\(\int \frac {(2+3 x+5 x^2)^3}{(3-x+2 x^2)^{3/2}} \, dx\) [121]

Optimal result

Integrand size = 27, antiderivative size = 124 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx=-\frac {1331 (17-45 x)}{368 \sqrt {3-x+2 x^2}}-\frac {181561 \sqrt {3-x+2 x^2}}{2048}+\frac {15565}{512} x \sqrt {3-x+2 x^2}+\frac {1825}{64} x^2 \sqrt {3-x+2 x^2}+\frac {125}{16} x^3 \sqrt {3-x+2 x^2}+\frac {1168881 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4096 \sqrt {2}} \] Output:

1/368*(-22627+59895*x)/(2*x^2-x+3)^(1/2)-181561/2048*(2*x^2-x+3)^(1/2)+155 
65/512*x*(2*x^2-x+3)^(1/2)+1825/64*x^2*(2*x^2-x+3)^(1/2)+125/16*x^3*(2*x^2 
-x+3)^(1/2)+1168881/8192*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {\frac {4 \left (-15423965+16138403 x-5754186 x^2+2624760 x^3+2318400 x^4+736000 x^5\right )}{\sqrt {3-x+2 x^2}}[phone] \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{188416} \] Input:

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

Output:

((4*(-15423965 + 16138403*x - 5754186*x^2 + 2624760*x^3 + 2318400*x^4 + 73 
6000*x^5))/Sqrt[3 - x + 2*x^2] + 26884263*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 
 2*x + 4*x^2]])/188416
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {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.

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

\(\Big \downarrow \) 2191

\(\displaystyle \frac {2}{23} \int -\frac {23 \left (-2000 x^4-4600 x^3-3860 x^2+1658 x+4795\right )}{64 \sqrt {2 x^2-x+3}}dx-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{32} \int \frac {-2000 x^4-4600 x^3-3860 x^2+1658 x+4795}{\sqrt {2 x^2-x+3}}dx-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (250 x^3 \sqrt {2 x^2-x+3}-\frac {1}{8} \int \frac {8 \left (-5475 x^3-1610 x^2+1658 x+4795\right )}{\sqrt {2 x^2-x+3}}dx\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (250 x^3 \sqrt {2 x^2-x+3}-\int \frac {-5475 x^3-1610 x^2+1658 x+4795}{\sqrt {2 x^2-x+3}}dx\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (-\frac {1}{6} \int \frac {3 \left (-15565 x^2+28532 x+19180\right )}{2 \sqrt {2 x^2-x+3}}dx+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (-\frac {1}{4} \int \frac {-15565 x^2+28532 x+19180}{\sqrt {2 x^2-x+3}}dx+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {15565}{4} x \sqrt {2 x^2-x+3}-\frac {1}{4} \int \frac {181561 x+246830}{2 \sqrt {2 x^2-x+3}}dx\right )+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {15565}{4} x \sqrt {2 x^2-x+3}-\frac {1}{8} \int \frac {181561 x+246830}{\sqrt {2 x^2-x+3}}dx\right )+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{8} \left (-\frac {1168881}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {181561}{2} \sqrt {2 x^2-x+3}\right )+\frac {15565}{4} \sqrt {2 x^2-x+3} x\right )+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{8} \left (-\frac {1168881 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}-\frac {181561}{2} \sqrt {2 x^2-x+3}\right )+\frac {15565}{4} \sqrt {2 x^2-x+3} x\right )+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{8} \left (-\frac {1168881 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}-\frac {181561}{2} \sqrt {2 x^2-x+3}\right )+\frac {15565}{4} \sqrt {2 x^2-x+3} x\right )+\frac {1825}{2} \sqrt {2 x^2-x+3} x^2+250 \sqrt {2 x^2-x+3} x^3\right )-\frac {1331 (17-45 x)}{368 \sqrt {2 x^2-x+3}}\)

Input:

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

Output:

(-1331*(17 - 45*x))/(368*Sqrt[3 - x + 2*x^2]) + ((1825*x^2*Sqrt[3 - x + 2* 
x^2])/2 + 250*x^3*Sqrt[3 - x + 2*x^2] + ((15565*x*Sqrt[3 - x + 2*x^2])/4 + 
 ((-181561*Sqrt[3 - x + 2*x^2])/2 - (1168881*ArcSinh[(-1 + 4*x)/Sqrt[23]]) 
/(4*Sqrt[2]))/8)/4)/32
 

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 = 2.67 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.44

method	result	size
risch	\(\frac {736000 x^{5}+2318400 x^{4}+2624760 x^{3}-5754186 x^{2}[phone] x -15423965}{47104 \sqrt {2 x^{2}-x +3}}-\frac {1168881 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8192}\)	\(55\)
trager	\(\frac {736000 x^{5}+2318400 x^{4}+2624760 x^{3}-5754186 x^{2}[phone] x -15423965}{47104 \sqrt {2 x^{2}-x +3}}-\frac {1168881 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{8192}\)	\(82\)
default	\(\frac {\frac {5392543 x}{94208}-\frac {5392543}{376832}}{\sqrt {2 x^{2}-x +3}}-\frac {5130399}{16384 \sqrt {2 x^{2}-x +3}}+\frac {1168881 x}{4096 \sqrt {2 x^{2}-x +3}}-\frac {1168881 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8192}-\frac {125091 x^{2}}{1024 \sqrt {2 x^{2}-x +3}}+\frac {14265 x^{3}}{256 \sqrt {2 x^{2}-x +3}}+\frac {1575 x^{4}}{32 \sqrt {2 x^{2}-x +3}}+\frac {125 x^{5}}{8 \sqrt {2 x^{2}-x +3}}\)	\(132\)

Input:

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

Output:

1/47104*(736000*x^5+2318400*x^4+2624760*x^3-5754186*x^2+16138403*x-1542396 
5)/(2*x^2-x+3)^(1/2)-1168881/8192*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {26884263 \, \sqrt {2} {\left (2 \, x^{2} - x + 3\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (736000 \, x^{5} + 2318400 \, x^{4} + 2624760 \, x^{3} - 5754186 \, x^{2} + 16138403 \, x - 15423965\right )} \sqrt {2 \, x^{2} - x + 3}}{376832 \, {\left (2 \, x^{2} - x + 3\right )}} \] Input:

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

Output:

1/376832*(26884263*sqrt(2)*(2*x^2 - x + 3)*log(4*sqrt(2)*sqrt(2*x^2 - x + 
3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 8*(736000*x^5 + 2318400*x^4 + 2624760 
*x^3 - 5754186*x^2 + 16138403*x - 15423965)*sqrt(2*x^2 - x + 3))/(2*x^2 - 
x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {125 \, x^{5}}{8 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {1575 \, x^{4}}{32 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {14265 \, x^{3}}{256 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {125091 \, x^{2}}{1024 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {1168881}{8192} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {16138403 \, x}{47104 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {15423965}{47104 \, \sqrt {2 \, x^{2} - x + 3}} \] Input:

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

Output:

125/8*x^5/sqrt(2*x^2 - x + 3) + 1575/32*x^4/sqrt(2*x^2 - x + 3) + 14265/25 
6*x^3/sqrt(2*x^2 - x + 3) - 125091/1024*x^2/sqrt(2*x^2 - x + 3) - 1168881/ 
8192*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 16138403/47104*x/sqrt(2*x^ 
2 - x + 3) - 15423965/47104/sqrt(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.58 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {1168881}{8192} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {{\left (46 \, {\left (20 \, {\left (40 \, {\left (20 \, x + 63\right )} x + 2853\right )} x - 125091\right )} x + 16138403\right )} x - 15423965}{47104 \, \sqrt {2 \, x^{2} - x + 3}} \] Input:

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

Output:

1168881/8192*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) 
 + 1/47104*((46*(20*(40*(20*x + 63)*x + 2853)*x - 125091)*x + 16138403)*x 
- 15423965)/sqrt(2*x^2 - x + 3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.73 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {2944000 \sqrt {2 x^{2}-x +3}\, x^{5}+9273600 \sqrt {2 x^{2}-x +3}\, x^{4}[phone] \sqrt {2 x^{2}-x +3}\, x^{3}-23016744 \sqrt {2 x^{2}-x +3}\, x^{2}[phone] \sqrt {2 x^{2}-x +3}\, x -61695860 \sqrt {2 x^{2}-x +3}-53768526 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{2}[phone] \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x -80652789 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )[phone] \sqrt {2}\, x^{2}-18075985 \sqrt {2}\, x [phone] \sqrt {2}}{376832 x^{2}-188416 x +565248} \] Input:

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

Output:

(2944000*sqrt(2*x**2 - x + 3)*x**5 + 9273600*sqrt(2*x**2 - x + 3)*x**4 + 1 
0499040*sqrt(2*x**2 - x + 3)*x**3 - 23016744*sqrt(2*x**2 - x + 3)*x**2 + 6 
4553612*sqrt(2*x**2 - x + 3)*x - 61695860*sqrt(2*x**2 - x + 3) - 53768526* 
sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**2 + 26 
884263*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x 
- 80652789*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23) 
) + 36151970*sqrt(2)*x**2 - 18075985*sqrt(2)*x + 54227955*sqrt(2))/(188416 
*(2*x**2 - x + 3))