3.326 \(\int \frac {\sqrt {3-x+2 x^2} (2+x+3 x^2-x^3+5 x^4)}{5+2 x} \, dx\)

Optimal. Leaf size=149 \[ \frac {1}{16} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2-\frac {127}{128} \left (2 x^2-x+3\right )^{3/2} (2 x+5)+\frac {4535}{768} \left (2 x^2-x+3\right )^{3/2}+\frac {(489587-80844 x) \sqrt {2 x^2-x+3}}{4096}-\frac {11001 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{16 \sqrt {2}}+\frac {5627989 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}} \]

[Out]

4535/768*(2*x^2-x+3)^(3/2)-127/128*(5+2*x)*(2*x^2-x+3)^(3/2)+1/16*(5+2*x)^2*(2*x^2-x+3)^(3/2)+5627989/16384*ar
csinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-11001/32*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)+1/4096
*(489587-80844*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1653, 814, 843, 619, 215, 724, 206} \[ \frac {1}{16} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2-\frac {127}{128} \left (2 x^2-x+3\right )^{3/2} (2 x+5)+\frac {4535}{768} \left (2 x^2-x+3\right )^{3/2}+\frac {(489587-80844 x) \sqrt {2 x^2-x+3}}{4096}-\frac {11001 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{16 \sqrt {2}}+\frac {5627989 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((489587 - 80844*x)*Sqrt[3 - x + 2*x^2])/4096 + (4535*(3 - x + 2*x^2)^(3/2))/768 - (127*(5 + 2*x)*(3 - x + 2*x
^2)^(3/2))/128 + ((5 + 2*x)^2*(3 - x + 2*x^2)^(3/2))/16 + (5627989*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8192*Sqrt[2])
- (11001*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(16*Sqrt[2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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]

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 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

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 \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{5+2 x} \, dx &=\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}+\frac {1}{160} \int \frac {\sqrt {3-x+2 x^2} \left (-805-6490 x-9300 x^2-5080 x^3\right )}{5+2 x} \, dx\\ &=-\frac {127}{128} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \frac {\sqrt {3-x+2 x^2} \left (-127720+824160 x+725600 x^2\right )}{5+2 x} \, dx}{10240}\\ &=\frac {4535}{768} \left (3-x+2 x^2\right )^{3/2}-\frac {127}{128} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \frac {(7818720-19402560 x) \sqrt {3-x+2 x^2}}{5+2 x} \, dx}{245760}\\ &=\frac {(489587-80844 x) \sqrt {3-x+2 x^2}}{4096}+\frac {4535}{768} \left (3-x+2 x^2\right )^{3/2}-\frac {127}{128} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}-\frac {\int \frac {-5428921920+10805738880 x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{7864320}\\ &=\frac {(489587-80844 x) \sqrt {3-x+2 x^2}}{4096}+\frac {4535}{768} \left (3-x+2 x^2\right )^{3/2}-\frac {127}{128} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}-\frac {5627989 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{8192}+\frac {33003}{8} \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx\\ &=\frac {(489587-80844 x) \sqrt {3-x+2 x^2}}{4096}+\frac {4535}{768} \left (3-x+2 x^2\right )^{3/2}-\frac {127}{128} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}-\frac {33003}{4} \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )-\frac {5627989 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{8192 \sqrt {46}}\\ &=\frac {(489587-80844 x) \sqrt {3-x+2 x^2}}{4096}+\frac {4535}{768} \left (3-x+2 x^2\right )^{3/2}-\frac {127}{128} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}+\frac {5627989 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}}-\frac {11001 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{16 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 91, normalized size = 0.61 \[ \frac {-16897536 \sqrt {2} \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )+4 \sqrt {2 x^2-x+3} \left (6144 x^4-21120 x^3+79840 x^2-300404 x+1561161\right )[phone] \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{49152} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(1561161 - 300404*x + 79840*x^2 - 21120*x^3 + 6144*x^4) + 16883967*Sqrt[2]*ArcSinh[(1 -
4*x)/Sqrt[23]] - 16897536*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/49152

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fricas [A]  time = 0.89, size = 125, normalized size = 0.84 \[ \frac {1}{12288} \, {\left (6144 \, x^{4} - 21120 \, x^{3} + 79840 \, x^{2} - 300404 \, x + 1561161\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {5627989}{32768} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + \frac {11001}{64} \, \sqrt {2} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12288*(6144*x^4 - 21120*x^3 + 79840*x^2 - 300404*x + 1561161)*sqrt(2*x^2 - x + 3) + 5627989/32768*sqrt(2)*lo
g(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 11001/64*sqrt(2)*log(-(24*sqrt(2)*sqrt(2*x^2
- x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25))

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giac [A]  time = 0.22, size = 129, normalized size = 0.87 \[ \frac {1}{12288} \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (16 \, x - 55\right )} x + 2495\right )} x - 75101\right )} x + 1561161\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {5627989}{16384} \, \sqrt {2} \log \left (-4 \, \sqrt {2} x + \sqrt {2} + 4 \, \sqrt {2 \, x^{2} - x + 3}\right ) - \frac {11001}{32} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {11001}{32} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12288*(4*(8*(12*(16*x - 55)*x + 2495)*x - 75101)*x + 1561161)*sqrt(2*x^2 - x + 3) + 5627989/16384*sqrt(2)*lo
g(-4*sqrt(2)*x + sqrt(2) + 4*sqrt(2*x^2 - x + 3)) - 11001/32*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2
*x^2 - x + 3))) + 11001/32*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3)))

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maple [A]  time = 0.01, size = 127, normalized size = 0.85 \[ \frac {\left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{2}}{4}-\frac {47 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x}{64}-\frac {5627989 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16384}-\frac {11001 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{32}+\frac {1925 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{768}-\frac {20211 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{4096}+\frac {3667 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x^2-x+3)^(3/2)*x^2-47/64*(2*x^2-x+3)^(3/2)*x+1925/768*(2*x^2-x+3)^(3/2)-20211/4096*(4*x-1)*(2*x^2-x+3)^
(1/2)-5627989/16384*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+3667/32*(2*(x+5/2)^2-11*x-19/2)^(1/2)-11001/32*2^(1
/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))

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maxima [A]  time = 1.01, size = 128, normalized size = 0.86 \[ \frac {1}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {47}{64} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1925}{768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {20211}{1024} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {5627989}{16384} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {11001}{32} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {489587}{4096} \, \sqrt {2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x^2 - x + 3)^(3/2)*x^2 - 47/64*(2*x^2 - x + 3)^(3/2)*x + 1925/768*(2*x^2 - x + 3)^(3/2) - 20211/1024*sq
rt(2*x^2 - x + 3)*x - 5627989/16384*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) + 11001/32*sqrt(2)*arcsin
h(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 489587/4096*sqrt(2*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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