### 3.214 $$\int (1+2 x)^3 (2-x+3 x^2)^{3/2} (1+3 x+4 x^2) \, dx$$

Optimal. Leaf size=158 $\frac{2}{27} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^4+\frac{77}{81} x^3 \left (3 x^2-x+2\right )^{5/2}+\frac{913}{486} x^2 \left (3 x^2-x+2\right )^{5/2}-\frac{11 (283-5850 x) \left (3 x^2-x+2\right )^{5/2}}{58320}+\frac{54593 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}+\frac{1255639 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}+\frac{28879697 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}$

[Out]

(1255639*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/4478976 + (54593*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/559872 - (11*(283 -
5850*x)*(2 - x + 3*x^2)^(5/2))/58320 + (913*x^2*(2 - x + 3*x^2)^(5/2))/486 + (77*x^3*(2 - x + 3*x^2)^(5/2))/81
+ (2*(1 + 2*x)^4*(2 - x + 3*x^2)^(5/2))/27 + (28879697*ArcSinh[(1 - 6*x)/Sqrt[23]])/(8957952*Sqrt[3])

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Rubi [A]  time = 0.200635, antiderivative size = 158, 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, 12, 779, 612, 619, 215} $\frac{2}{27} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^4+\frac{77}{81} x^3 \left (3 x^2-x+2\right )^{5/2}+\frac{913}{486} x^2 \left (3 x^2-x+2\right )^{5/2}-\frac{11 (283-5850 x) \left (3 x^2-x+2\right )^{5/2}}{58320}+\frac{54593 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}+\frac{1255639 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}+\frac{28879697 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1255639*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/4478976 + (54593*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/559872 - (11*(283 -
5850*x)*(2 - x + 3*x^2)^(5/2))/58320 + (913*x^2*(2 - x + 3*x^2)^(5/2))/486 + (77*x^3*(2 - x + 3*x^2)^(5/2))/81
+ (2*(1 + 2*x)^4*(2 - x + 3*x^2)^(5/2))/27 + (28879697*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 )^{3/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{108} \int 308 x (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{27} \int x (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{648} \int x \left (2-x+3 x^2\right )^{3/2} \left (24+96 x+332 x^2\right ) \, dx\\ &=\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{11 \int x (-824+3510 x) \left (2-x+3 x^2\right )^{3/2} \, dx}{1944}\\ &=-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{54593 \int \left (2-x+3 x^2\right )^{3/2} \, dx}{23328}\\ &=\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{1255639 \int \sqrt{2-x+3 x^2} \, dx}{373248}\\ &=\frac{1255639 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}+\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{28879697 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{8957952}\\ &=\frac{1255639 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}+\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{\left (1255639 \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{1255639 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}+\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{28879697 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0498844, size = 80, normalized size = 0.51 $\frac{6 \sqrt{3 x^2-x+2} \left (238878720 x^8+510105600 x^7+635765760 x^6+711210240 x^5+649452672 x^4+421626672 x^3+201289704 x^2+84014278 x+12499587\right )-144398485 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{134369280}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(12499587 + 84014278*x + 201289704*x^2 + 421626672*x^3 + 649452672*x^4 + 711210240*x^5
+ 635765760*x^6 + 510105600*x^7 + 238878720*x^8) - 144398485*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/134369280

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Maple [A]  time = 0.055, size = 134, normalized size = 0.9 \begin{align*}{\frac{32\,{x}^{4}}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{269\,{x}^{3}}{81} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1777\,{x}^{2}}{486} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1099\,x}{648} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{-1255639+7533834\,x}{4478976}\sqrt{3\,{x}^{2}-x+2}}-{\frac{28879697\,\sqrt{3}}{26873856}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{-54593+327558\,x}{559872} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1207}{58320} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/27*x^4*(3*x^2-x+2)^(5/2)+269/81*x^3*(3*x^2-x+2)^(5/2)+1777/486*x^2*(3*x^2-x+2)^(5/2)+1099/648*x*(3*x^2-x+2)
^(5/2)-1255639/4478976*(-1+6*x)*(3*x^2-x+2)^(1/2)-28879697/26873856*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-545
93/559872*(-1+6*x)*(3*x^2-x+2)^(3/2)+1207/58320*(3*x^2-x+2)^(5/2)

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Maxima [A]  time = 1.54024, size = 209, normalized size = 1.32 \begin{align*} \frac{32}{27} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{4} + \frac{269}{81} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{1777}{486} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{1099}{648} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{1207}{58320} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} - \frac{54593}{93312} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{54593}{559872} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{1255639}{746496} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{28879697}{26873856} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{1255639}{4478976} \, \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)^(3/2)*(4*x^2+3*x+1),x, algorithm="maxima")

[Out]

32/27*(3*x^2 - x + 2)^(5/2)*x^4 + 269/81*(3*x^2 - x + 2)^(5/2)*x^3 + 1777/486*(3*x^2 - x + 2)^(5/2)*x^2 + 1099
/648*(3*x^2 - x + 2)^(5/2)*x + 1207/58320*(3*x^2 - x + 2)^(5/2) - 54593/93312*(3*x^2 - x + 2)^(3/2)*x + 54593/
559872*(3*x^2 - x + 2)^(3/2) - 1255639/746496*sqrt(3*x^2 - x + 2)*x - 28879697/26873856*sqrt(3)*arcsinh(1/23*s
qrt(23)*(6*x - 1)) + 1255639/4478976*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.40745, size = 354, normalized size = 2.24 \begin{align*} \frac{1}{22394880} \,{\left (238878720 \, x^{8} + 510105600 \, x^{7} + 635765760 \, x^{6} + 711210240 \, x^{5} + 649452672 \, x^{4} + 421626672 \, x^{3} + 201289704 \, x^{2} + 84014278 \, x + 12499587\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{28879697}{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)^3*(3*x^2-x+2)^(3/2)*(4*x^2+3*x+1),x, algorithm="fricas")

[Out]

1/22394880*(238878720*x^8 + 510105600*x^7 + 635765760*x^6 + 711210240*x^5 + 649452672*x^4 + 421626672*x^3 + 20
1289704*x^2 + 84014278*x + 12499587)*sqrt(3*x^2 - x + 2) + 28879697/53747712*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 )^{3} \left (3 x^{2} - x + 2\right )^{\frac{3}{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)**(3/2)*(4*x**2+3*x+1),x)

[Out]

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

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Giac [A]  time = 1.17993, size = 119, normalized size = 0.75 \begin{align*} \frac{1}{22394880} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (36 \,{\left (2 \,{\left (96 \, x + 205\right )} x + 511\right )} x + 20579\right )} x + 563761\right )} x + 2927963\right )} x + 8387071\right )} x + 42007139\right )} x + 12499587\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{28879697}{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)^3*(3*x^2-x+2)^(3/2)*(4*x^2+3*x+1),x, algorithm="giac")

[Out]

1/22394880*(2*(12*(6*(8*(30*(36*(2*(96*x + 205)*x + 511)*x + 20579)*x + 563761)*x + 2927963)*x + 8387071)*x +
42007139)*x + 12499587)*sqrt(3*x^2 - x + 2) + 28879697/26873856*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
- x + 2)) + 1)