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

Optimal. Leaf size=174 \[ \frac{438065 \left (2 x^2-x+3\right )^{5/2}}{82944 (2 x+5)}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{1152 (2 x+5)^2}+\frac{1}{16} \left (2 x^2-x+3\right )^{5/2}+\frac{(2154633-534617 x) \left (2 x^2-x+3\right )^{3/2}}{82944}+\frac{(33741483-5623292 x) \sqrt{2 x^2-x+3}}{24576}-\frac{8083915 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1024 \sqrt{2}}+\frac{129342063 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16384 \sqrt{2}} \]

[Out]

((33741483 - 5623292*x)*Sqrt[3 - x + 2*x^2])/24576 + ((2154633 - 534617*x)*(3 - x + 2*x^2)^(3/2))/82944 + (3 -
x + 2*x^2)^(5/2)/16 - (3667*(3 - x + 2*x^2)^(5/2))/(1152*(5 + 2*x)^2) + (438065*(3 - x + 2*x^2)^(5/2))/(82944
*(5 + 2*x)) + (129342063*ArcSinh[(1 - 4*x)/Sqrt[23]])/(16384*Sqrt[2]) - (8083915*ArcTanh[(17 - 22*x)/(12*Sqrt[
2]*Sqrt[3 - x + 2*x^2])])/(1024*Sqrt[2])

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Rubi [A]  time = 0.272723, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1650, 1653, 814, 843, 619, 215, 724, 206} \[ \frac{438065 \left (2 x^2-x+3\right )^{5/2}}{82944 (2 x+5)}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{1152 (2 x+5)^2}+\frac{1}{16} \left (2 x^2-x+3\right )^{5/2}+\frac{(2154633-534617 x) \left (2 x^2-x+3\right )^{3/2}}{82944}+\frac{(33741483-5623292 x) \sqrt{2 x^2-x+3}}{24576}-\frac{8083915 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1024 \sqrt{2}}+\frac{129342063 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16384 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((33741483 - 5623292*x)*Sqrt[3 - x + 2*x^2])/24576 + ((2154633 - 534617*x)*(3 - x + 2*x^2)^(3/2))/82944 + (3 -
x + 2*x^2)^(5/2)/16 - (3667*(3 - x + 2*x^2)^(5/2))/(1152*(5 + 2*x)^2) + (438065*(3 - x + 2*x^2)^(5/2))/(82944
*(5 + 2*x)) + (129342063*ArcSinh[(1 - 4*x)/Sqrt[23]])/(16384*Sqrt[2]) - (8083915*ArcTanh[(17 - 22*x)/(12*Sqrt[
2]*Sqrt[3 - x + 2*x^2])])/(1024*Sqrt[2])

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -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]))

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 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]

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 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])

Rubi steps

\begin{align*} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}-\frac{1}{144} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{35015}{16}-\frac{21585 x}{4}+972 x^2-360 x^3\right )}{(5+2 x)^2} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{2737465}{16}-505457 x+12960 x^2\right )}{5+2 x} \, dx}{10368}\\ &=\frac{1}{16} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}+\frac{\int \frac{\left (\frac{14335325}{2}-21384680 x\right ) \left (3-x+2 x^2\right )^{3/2}}{5+2 x} \, dx}{414720}\\ &=\frac{(2154633-534617 x) \left (3-x+2 x^2\right )^{3/2}}{82944}+\frac{1}{16} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}-\frac{\int \frac{(-9113472000+24292621440 x) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{26542080}\\ &=\frac{(33741483-5623292 x) \sqrt{3-x+2 x^2}}{24576}+\frac{(2154633-534617 x) \left (3-x+2 x^2\right )^{3/2}}{82944}+\frac{1}{16} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}+\frac{\int \frac{6705272016000-13410185091840 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{849346560}\\ &=\frac{(33741483-5623292 x) \sqrt{3-x+2 x^2}}{24576}+\frac{(2154633-534617 x) \left (3-x+2 x^2\right )^{3/2}}{82944}+\frac{1}{16} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}-\frac{129342063 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{16384}+\frac{24251745}{512} \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{(33741483-5623292 x) \sqrt{3-x+2 x^2}}{24576}+\frac{(2154633-534617 x) \left (3-x+2 x^2\right )^{3/2}}{82944}+\frac{1}{16} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}-\frac{24251745}{256} \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )-\frac{129342063 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{16384 \sqrt{46}}\\ &=\frac{(33741483-5623292 x) \sqrt{3-x+2 x^2}}{24576}+\frac{(2154633-534617 x) \left (3-x+2 x^2\right )^{3/2}}{82944}+\frac{1}{16} \left (3-x+2 x^2\right )^{5/2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1152 (5+2 x)^2}+\frac{438065 \left (3-x+2 x^2\right )^{5/2}}{82944 (5+2 x)}+\frac{129342063 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16384 \sqrt{2}}-\frac{8083915 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1024 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.212224, size = 108, normalized size = 0.62 \[ \frac{\frac{4 \sqrt{2 x^2-x+3} \left (8192 x^6-43520 x^5+253312 x^4-1620944 x^3+16667188 x^2+181223072 x+298966737\right )}{(2 x+5)^2}-129342640 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Sqrt[3 - x + 2*x^2]*(298966737 + 181223072*x + 16667188*x^2 - 1620944*x^3 + 253312*x^4 - 43520*x^5 + 8192*
x^6))/(5 + 2*x)^2 + 129342063*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 129342640*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*
Sqrt[6 - 2*x + 4*x^2])])/32768

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Maple [A]  time = 0.061, size = 214, normalized size = 1.2 \begin{align*} -{\frac{-343745+1374980\,x}{6144}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3667}{4608} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}-{\frac{8083915\,\sqrt{2}}{2048}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }-{\frac{-10281+41124\,x}{8192}\sqrt{2\,{x}^{2}-x+3}}-{\frac{129342063\,\sqrt{2}}{32768}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{-149+596\,x}{512} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{8083915}{331776} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{8083915}{6144}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{1}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{-438065+1752260\,x}{331776} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{438065}{165888} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343745/6144*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(1/2)-3667/4608/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(5/2)-8083915/
2048*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))-10281/8192*(-1+4*x)*(2*x^2-x+3)^(
1/2)-129342063/32768*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-149/512*(-1+4*x)*(2*x^2-x+3)^(3/2)+8083915/331776*
(2*(x+5/2)^2-11*x-19/2)^(3/2)+8083915/6144*(2*(x+5/2)^2-11*x-19/2)^(1/2)+1/16*(2*x^2-x+3)^(5/2)-438065/331776*
(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(3/2)+438065/165888/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(5/2)

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Maxima [A]  time = 1.58913, size = 232, normalized size = 1.33 \begin{align*} \frac{1}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} - \frac{149}{128} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{46691}{4608} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1152 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{1405823}{6144} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{129342063}{32768} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{8083915}{2048} \, \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{11247161}{8192} \, \sqrt{2 \, x^{2} - x + 3} + \frac{438065 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4608 \,{\left (2 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*x^2 - x + 3)^(5/2) - 149/128*(2*x^2 - x + 3)^(3/2)*x + 46691/4608*(2*x^2 - x + 3)^(3/2) - 3667/1152*(2
*x^2 - x + 3)^(5/2)/(4*x^2 + 20*x + 25) - 1405823/6144*sqrt(2*x^2 - x + 3)*x - 129342063/32768*sqrt(2)*arcsinh
(4/23*sqrt(23)*x - 1/23*sqrt(23)) + 8083915/2048*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23
)/abs(2*x + 5)) + 11247161/8192*sqrt(2*x^2 - x + 3) + 438065/4608*(2*x^2 - x + 3)^(3/2)/(2*x + 5)

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Fricas [A]  time = 1.39816, size = 524, normalized size = 3.01 \begin{align*} \frac{129342063 \, \sqrt{2}{\left (4 \, x^{2} + 20 \, x + 25\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 129342640 \, \sqrt{2}{\left (4 \, x^{2} + 20 \, x + 25\right )} \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 ) + 8 \,{\left (8192 \, x^{6} - 43520 \, x^{5} + 253312 \, x^{4} - 1620944 \, x^{3} + 16667188 \, x^{2} + 181223072 \, x + 298966737\right )} \sqrt{2 \, x^{2} - x + 3}}{65536 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/65536*(129342063*sqrt(2)*(4*x^2 + 20*x + 25)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 2
5) + 129342640*sqrt(2)*(4*x^2 + 20*x + 25)*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)) + 8*(8192*x^6 - 43520*x^5 + 253312*x^4 - 1620944*x^3 + 16667188*x^2 + 181223072
*x + 298966737)*sqrt(2*x^2 - x + 3))/(4*x^2 + 20*x + 25)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25388, size = 362, normalized size = 2.08 \begin{align*} \frac{1}{8192} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \, x - 165\right )} x + 4879\right )} x - 263469\right )} x + 8460377\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{129342063}{32768} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{8083915}{2048} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{8083915}{2048} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{\sqrt{2}{\left (14243182 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 109906674 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 170996871 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 110506087\right )}}{512 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8192*(4*(8*(4*(16*x - 165)*x + 4879)*x - 263469)*x + 8460377)*sqrt(2*x^2 - x + 3) + 129342063/32768*sqrt(2)*
log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 8083915/2048*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) +
2*sqrt(2*x^2 - x + 3))) + 8083915/2048*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 1
/512*sqrt(2)*(14243182*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 109906674*(sqrt(2)*x - sqrt(2*x^2 - x + 3
))^2 - 170996871*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 110506087)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2
+ 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^2