### 3.216 $$\int (1+2 x) (2-x+3 x^2)^{3/2} (1+3 x+4 x^2) \, dx$$

Optimal. Leaf size=116 $\frac{2}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}+\frac{1}{378} (102 x+109) \left (3 x^2-x+2\right )^{5/2}-\frac{71 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{2592}-\frac{1633 (1-6 x) \sqrt{3 x^2-x+2}}{20736}-\frac{37559 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{41472 \sqrt{3}}$

[Out]

(-1633*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/20736 - (71*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/2592 + (2*(1 + 2*x)^2*(2 -
x + 3*x^2)^(5/2))/21 + ((109 + 102*x)*(2 - x + 3*x^2)^(5/2))/378 - (37559*ArcSinh[(1 - 6*x)/Sqrt[23]])/(41472*
Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0822707, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, 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} $\frac{2}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}+\frac{1}{378} (102 x+109) \left (3 x^2-x+2\right )^{5/2}-\frac{71 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{2592}-\frac{1633 (1-6 x) \sqrt{3 x^2-x+2}}{20736}-\frac{37559 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{41472 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-1633*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/20736 - (71*(1 - 6*x)*(2 - x + 3*x^2)^(3/2))/2592 + (2*(1 + 2*x)^2*(2 -
x + 3*x^2)^(5/2))/21 + ((109 + 102*x)*(2 - x + 3*x^2)^(5/2))/378 - (37559*ArcSinh[(1 - 6*x)/Sqrt[23]])/(41472*
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 )^{3/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{84} \int (1+2 x) (40+204 x) \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{71}{108} \int \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{1633 \int \sqrt{2-x+3 x^2} \, dx}{1728}\\ &=-\frac{1633 (1-6 x) \sqrt{2-x+3 x^2}}{20736}-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{37559 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{41472}\\ &=-\frac{1633 (1-6 x) \sqrt{2-x+3 x^2}}{20736}-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{\left (1633 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{41472}\\ &=-\frac{1633 (1-6 x) \sqrt{2-x+3 x^2}}{20736}-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}-\frac{37559 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{41472 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.035977, size = 70, normalized size = 0.6 $\frac{6 \sqrt{3 x^2-x+2} \left (497664 x^6+518400 x^5+653184 x^4+744336 x^3+531384 x^2+275410 x+203337\right )+262913 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{870912}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*Sqrt[2 - x + 3*x^2]*(203337 + 275410*x + 531384*x^2 + 744336*x^3 + 653184*x^4 + 518400*x^5 + 497664*x^6) +
262913*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/870912

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 100, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{2}}{21} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{41\,x}{63} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{145}{378} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-71+426\,x}{2592} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-1633+9798\,x}{20736}\sqrt{3\,{x}^{2}-x+2}}+{\frac{37559\,\sqrt{3}}{124416}{\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)^(3/2)*(4*x^2+3*x+1),x)

[Out]

8/21*x^2*(3*x^2-x+2)^(5/2)+41/63*x*(3*x^2-x+2)^(5/2)+145/378*(3*x^2-x+2)^(5/2)+71/2592*(-1+6*x)*(3*x^2-x+2)^(3
/2)+1633/20736*(-1+6*x)*(3*x^2-x+2)^(1/2)+37559/124416*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))

________________________________________________________________________________________

Maxima [A]  time = 1.50264, size = 163, normalized size = 1.41 \begin{align*} \frac{8}{21} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{41}{63} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{145}{378} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{71}{432} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{71}{2592} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{1633}{3456} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{37559}{124416} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{1633}{20736} \, \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)^(3/2)*(4*x^2+3*x+1),x, algorithm="maxima")

[Out]

8/21*(3*x^2 - x + 2)^(5/2)*x^2 + 41/63*(3*x^2 - x + 2)^(5/2)*x + 145/378*(3*x^2 - x + 2)^(5/2) + 71/432*(3*x^2
- x + 2)^(3/2)*x - 71/2592*(3*x^2 - x + 2)^(3/2) + 1633/3456*sqrt(3*x^2 - x + 2)*x + 37559/124416*sqrt(3)*arc
sinh(1/23*sqrt(23)*(6*x - 1)) - 1633/20736*sqrt(3*x^2 - x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.56758, size = 277, normalized size = 2.39 \begin{align*} \frac{1}{145152} \,{\left (497664 \, x^{6} + 518400 \, x^{5} + 653184 \, x^{4} + 744336 \, x^{3} + 531384 \, x^{2} + 275410 \, x + 203337\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{37559}{248832} \, \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)^(3/2)*(4*x^2+3*x+1),x, algorithm="fricas")

[Out]

1/145152*(497664*x^6 + 518400*x^5 + 653184*x^4 + 744336*x^3 + 531384*x^2 + 275410*x + 203337)*sqrt(3*x^2 - x +
2) + 37559/248832*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17022, size = 105, normalized size = 0.91 \begin{align*} \frac{1}{145152} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (24 \,{\left (2 \,{\left (24 \, x + 25\right )} x + 63\right )} x + 1723\right )} x + 22141\right )} x + 137705\right )} x + 203337\right )} \sqrt{3 \, x^{2} - x + 2} - \frac{37559}{124416} \, \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)^(3/2)*(4*x^2+3*x+1),x, algorithm="giac")

[Out]

1/145152*(2*(12*(18*(24*(2*(24*x + 25)*x + 63)*x + 1723)*x + 22141)*x + 137705)*x + 203337)*sqrt(3*x^2 - x + 2
) - 37559/124416*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)