\(\int \frac {(2+3 x+5 x^2)^4}{(3-x+2 x^2)^{5/2}} \, dx\) [127]

Optimal result

Integrand size = 27, antiderivative size = 147 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}-\frac {1308645 \sqrt {3-x+2 x^2}}{4096}+\frac {526075 x \sqrt {3-x+2 x^2}}{3072}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}+\frac {16955197 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}} \] Output:

1/4416*(-1478741-1156639*x)/(2*x^2-x+3)^(3/2)+1331/101568*(7409+116368*x)/ 
(2*x^2-x+3)^(1/2)-1308645/4096*(2*x^2-x+3)^(1/2)+526075/3072*x*(2*x^2-x+3) 
^(1/2)+38375/384*x^2*(2*x^2-x+3)^(1/2)+625/32*x^3*(2*x^2-x+3)^(1/2)+169551 
97/16384*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.58 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {-18974698519+49883864262 x-36481630395 x^2+39848900984 x^3-5076781260 x^4+3504730800 x^5+2090608000 x^6+507840000 x^7}{6500352 \left (3-x+2 x^2\right )^{3/2}}+\frac {16955197 \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{8192 \sqrt {2}} \] Input:

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

Output:

(-18974698519 + 49883864262*x - 36481630395*x^2 + 39848900984*x^3 - 507678 
1260*x^4 + 3504730800*x^5 + 2090608000*x^6 + 507840000*x^7)/(6500352*(3 - 
x + 2*x^2)^(3/2)) + (16955197*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/(819 
2*Sqrt[2])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2191, 27, 2191, 27, 2192, 27, 2192, 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.

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

\(\Big \downarrow \) 2191

\(\displaystyle \frac {2}{69} \int \frac {2760000 x^6+8004000 x^5+10239600 x^4+3447240 x^3-5859204 x^2-3967086 x+3839123}{256 \left (2 x^2-x+3\right )^{3/2}}dx-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2760000 x^6+8004000 x^5+10239600 x^4+3447240 x^3-5859204 x^2-3967086 x+3839123}{\left (2 x^2-x+3\right )^{3/2}}dx}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2191

\(\displaystyle \frac {\frac {2}{23} \int -\frac {1587 \left (-10000 x^4-34000 x^3-39100 x^2+18960 x+89359\right )}{\sqrt {2 x^2-x+3}}dx+\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \int \frac {-10000 x^4-34000 x^3-39100 x^2+18960 x+89359}{\sqrt {2 x^2-x+3}}dx}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{8} \int \frac {8 \left (-38375 x^3-27850 x^2+18960 x+89359\right )}{\sqrt {2 x^2-x+3}}dx-1250 x^3 \sqrt {2 x^2-x+3}\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\int \frac {-38375 x^3-27850 x^2+18960 x+89359}{\sqrt {2 x^2-x+3}}dx-1250 x^3 \sqrt {2 x^2-x+3}\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{6} \int \frac {-526075 x^2+688020 x+1072308}{2 \sqrt {2 x^2-x+3}}dx-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{12} \int \frac {-526075 x^2+688020 x+1072308}{\sqrt {2 x^2-x+3}}dx-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{12} \left (\frac {1}{4} \int \frac {3 (1308645 x+3911638)}{2 \sqrt {2 x^2-x+3}}dx-\frac {526075}{4} x \sqrt {2 x^2-x+3}\right )-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{12} \left (\frac {3}{8} \int \frac {1308645 x+3911638}{\sqrt {2 x^2-x+3}}dx-\frac {526075}{4} x \sqrt {2 x^2-x+3}\right )-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{12} \left (\frac {3}{8} \left (\frac {16955197}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1308645}{2} \sqrt {2 x^2-x+3}\right )-\frac {526075}{4} x \sqrt {2 x^2-x+3}\right )-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{12} \left (\frac {3}{8} \left (\frac {16955197 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}+\frac {1308645}{2} \sqrt {2 x^2-x+3}\right )-\frac {526075}{4} x \sqrt {2 x^2-x+3}\right )-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\frac {2662 (116368 x+7409)}{23 \sqrt {2 x^2-x+3}}-138 \left (\frac {1}{12} \left (\frac {3}{8} \left (\frac {16955197 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}+\frac {1308645}{2} \sqrt {2 x^2-x+3}\right )-\frac {526075}{4} x \sqrt {2 x^2-x+3}\right )-\frac {38375}{6} \sqrt {2 x^2-x+3} x^2-1250 \sqrt {2 x^2-x+3} x^3\right )}{8832}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}\)

Input:

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

Output:

(-14641*(101 + 79*x))/(4416*(3 - x + 2*x^2)^(3/2)) + ((2662*(7409 + 116368 
*x))/(23*Sqrt[3 - x + 2*x^2]) - 138*((-38375*x^2*Sqrt[3 - x + 2*x^2])/6 - 
1250*x^3*Sqrt[3 - x + 2*x^2] + ((-526075*x*Sqrt[3 - x + 2*x^2])/4 + (3*((1 
308645*Sqrt[3 - x + 2*x^2])/2 + (16955197*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4 
*Sqrt[2])))/8)/12))/8832
 

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 = 2.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44

method	result	size
risch	\(\frac {507840000 x^{7}[phone] x^{6}[phone] x^{5}-5076781260 x^{4}[phone] x^{3}-36481630395 x^{2}[phone] x -18974698519}{6500352 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {16955197 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16384}\)	\(65\)
trager	\(\frac {507840000 x^{7}[phone] x^{6}[phone] x^{5}-5076781260 x^{4}[phone] x^{3}-36481630395 x^{2}[phone] x -18974698519}{6500352 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {16955197 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{16384}\)	\(90\)
default	\(\frac {16955197}{32768 \sqrt {2 x^{2}-x +3}}-\frac {2149616639}{524288 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {138025 x^{5}}{256 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {5141612725 x}{9043968}-\frac {5141612725}{36175872}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {67488035 x^{2}}{16384 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {16955197 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16384}+\frac {30875 x^{6}}{96 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {625 x^{7}}{8 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {16955197 x^{3}}{12288 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {992926033 x}{3250176}-\frac {992926033}{13000704}}{\sqrt {2 x^{2}-x +3}}-\frac {799745 x^{4}}{1024 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {55167267 x}{131072 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {16955197 x}{8192 \sqrt {2 x^{2}-x +3}}\)	\(214\)

Input:

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

Output:

1/6500352*(507840000*x^7+2090608000*x^6+3504730800*x^5-5076781260*x^4+3984 
8900984*x^3-36481630395*x^2+49883864262*x-18974698519)/(2*x^2-x+3)^(3/2)-1 
6955197/16384*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {26907897639 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (507840000 \, x^{7} + 2090608000 \, x^{6} + 3504730800 \, x^{5} - 5076781260 \, x^{4} + 39848900984 \, x^{3} - 36481630395 \, x^{2} + 49883864262 \, x - 18974698519\right )} \sqrt {2 \, x^{2} - x + 3}}{52002816 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:

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

Output:

1/52002816*(26907897639*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(4*s 
qrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 8*(507840000* 
x^7 + 2090608000*x^6 + 3504730800*x^5 - 5076781260*x^4 + 39848900984*x^3 - 
 36481630395*x^2 + 49883864262*x - 18974698519)*sqrt(2*x^2 - x + 3))/(4*x^ 
4 - 4*x^3 + 13*x^2 - 6*x + 9)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (118) = 236\).

Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.72 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {625 \, x^{7}}{8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {30875 \, x^{6}}{96 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {138025 \, x^{5}}{256 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {799745 \, x^{4}}{1024 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {16955197}{13000704} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {16955197}{16384} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {1203818987}{6500352} \, \sqrt {2 \, x^{2} - x + 3} + \frac {3536205583 \, x}{3250176 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {2638851 \, x^{2}}{512 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {257773037}{1083392 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {29484067 \, x}{23552 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {374445479}{70656 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:

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

Output:

625/8*x^7/(2*x^2 - x + 3)^(3/2) + 30875/96*x^6/(2*x^2 - x + 3)^(3/2) + 138 
025/256*x^5/(2*x^2 - x + 3)^(3/2) - 799745/1024*x^4/(2*x^2 - x + 3)^(3/2) 
- 16955197/13000704*x*(284*x/sqrt(2*x^2 - x + 3) - 3174*x^2/(2*x^2 - x + 3 
)^(3/2) - 71/sqrt(2*x^2 - x + 3) + 805*x/(2*x^2 - x + 3)^(3/2) - 3243/(2*x 
^2 - x + 3)^(3/2)) - 16955197/16384*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1 
)) + 1203818987/6500352*sqrt(2*x^2 - x + 3) + 3536205583/3250176*x/sqrt(2* 
x^2 - x + 3) - 2638851/512*x^2/(2*x^2 - x + 3)^(3/2) + 257773037/1083392/s 
qrt(2*x^2 - x + 3) + 29484067/23552*x/(2*x^2 - x + 3)^(3/2) - 374445479/70 
656/(2*x^2 - x + 3)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.55 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {16955197}{16384} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {{\left ({\left (4 \, {\left (2645 \, {\left (20 \, {\left (40 \, {\left (60 \, x + 247\right )} x + 16563\right )} x - 479847\right )} x + 9962225246\right )} x - 36481630395\right )} x + 49883864262\right )} x - 18974698519}{6500352 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:

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

Output:

16955197/16384*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 
1) + 1/6500352*(((4*(2645*(20*(40*(60*x + 247)*x + 16563)*x - 479847)*x + 
9962225246)*x - 36481630395)*x + 49883864262)*x - 18974698519)/(2*x^2 - x 
+ 3)^(3/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.29 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {16250880000 \sqrt {2 x^{2}-x +3}\, x^{7}[phone] \sqrt {2 x^{2}-x +3}\, x^{6}[phone] \sqrt {2 x^{2}-x +3}\, x^{5}-162457000320 \sqrt {2 x^{2}-x +3}\, x^{4}[phone] \sqrt {2 x^{2}-x +3}\, x^{3}-1167412172640 \sqrt {2 x^{2}-x +3}\, x^{2}[phone] \sqrt {2 x^{2}-x +3}\, x -607190352608 \sqrt {2 x^{2}-x +3}-861052724448 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{4}[phone] \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{3}-2798421354456 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{2}[phone] \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x -1937368630008 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )-203665714820 \sqrt {2}\, x^{4}[phone] \sqrt {2}\, x^{3}-661913573165 \sqrt {2}\, x^{2}[phone] \sqrt {2}\, x -458247858345 \sqrt {2}}{832045056 x^{4}-832045056 x^{3}[phone] x^{2}-1248067584 x [phone]} \] Input:

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

Output:

(16250880000*sqrt(2*x**2 - x + 3)*x**7 + 66899456000*sqrt(2*x**2 - x + 3)* 
x**6 + 112151385600*sqrt(2*x**2 - x + 3)*x**5 - 162457000320*sqrt(2*x**2 - 
 x + 3)*x**4 + 1275164831488*sqrt(2*x**2 - x + 3)*x**3 - 1167412172640*sqr 
t(2*x**2 - x + 3)*x**2 + 1596283656384*sqrt(2*x**2 - x + 3)*x - 6071903526 
08*sqrt(2*x**2 - x + 3) - 861052724448*sqrt(2)*log((2*sqrt(2*x**2 - x + 3) 
*sqrt(2) + 4*x - 1)/sqrt(23))*x**4 + 861052724448*sqrt(2)*log((2*sqrt(2*x* 
*2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**3 - 2798421354456*sqrt(2)*log( 
(2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**2 + 1291579086672* 
sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x - 19373 
68630008*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)) 
- 203665714820*sqrt(2)*x**4 + 203665714820*sqrt(2)*x**3 - 661913573165*sqr 
t(2)*x**2 + 305498572230*sqrt(2)*x - 458247858345*sqrt(2))/(208011264*(4*x 
**4 - 4*x**3 + 13*x**2 - 6*x + 9))