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]

-92727/131072*(1-4*x)*(2*x^2-x+3)^(3/2)+69415/32256*(5+2*x)^2*(2*x^2-x+3)^(5/2)-1121/2304*(5+2*x)^3*(2*x^2-x+3
)^(5/2)+5/144*(5+2*x)^4*(2*x^2-x+3)^(5/2)-3/143360*(661397+215900*x)*(2*x^2-x+3)^(5/2)-147157749/8388608*arcsi
nh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-6398163/2097152*(1-4*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 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]

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 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 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]))

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*}

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Mathematica [A]  time = 0.18, 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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fricas [A]  time = 0.81, size = 93, normalized size = 0.56 \[ \frac {1}{660602880} \, {\left (1468006400 \, x^{8} + 2926837760 \, x^{7} + 1033175040 \, x^{6} + 12117893120 \, x^{5} + 379086848 \, x^{4} + 12669290112 \, x^{3} + 4870637856 \, x^{2} + 12357760788 \, x + 1592737263\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {147157749}{16777216} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]

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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giac [A]  time = 0.19, size = 88, normalized size = 0.53 \[ \frac {1}{660602880} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (8 \, {\left (28 \, {\left (160 \, x + 319\right )} x + 3153\right )} x + 295847\right )} x + 185101\right )} x + 98978829\right )} x + 152207433\right )} x + 3089440197\right )} x + 1592737263\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {147157749}{8388608} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]

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)

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maple [A]  time = 0.02, size = 134, normalized size = 0.81 \[ \frac {5 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x^{4}}{9}+\frac {479 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x^{3}}{288}+\frac {2005 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x^{2}}{8064}+\frac {5645 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x}{21504}+\frac {147157749 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}+\frac {120809 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{143360}+\frac {92727 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{131072}+\frac {6398163 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{2097152} \]

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]

120809/143360*(2*x^2-x+3)^(5/2)+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)+92727/131072*(4*x-1)*(2*x^2-x+3)^(3/2)[phone]/8388608*2^(1/2)*arcs
inh(4/23*23^(1/2)*(x-1/4))+6398163/2097152*(4*x-1)*(2*x^2-x+3)^(1/2)

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maxima [A]  time = 0.98, size = 155, normalized size = 0.93 \[ \frac {5}{9} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{4} + \frac {479}{288} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {2005}{8064} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {5645}{21504} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {120809}{143360} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {92727}{32768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {92727}{131072} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {6398163}{524288} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {147157749}{8388608} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {6398163}{2097152} \, \sqrt {2 \, x^{2} - x + 3} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (2\,x+5\right )\,{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2 x + 5\right ) \left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )\, dx \]

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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