\(\int \frac {(4+x-2 x^2)^3}{(2+3 x+5 x^2)^{5/2}} \, dx\) [158]

Optimal result

Integrand size = 25, antiderivative size = 105 \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=-\frac {2 (573358-1635093 x)}{290625 \left (2+3 x+5 x^2\right )^{3/2}}+\frac {2 (10505303+15267525 x)}{3003125 \sqrt {2+3 x+5 x^2}}+\frac {126}{625} \sqrt {2+3 x+5 x^2}-\frac {4}{125} x \sqrt {2+3 x+5 x^2}+\frac {97 \text {arcsinh}\left (\frac {3+10 x}{\sqrt {31}}\right )}{125 \sqrt {5}} \] Output:

1/290625*(-1146716+3270186*x)/(5*x^2+3*x+2)^(3/2)+2/3003125*(10505303+1526 
7525*x)/(5*x^2+3*x+2)^(1/2)+126/625*(5*x^2+3*x+2)^(1/2)-4/125*x*(5*x^2+3*x 
+2)^(1/2)+97/625*arcsinh(1/31*(3+10*x)*31^(1/2))*5^(1/2)
 

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76 \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=-\frac {2 \left (-1955612-9886476 x-12783747 x^2-10083075 x^3-735165 x^4+144150 x^5\right )}{360375 \left (2+3 x+5 x^2\right )^{3/2}}-\frac {97 \log \left (-3-10 x+2 \sqrt {5} \sqrt {2+3 x+5 x^2}\right )}{125 \sqrt {5}} \] Input:

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

Output:

(-2*(-1955612 - 9886476*x - 12783747*x^2 - 10083075*x^3 - 735165*x^4 + 144 
150*x^5))/(360375*(2 + 3*x + 5*x^2)^(3/2)) - (97*Log[-3 - 10*x + 2*Sqrt[5] 
*Sqrt[2 + 3*x + 5*x^2]])/(125*Sqrt[5])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2191, 27, 2191, 27, 2192, 27, 1160, 1090, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2191

\(\displaystyle \frac {2}{93} \int \frac {3 \left (-155000 x^4+325500 x^3+680450 x^2-1449095 x+1149901\right )}{6250 \left (5 x^2+3 x+2\right )^{3/2}}dx-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-155000 x^4+325500 x^3+680450 x^2-1449095 x+1149901}{\left (5 x^2+3 x+2\right )^{3/2}}dx}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2191

\(\displaystyle \frac {\frac {2}{31} \int \frac {4805 \left (-100 x^2+270 x+317\right )}{\sqrt {5 x^2+3 x+2}}dx+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {310 \int \frac {-100 x^2+270 x+317}{\sqrt {5 x^2+3 x+2}}dx+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {310 \left (\frac {1}{10} \int \frac {10 (315 x+337)}{\sqrt {5 x^2+3 x+2}}dx-10 x \sqrt {5 x^2+3 x+2}\right )+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {310 \left (\int \frac {315 x+337}{\sqrt {5 x^2+3 x+2}}dx-10 x \sqrt {5 x^2+3 x+2}\right )+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {310 \left (\frac {485}{2} \int \frac {1}{\sqrt {5 x^2+3 x+2}}dx-10 \sqrt {5 x^2+3 x+2} x+63 \sqrt {5 x^2+3 x+2}\right )+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {310 \left (\frac {97}{2} \sqrt {\frac {5}{31}} \int \frac {1}{\sqrt {\frac {1}{31} (10 x+3)^2+1}}d(10 x+3)-10 \sqrt {5 x^2+3 x+2} x+63 \sqrt {5 x^2+3 x+2}\right )+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {310 \left (\frac {97}{2} \sqrt {5} \text {arcsinh}\left (\frac {10 x+3}{\sqrt {31}}\right )-10 \sqrt {5 x^2+3 x+2} x+63 \sqrt {5 x^2+3 x+2}\right )+\frac {2 (15267525 x+10505303)}{31 \sqrt {5 x^2+3 x+2}}}{96875}-\frac {2 (573358-1635093 x)}{290625 \left (5 x^2+3 x+2\right )^{3/2}}\)

Input:

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

Output:

(-2*(573358 - 1635093*x))/(290625*(2 + 3*x + 5*x^2)^(3/2)) + ((2*(10505303 
 + 15267525*x))/(31*Sqrt[2 + 3*x + 5*x^2]) + 310*(63*Sqrt[2 + 3*x + 5*x^2] 
 - 10*x*Sqrt[2 + 3*x + 5*x^2] + (97*Sqrt[5]*ArcSinh[(3 + 10*x)/Sqrt[31]])/ 
2))/96875
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

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 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[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 1160

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

rule 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 2.64 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {2 \left (144150 x^{5}-735165 x^{4}-10083075 x^{3}-12783747 x^{2}-9886476 x -1955612\right )}{360375 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}+\frac {97 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {10 \sqrt {31}\, \left (x +\frac {3}{10}\right )}{31}\right )}{625}\)	\(55\)
trager	\(-\frac {2 \left (144150 x^{5}-735165 x^{4}-10083075 x^{3}-12783747 x^{2}-9886476 x -1955612\right )}{360375 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}+\frac {97 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (10 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {5 x^{2}+3 x +2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )\right )}{625}\)	\(82\)
default	\(\frac {\frac {4324909 x}{465000}+\frac {4324909}{1550000}}{\left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}+\frac {\frac {4406098 x}{360375}+\frac {2203049}{600625}}{\sqrt {5 x^{2}+3 x +2}}+\frac {39223}{150000 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}+\frac {54819 x}{5000 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}+\frac {4273 x^{2}}{250 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}-\frac {97 x^{3}}{75 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}-\frac {97 x}{125 \sqrt {5 x^{2}+3 x +2}}+\frac {291}{1250 \sqrt {5 x^{2}+3 x +2}}+\frac {97 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {10 \sqrt {31}\, \left (x +\frac {3}{10}\right )}{31}\right )}{625}+\frac {102 x^{4}}{25 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}-\frac {4 x^{5}}{5 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}\)	\(180\)

Input:

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

Output:

-2/360375*(144150*x^5-735165*x^4-10083075*x^3-12783747*x^2-9886476*x-19556 
12)/(5*x^2+3*x+2)^(3/2)+97/625*5^(1/2)*arcsinh(10/31*31^(1/2)*(x+3/10))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {279651 \, \sqrt {5} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (-4 \, \sqrt {5} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (10 \, x + 3\right )} - 200 \, x^{2} - 120 \, x - 49\right ) - 20 \, {\left (144150 \, x^{5} - 735165 \, x^{4} - 10083075 \, x^{3} - 12783747 \, x^{2} - 9886476 \, x - 1955612\right )} \sqrt {5 \, x^{2} + 3 \, x + 2}}{3603750 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/3603750*(279651*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(-4*sqr 
t(5)*sqrt(5*x^2 + 3*x + 2)*(10*x + 3) - 200*x^2 - 120*x - 49) - 20*(144150 
*x^5 - 735165*x^4 - 10083075*x^3 - 12783747*x^2 - 9886476*x - 1955612)*sqr 
t(5*x^2 + 3*x + 2))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
 

Sympy [F]

\[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {48 x}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\right )\, dx - \int \frac {84 x^{2}}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\, dx - \int \frac {47 x^{3}}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {42 x^{4}}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\right )\, dx - \int \left (- \frac {12 x^{5}}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\right )\, dx - \int \frac {8 x^{6}}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {64}{25 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 30 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 29 x^{2} \sqrt {5 x^{2} + 3 x + 2} + 12 x \sqrt {5 x^{2} + 3 x + 2} + 4 \sqrt {5 x^{2} + 3 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(-48*x/(25*x**4*sqrt(5*x**2 + 3*x + 2) + 30*x**3*sqrt(5*x**2 + 3* 
x + 2) + 29*x**2*sqrt(5*x**2 + 3*x + 2) + 12*x*sqrt(5*x**2 + 3*x + 2) + 4* 
sqrt(5*x**2 + 3*x + 2)), x) - Integral(84*x**2/(25*x**4*sqrt(5*x**2 + 3*x 
+ 2) + 30*x**3*sqrt(5*x**2 + 3*x + 2) + 29*x**2*sqrt(5*x**2 + 3*x + 2) + 1 
2*x*sqrt(5*x**2 + 3*x + 2) + 4*sqrt(5*x**2 + 3*x + 2)), x) - Integral(47*x 
**3/(25*x**4*sqrt(5*x**2 + 3*x + 2) + 30*x**3*sqrt(5*x**2 + 3*x + 2) + 29* 
x**2*sqrt(5*x**2 + 3*x + 2) + 12*x*sqrt(5*x**2 + 3*x + 2) + 4*sqrt(5*x**2 
+ 3*x + 2)), x) - Integral(-42*x**4/(25*x**4*sqrt(5*x**2 + 3*x + 2) + 30*x 
**3*sqrt(5*x**2 + 3*x + 2) + 29*x**2*sqrt(5*x**2 + 3*x + 2) + 12*x*sqrt(5* 
x**2 + 3*x + 2) + 4*sqrt(5*x**2 + 3*x + 2)), x) - Integral(-12*x**5/(25*x* 
*4*sqrt(5*x**2 + 3*x + 2) + 30*x**3*sqrt(5*x**2 + 3*x + 2) + 29*x**2*sqrt( 
5*x**2 + 3*x + 2) + 12*x*sqrt(5*x**2 + 3*x + 2) + 4*sqrt(5*x**2 + 3*x + 2) 
), x) - Integral(8*x**6/(25*x**4*sqrt(5*x**2 + 3*x + 2) + 30*x**3*sqrt(5*x 
**2 + 3*x + 2) + 29*x**2*sqrt(5*x**2 + 3*x + 2) + 12*x*sqrt(5*x**2 + 3*x + 
 2) + 4*sqrt(5*x**2 + 3*x + 2)), x) - Integral(-64/(25*x**4*sqrt(5*x**2 + 
3*x + 2) + 30*x**3*sqrt(5*x**2 + 3*x + 2) + 29*x**2*sqrt(5*x**2 + 3*x + 2) 
 + 12*x*sqrt(5*x**2 + 3*x + 2) + 4*sqrt(5*x**2 + 3*x + 2)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (84) = 168\).

Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.09 \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=-\frac {4 \, x^{5}}{5 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + \frac {102 \, x^{4}}{25 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} - \frac {97}{360375} \, x {\left (\frac {3330 \, x}{\sqrt {5 \, x^{2} + 3 \, x + 2}} + \frac {14415 \, x^{2}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + \frac {999}{\sqrt {5 \, x^{2} + 3 \, x + 2}} + \frac {4743 \, x}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + \frac {4402}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}}\right )} + \frac {97}{625} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {21534}{120125} \, \sqrt {5 \, x^{2} + 3 \, x + 2} + \frac {1405326 \, x}{120125 \, \sqrt {5 \, x^{2} + 3 \, x + 2}} + \frac {1991 \, x^{2}}{125 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + \frac {2234283}{600625 \, \sqrt {5 \, x^{2} + 3 \, x + 2}} + \frac {1155623 \, x}{58125 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + \frac {156722}{58125 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

-4/5*x^5/(5*x^2 + 3*x + 2)^(3/2) + 102/25*x^4/(5*x^2 + 3*x + 2)^(3/2) - 97 
/360375*x*(3330*x/sqrt(5*x^2 + 3*x + 2) + 14415*x^2/(5*x^2 + 3*x + 2)^(3/2 
) + 999/sqrt(5*x^2 + 3*x + 2) + 4743*x/(5*x^2 + 3*x + 2)^(3/2) + 4402/(5*x 
^2 + 3*x + 2)^(3/2)) + 97/625*sqrt(5)*arcsinh(1/31*sqrt(31)*(10*x + 3)) + 
21534/120125*sqrt(5*x^2 + 3*x + 2) + 1405326/120125*x/sqrt(5*x^2 + 3*x + 2 
) + 1991/125*x^2/(5*x^2 + 3*x + 2)^(3/2) + 2234283/600625/sqrt(5*x^2 + 3*x 
 + 2) + 1155623/58125*x/(5*x^2 + 3*x + 2)^(3/2) + 156722/58125/(5*x^2 + 3* 
x + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69 \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=-\frac {97}{625} \, \sqrt {5} \log \left (-2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 3\right ) - \frac {2 \, {\left (3 \, {\left ({\left (5 \, {\left (961 \, {\left (10 \, x - 51\right )} x - 672205\right )} x - 4261249\right )} x - 3295492\right )} x - 1955612\right )}}{360375 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

-97/625*sqrt(5)*log(-2*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) - 3) - 
2/360375*(3*((5*(961*(10*x - 51)*x - 672205)*x - 4261249)*x - 3295492)*x - 
 1955612)/(5*x^2 + 3*x + 2)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=\int \frac {{\left (-2\,x^2+x+4\right )}^3}{{\left (5\,x^2+3\,x+2\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.90 \[ \int \frac {\left (4+x-2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {-14415000 \sqrt {5 x^{2}+3 x +2}\, x^{5}[phone] \sqrt {5 x^{2}+3 x +2}\, x^{4}[phone] \sqrt {5 x^{2}+3 x +2}\, x^{3}[phone] \sqrt {5 x^{2}+3 x +2}\, x^{2}[phone] \sqrt {5 x^{2}+3 x +2}\, x [phone] \sqrt {5 x^{2}+3 x +2}[phone] \sqrt {5}\, \mathrm {log}\left (\frac {2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}+10 x +3}{\sqrt {31}}\right ) x^{4}[phone] \sqrt {5}\, \mathrm {log}\left (\frac {2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}+10 x +3}{\sqrt {31}}\right ) x^{3}[phone] \sqrt {5}\, \mathrm {log}\left (\frac {2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}+10 x +3}{\sqrt {31}}\right ) x^{2}[phone] \sqrt {5}\, \mathrm {log}\left (\frac {2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}+10 x +3}{\sqrt {31}}\right ) x [phone] \sqrt {5}\, \mathrm {log}\left (\frac {2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}+10 x +3}{\sqrt {31}}\right )-1176655325 \sqrt {5}\, x^{4}-1411986390 \sqrt {5}\, x^{3}-1364920177 \sqrt {5}\, x^{2}-564794556 \sqrt {5}\, x -188264852 \sqrt {5}}{450468750 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 14415000*sqrt(5*x**2 + 3*x + 2)*x**5 + 73516500*sqrt(5*x**2 + 3*x + 2) 
*x**4 + 1008307500*sqrt(5*x**2 + 3*x + 2)*x**3 + 1278374700*sqrt(5*x**2 + 
3*x + 2)*x**2 + 988647600*sqrt(5*x**2 + 3*x + 2)*x + 195561200*sqrt(5*x**2 
 + 3*x + 2) + 69912750*sqrt(5)*log((2*sqrt(5*x**2 + 3*x + 2)*sqrt(5) + 10* 
x + 3)/sqrt(31))*x**4 + 83895300*sqrt(5)*log((2*sqrt(5*x**2 + 3*x + 2)*sqr 
t(5) + 10*x + 3)/sqrt(31))*x**3 + 81098790*sqrt(5)*log((2*sqrt(5*x**2 + 3* 
x + 2)*sqrt(5) + 10*x + 3)/sqrt(31))*x**2 + 33558120*sqrt(5)*log((2*sqrt(5 
*x**2 + 3*x + 2)*sqrt(5) + 10*x + 3)/sqrt(31))*x + 11186040*sqrt(5)*log((2 
*sqrt(5*x**2 + 3*x + 2)*sqrt(5) + 10*x + 3)/sqrt(31)) - 1176655325*sqrt(5) 
*x**4 - 1411986390*sqrt(5)*x**3 - 1364920177*sqrt(5)*x**2 - 564794556*sqrt 
(5)*x - 188264852*sqrt(5))/(18018750*(25*x**4 + 30*x**3 + 29*x**2 + 12*x + 
 4))