### 3.330 $$\int \frac{\sqrt{3-x+2 x^2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^5} \, dx$$

Optimal. Leaf size=165 $-\frac{9363383 \left (2 x^2-x+3\right )^{3/2}}{23887872 (2 x+5)^2}+\frac{593771 \left (2 x^2-x+3\right )^{3/2}}{497664 (2 x+5)^3}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{2304 (2 x+5)^4}+\frac{7 (9616196 x+52836655) \sqrt{2 x^2-x+3}}{95551488 (2 x+5)}-\frac{4640586097 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1146617856 \sqrt{2}}+\frac{259 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}$

[Out]

(7*(52836655 + 9616196*x)*Sqrt[3 - x + 2*x^2])/(95551488*(5 + 2*x)) - (3667*(3 - x + 2*x^2)^(3/2))/(2304*(5 +
2*x)^4) + (593771*(3 - x + 2*x^2)^(3/2))/(497664*(5 + 2*x)^3) - (9363383*(3 - x + 2*x^2)^(3/2))/(23887872*(5 +
2*x)^2) + (259*ArcSinh[(1 - 4*x)/Sqrt[23]])/(64*Sqrt[2]) - (4640586097*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3
- x + 2*x^2])])/(1146617856*Sqrt[2])

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Rubi [A]  time = 0.234099, antiderivative size = 165, normalized size of antiderivative = 1., 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 = {1650, 812, 843, 619, 215, 724, 206} $-\frac{9363383 \left (2 x^2-x+3\right )^{3/2}}{23887872 (2 x+5)^2}+\frac{593771 \left (2 x^2-x+3\right )^{3/2}}{497664 (2 x+5)^3}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{2304 (2 x+5)^4}+\frac{7 (9616196 x+52836655) \sqrt{2 x^2-x+3}}{95551488 (2 x+5)}-\frac{4640586097 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1146617856 \sqrt{2}}+\frac{259 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}$

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)^5,x]

[Out]

(7*(52836655 + 9616196*x)*Sqrt[3 - x + 2*x^2])/(95551488*(5 + 2*x)) - (3667*(3 - x + 2*x^2)^(3/2))/(2304*(5 +
2*x)^4) + (593771*(3 - x + 2*x^2)^(3/2))/(497664*(5 + 2*x)^3) - (9363383*(3 - x + 2*x^2)^(3/2))/(23887872*(5 +
2*x)^2) + (259*ArcSinh[(1 - 4*x)/Sqrt[23]])/(64*Sqrt[2]) - (4640586097*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3
- x + 2*x^2])])/(1146617856*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 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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{\sqrt{3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^5} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}-\frac{1}{288} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{44361}{16}-\frac{17501 x}{4}+1944 x^2-720 x^3\right )}{(5+2 x)^4} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{4140069}{16}-404352 x+77760 x^2\right )}{(5+2 x)^3} \, dx}{62208}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}-\frac{\int \frac{\left (\frac{99869175}{16}-\frac{50485029 x}{4}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^2} \, dx}{8957952}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}+\frac{\int \frac{\frac{2321210451}{8}-580027392 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{71663616}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}-\frac{259}{64} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{4640586097 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{191102976}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}-\frac{4640586097 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{95551488}-\frac{259 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{64 \sqrt{46}}\\ &=\frac{7 (52836655+9616196 x) \sqrt{3-x+2 x^2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{2304 (5+2 x)^4}+\frac{593771 \left (3-x+2 x^2\right )^{3/2}}{497664 (5+2 x)^3}-\frac{9363383 \left (3-x+2 x^2\right )^{3/2}}{23887872 (5+2 x)^2}+\frac{259 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}-\frac{4640586097 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1146617856 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.178584, size = 98, normalized size = 0.59 $\frac{\frac{24 \sqrt{2 x^2-x+3} \left (238878720 x^4+6105343976 x^3+31323229164 x^2+62847867486 x+44676885233\right )}{(2 x+5)^4}-4640586097 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+4640219136 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2293235712}$

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)^5,x]

[Out]

((24*Sqrt[3 - x + 2*x^2]*(44676885233 + 62847867486*x + 31323229164*x^2 + 6105343976*x^3 + 238878720*x^4))/(5
+ 2*x)^4 + 4640219136*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 4640586097*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6
- 2*x + 4*x^2])])/2293235712

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Maple [A]  time = 0.066, size = 167, normalized size = 1. \begin{align*}{\frac{201573155}{3439853568} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}-{\frac{-201573155+806292620\,x}{6879707136}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{9363383}{95551488} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}-{\frac{4640586097\,\sqrt{2}}{2293235712}{\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{259\,\sqrt{2}}{128}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{4640586097}{6879707136}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3667}{36864} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}}+{\frac{593771}{3981312} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}} \end{align*}

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)^5,x)

[Out]

201573155/3439853568/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(3/2)-201573155/6879707136*(-1+4*x)*(2*(x+5/2)^2-11*x-19/
2)^(1/2)-9363383/95551488/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(3/2)-4640586097/2293235712*2^(1/2)*arctanh(1/12*(
17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))-259/128*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+4640586097/68
79707136*(2*(x+5/2)^2-11*x-19/2)^(1/2)-3667/36864/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(3/2)+593771/3981312/(x+5/
2)^3*(2*(x+5/2)^2-11*x-19/2)^(3/2)

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Maxima [A]  time = 1.59831, size = 244, normalized size = 1.48 \begin{align*} -\frac{259}{128} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{4640586097}{2293235712} \, \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{16828343}{47775744} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2304 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{593771 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{497664 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{9363383 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{23887872 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{201573155 \, \sqrt{2 \, x^{2} - x + 3}}{95551488 \,{\left (2 \, x + 5\right )}} \end{align*}

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)^5,x, algorithm="maxima")

[Out]

-259/128*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) + 4640586097/2293235712*sqrt(2)*arcsinh(22/23*sqrt(2
3)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 16828343/47775744*sqrt(2*x^2 - x + 3) - 3667/2304*(2*x^2 -
x + 3)^(3/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 593771/497664*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2
+ 150*x + 125) - 9363383/23887872*(2*x^2 - x + 3)^(3/2)/(4*x^2 + 20*x + 25) + 201573155/95551488*sqrt(2*x^2 -
x + 3)/(2*x + 5)

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Fricas [A]  time = 1.4576, size = 618, normalized size = 3.75 \begin{align*} \frac{4640219136 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 4640586097 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 ) + 48 \,{\left (238878720 \, x^{4} + 6105343976 \, x^{3} + 31323229164 \, x^{2} + 62847867486 \, x + 44676885233\right )} \sqrt{2 \, x^{2} - x + 3}}{4586471424 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} \end{align*}

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)^5,x, algorithm="fricas")

[Out]

1/4586471424*(4640219136*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)
*(4*x - 1) - 32*x^2 + 16*x - 25) + 4640586097*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)*log(-(24*sqr
t(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) + 48*(238878720*x^4 + 61
05343976*x^3 + 31323229164*x^2 + 62847867486*x + 44676885233)*sqrt(2*x^2 - x + 3))/(16*x^4 + 160*x^3 + 600*x^2
+ 1000*x + 625)

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

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)**5,x)

[Out]

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

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Giac [B]  time = 1.33849, size = 441, normalized size = 2.67 \begin{align*} -\frac{1}{2293235712} \, \sqrt{2}{\left (4640586097 \, \log \left (12 \, \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{72}{2 \, x + 5} - 11\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) + 4640219136 \, \log \left ({\left | \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5} + 1 \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) - 4640219136 \, \log \left ({\left | \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5} - 1 \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) + 12 \,{\left (\frac{24 \,{\left (\frac{144 \,{\left (\frac{792072 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )}{2 \, x + 5} - 835793 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )}}{2 \, x + 5} + 57384361 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )}}{2 \, x + 5} - 464569597 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )} \sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{179159040 \,{\left (11 \,{\left (\sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5}\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right ) - 12 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 5}\right )\right )}}{{\left (\sqrt{-\frac{11}{2 \, x + 5} + \frac{36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac{6}{2 \, x + 5}\right )}^{2} - 1}\right )} \end{align*}

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)^5,x, algorithm="giac")

[Out]

-1/2293235712*sqrt(2)*(4640586097*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)*sgn(1/(
2*x + 5)) + 4640219136*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) + 1))*sgn(1/(2*x + 5)) -
4640219136*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) - 1))*sgn(1/(2*x + 5)) + 12*(24*(14
4*(792072*sgn(1/(2*x + 5))/(2*x + 5) - 835793*sgn(1/(2*x + 5)))/(2*x + 5) + 57384361*sgn(1/(2*x + 5)))/(2*x +
5) - 464569597*sgn(1/(2*x + 5)))*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 179159040*(11*(sqrt(-11/(2*x + 5)
+ 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))*sgn(1/(2*x + 5)) - 12*sgn(1/(2*x + 5)))/((sqrt(-11/(2*x + 5) + 36/(2*x +
5)^2 + 1) + 6/(2*x + 5))^2 - 1))