### 3.334 $$\int (5+2 x) (3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4) \, dx$$

Optimal. Leaf size=166 $\frac{5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4-\frac{1121 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^3}{2304}+\frac{69415 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^2}{32256}-\frac{3 (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}}{143360}-\frac{92727 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{131072}-\frac{6398163 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{147157749 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}}$

[Out]

(-6398163*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (92727*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/131072 + (69415*(5
+ 2*x)^2*(3 - x + 2*x^2)^(5/2))/32256 - (1121*(5 + 2*x)^3*(3 - x + 2*x^2)^(5/2))/2304 + (5*(5 + 2*x)^4*(3 - x
+ 2*x^2)^(5/2))/144 - (3*(661397 + 215900*x)*(3 - x + 2*x^2)^(5/2))/143360 - (147157749*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(4194304*Sqrt[2])

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Rubi [A]  time = 0.194405, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 38, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.132, Rules used = {1653, 779, 612, 619, 215} $\frac{5}{144} \left (2 x^2-x+3\right )^{5/2} (2 x+5)^4-\frac{1121 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^3}{2304}+\frac{69415 \left (2 x^2-x+3\right )^{5/2} (2 x+5)^2}{32256}-\frac{3 (215900 x+661397) \left (2 x^2-x+3\right )^{5/2}}{143360}-\frac{92727 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{131072}-\frac{6398163 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{147157749 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-6398163*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (92727*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/131072 + (69415*(5
+ 2*x)^2*(3 - x + 2*x^2)^(5/2))/32256 - (1121*(5 + 2*x)^3*(3 - x + 2*x^2)^(5/2))/2304 + (5*(5 + 2*x)^4*(3 - x
+ 2*x^2)^(5/2))/144 - (3*(661397 + 215900*x)*(3 - x + 2*x^2)^(5/2))/143360 - (147157749*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(4194304*Sqrt[2])

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 (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{288} \int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (-2299-11262 x-15996 x^2-8968 x^3\right ) \, dx\\ &=-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}+\frac{\int (5+2 x) \left (3-x+2 x^2\right )^{3/2} \left (198968+2253280 x+2221280 x^2\right ) \, dx}{36864}\\ &=\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}+\frac{\int (13363488-55961280 x) (5+2 x) \left (3-x+2 x^2\right )^{3/2} \, dx}{2064384}\\ &=\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{92727 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{8192}\\ &=-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{6398163 \int \sqrt{3-x+2 x^2} \, dx}{262144}\\ &=-\frac{6398163 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{147157749 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4194304}\\ &=-\frac{6398163 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}+\frac{\left (6398163 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4194304}\\ &=-\frac{6398163 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{92727 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{131072}+\frac{69415 (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}}{32256}-\frac{1121 (5+2 x)^3 \left (3-x+2 x^2\right )^{5/2}}{2304}+\frac{5}{144} (5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}-\frac{3 (661397+215900 x) \left (3-x+2 x^2\right )^{5/2}}{143360}-\frac{147157749 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.190871, size = 80, normalized size = 0.48 $\frac{4 \sqrt{2 x^2-x+3} \left (1468006400 x^8+2926837760 x^7+1033175040 x^6+12117893120 x^5+379086848 x^4+12669290112 x^3+4870637856 x^2+12357760788 x+1592737263\right )-46354690935 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2642411520}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(1592737263 + 12357760788*x + 4870637856*x^2 + 12669290112*x^3 + 379086848*x^4 + 121178
93120*x^5 + 1033175040*x^6 + 2926837760*x^7 + 1468006400*x^8) - 46354690935*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]
])/2642411520

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Maple [A]  time = 0.059, size = 134, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{4}}{9} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{479\,{x}^{3}}{288} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{2005\,{x}^{2}}{8064} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{5645\,x}{21504} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-6398163+25592652\,x}{2097152}\sqrt{2\,{x}^{2}-x+3}}+{\frac{147157749\,\sqrt{2}}{8388608}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-92727+370908\,x}{131072} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{120809}{143360} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/9*x^4*(2*x^2-x+3)^(5/2)+479/288*x^3*(2*x^2-x+3)^(5/2)+2005/8064*x^2*(2*x^2-x+3)^(5/2)+5645/21504*x*(2*x^2-x+
3)^(5/2)+6398163/2097152*(-1+4*x)*(2*x^2-x+3)^(1/2)+147157749/8388608*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+9
2727/131072*(-1+4*x)*(2*x^2-x+3)^(3/2)+120809/143360*(2*x^2-x+3)^(5/2)

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Maxima [A]  time = 1.58366, size = 209, normalized size = 1.26 \begin{align*} \frac{5}{9} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{4} + \frac{479}{288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{2005}{8064} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{5645}{21504} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{120809}{143360} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{92727}{32768} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{92727}{131072} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{6398163}{524288} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{147157749}{8388608} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{6398163}{2097152} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/9*(2*x^2 - x + 3)^(5/2)*x^4 + 479/288*(2*x^2 - x + 3)^(5/2)*x^3 + 2005/8064*(2*x^2 - x + 3)^(5/2)*x^2 + 5645
/21504*(2*x^2 - x + 3)^(5/2)*x + 120809/143360*(2*x^2 - x + 3)^(5/2) + 92727/32768*(2*x^2 - x + 3)^(3/2)*x - 9
2727/131072*(2*x^2 - x + 3)^(3/2) + 6398163/524288*sqrt(2*x^2 - x + 3)*x + 147157749/8388608*sqrt(2)*arcsinh(1
/23*sqrt(23)*(4*x - 1)) - 6398163/2097152*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.32289, size = 375, normalized size = 2.26 \begin{align*} \frac{1}{660602880} \,{\left (1468006400 \, x^{8} + 2926837760 \, x^{7} + 1033175040 \, x^{6} + 12117893120 \, x^{5} + 379086848 \, x^{4} + 12669290112 \, x^{3} + 4870637856 \, x^{2} + 12357760788 \, x + 1592737263\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{147157749}{16777216} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/660602880*(1468006400*x^8 + 2926837760*x^7 + 1033175040*x^6 + 12117893120*x^5 + 379086848*x^4 + 12669290112*
x^3 + 4870637856*x^2 + 12357760788*x + 1592737263)*sqrt(2*x^2 - x + 3) + 147157749/16777216*sqrt(2)*log(-4*sqr
t(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x + 5\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22369, size = 119, normalized size = 0.72 \begin{align*} \frac{1}{660602880} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (8 \,{\left (28 \,{\left (160 \, x + 319\right )} x + 3153\right )} x + 295847\right )} x + 185101\right )} x + 98978829\right )} x + 152207433\right )} x + 3089440197\right )} x + 1592737263\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{147157749}{8388608} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/660602880*(4*(8*(4*(16*(20*(8*(28*(160*x + 319)*x + 3153)*x + 295847)*x + 185101)*x + 98978829)*x + 15220743
3)*x + 3089440197)*x + 1592737263)*sqrt(2*x^2 - x + 3) - 147157749/8388608*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x -
sqrt(2*x^2 - x + 3)) + 1)