860062 x+10960335\right )[phone] \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{35831808} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(10960335 + 19860062*x + 41031048*x^2 + 58946544*x^3 + 66969216*x^4 + 80034048*x^5 + 79
377408*x^6 + 30357504*x^7 + 47775744*x^8) + 27071575*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/35831808

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Maple [A]  time = 0.055, size = 119, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{2}}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{157\,x}{324} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{185}{648} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{-445+2670\,x}{15552} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-51175+307050\,x}{746496} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-1177025+7062150\,x}{5971968}\sqrt{3\,{x}^{2}-x+2}}+{\frac{27071575\,\sqrt{3}}{35831808}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/27*x^2*(3*x^2-x+2)^(7/2)+157/324*x*(3*x^2-x+2)^(7/2)+185/648*(3*x^2-x+2)^(7/2)+445/15552*(-1+6*x)*(3*x^2-x+2
)^(5/2)+51175/746496*(-1+6*x)*(3*x^2-x+2)^(3/2)+1177025/5971968*(-1+6*x)*(3*x^2-x+2)^(1/2)[phone]/35831808*3
^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))

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Maxima [A]  time = 1.56519, size = 203, normalized size = 1.46 \begin{align*} \frac{8}{27} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{157}{324} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} x + \frac{185}{648} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}} + \frac{445}{2592} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x - \frac{445}{15552} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{51175}{124416} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{51175}{746496} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{1177025}{995328} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{27071575}{35831808} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{1177025}{5971968} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/27*(3*x^2 - x + 2)^(7/2)*x^2 + 157/324*(3*x^2 - x + 2)^(7/2)*x + 185/648*(3*x^2 - x + 2)^(7/2) + 445/2592*(3
*x^2 - x + 2)^(5/2)*x - 445/15552*(3*x^2 - x + 2)^(5/2) + 51175/124416*(3*x^2 - x + 2)^(3/2)*x - 51175/746496*
(3*x^2 - x + 2)^(3/2) + 1177025/995328*sqrt(3*x^2 - x + 2)*x + 27071575/35831808*sqrt(3)*arcsinh(1/23*sqrt(23)
*(6*x - 1)) - 1177025/5971968*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.35887, size = 344, normalized size = 2.47 \begin{align*} \frac{1}{5971968} \,{\left (47775744 \, x^{8} + 30357504 \, x^{7} + 79377408 \, x^{6} + 80034048 \, x^{5} + 66969216 \, x^{4} + 58946544 \, x^{3} + 41031048 \, x^{2} + 19860062 \, x + 10960335\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{27071575}{71663616} \, \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*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="fricas")

[Out]

1/5971968*(47775744*x^8 + 30357504*x^7 + 79377408*x^6 + 80034048*x^5 + 66969216*x^4 + 58946544*x^3 + 41031048*
x^2 + 19860062*x + 10960335)*sqrt(3*x^2 - x + 2) + 27071575/71663616*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 ) \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*x**2-x+2)**(5/2)*(4*x**2+3*x+1),x)

[Out]

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

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Giac [A]  time = 1.2123, size = 119, normalized size = 0.86 \begin{align*} \frac{1}{5971968} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (96 \, x + 61\right )} x + 319\right )} x + 11579\right )} x + 58133\right )} x + 409351\right )} x + 1709627\right )} x + 9930031\right )} x + 10960335\right )} \sqrt{3 \, x^{2} - x + 2} - \frac{27071575}{35831808} \, \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*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="giac")

[Out]

1/5971968*(2*(12*(6*(8*(6*(36*(2*(96*x + 61)*x + 319)*x + 11579)*x + 58133)*x + 409351)*x + 1709627)*x + 99300
31)*x + 10960335)*sqrt(3*x^2 - x + 2) - 27071575/35831808*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x +
2)) + 1)