3.26.93 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^{13/2}} \, dx\) [2593]

3.26.93.1 Optimal result

Integrand size = 29, antiderivative size = 229 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=\frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac {5861 \sqrt {2+5 x+3 x^2}}{618750 \sqrt {3+2 x}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {5861 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{412500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {14807 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{577500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

output

1/495*(258+367*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2)-5861/1237500*Elliptic 
E(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5 
*x+2)^(1/2)+14807/1732500*EllipticF(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3 
*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+14807/866250*(3*x^2+5*x+2)^( 
1/2)/(3+2*x)^(3/2)-1/57750*(15647+14773*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(7/ 
2)+5861/618750*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)
 

3.26.93.2 Mathematica [A] (verified)

Time = 31.39 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=-\frac {-4 \left (2+5 x+3 x^2\right ) \left (9919671+42879355 x+65139670 x^2+41848650 x^3+11031040 x^4+1312864 x^5\right )+2 (3+2 x)^5 \left (82054 \left (2+5 x+3 x^2\right )+41027 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+3394 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{17325000 (3+2 x)^{11/2} \sqrt {2+5 x+3 x^2}} \]

input

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(13/2),x]
 

output

-1/17325000*(-4*(2 + 5*x + 3*x^2)*(9919671 + 42879355*x + 65139670*x^2 + 4 
1848650*x^3 + 11031040*x^4 + 1312864*x^5) + 2*(3 + 2*x)^5*(82054*(2 + 5*x 
+ 3*x^2) + 41027*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 
 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 3394*Sq 
rt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*El 
lipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/((3 + 2*x)^(11/2)*Sqrt[2 + 
 5*x + 3*x^2])
 

3.26.93.3 Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1229, 25, 1229, 25, 1237, 27, 1237, 27, 1269, 1172, 27, 321, 327}

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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{13/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}-\frac {1}{330} \int -\frac {(303 x+194) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{9/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{330} \int \frac {(303 x+194) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{9/2}}dx+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{330} \left (-\frac {1}{350} \int -\frac {13059 x+12185}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (14773 x+15647)}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \int \frac {13059 x+12185}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac {2}{15} \int -\frac {44421 x+46118}{2 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \int \frac {44421 x+46118}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {2}{5} \int \frac {3 (41027 x+24523)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \int \frac {41027 x+24523}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {41027}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {74035}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {41027 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {74035 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {41027 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {74035 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {41027 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {74035 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{330} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {82054 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {41027 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {74035 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {29614 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}\)

input

Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(13/2),x]
 

output

((258 + 367*x)*(2 + 5*x + 3*x^2)^(3/2))/(495*(3 + 2*x)^(11/2)) + (-1/175*( 
(15647 + 14773*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(7/2) + ((29614*Sqrt[2 
+ 5*x + 3*x^2])/(15*(3 + 2*x)^(3/2)) + ((82054*Sqrt[2 + 5*x + 3*x^2])/(5*S 
qrt[3 + 2*x]) - (3*((41027*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3] 
*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (74035*Sqrt[-2 - 5 
*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 
+ 5*x + 3*x^2])))/5)/15)/350)/330
 

3.26.93.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(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] + Simp[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*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(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] &&  !IGtQ[m, 0]
 

3.26.93.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.39

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5632 \left (x +\frac {3}{2}\right )^{6}}+\frac {1303 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{25344 \left (x +\frac {3}{2}\right )^{5}}-\frac {2701 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{40320 \left (x +\frac {3}{2}\right )^{4}}+\frac {34679 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1848000 \left (x +\frac {3}{2}\right )^{3}}+\frac {14807 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{3465000 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {5861}{206250} x^{2}+\frac {5861}{123750} x +\frac {5861}{309375}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {24523 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{43312500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {5861 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{6187500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(318\)
default	\(\frac {1312864 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}-1584384 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}+9846480 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-11882880 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-35648640 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-53472960 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}[phone] x^{7}[phone] \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-40104720 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}[phone] x^{6}+9969561 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-12031416 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{129937500 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {11}{2}}}\)	\(575\)

input

int((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x,method=_RETURNVERBOSE)
 

output

((3+2*x)*(3*x^2+5*x+2))^(1/2)/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-65/5632* 
(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^6+1303/25344*(6*x^3+19*x^2+19*x+6)^(1/ 
2)/(x+3/2)^5-2701/40320*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4+34679/184800 
0*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3+14807/3465000*(6*x^3+19*x^2+19*x+6 
)^(1/2)/(x+3/2)^2+5861/1237500*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/ 
2)+24523/43312500*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19 
*x^2+19*x+6)^(1/2)*EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))+5861/61875 
00*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1 
/2)*(1/3*EllipticE(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-EllipticF(1/5*(-20-3 
0*x)^(1/2),1/2*10^(1/2))))
 

3.26.93.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=\frac {338099 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 738486 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (1312864 \, x^{5} + 11031040 \, x^{4} + 41848650 \, x^{3} + 65139670 \, x^{2} + 42879355 \, x + 9919671\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{155925000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

input

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="fricas")
 

output

1/155925000*(338099*sqrt(6)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860 
*x^2 + 2916*x + 729)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 7384 
86*sqrt(6)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 7 
29)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x 
+ 19/18)) + 36*(1312864*x^5 + 11031040*x^4 + 41848650*x^3 + 65139670*x^2 + 
 42879355*x + 9919671)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(64*x^6 + 576* 
x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)
 

3.26.93.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=\text {Timed out} \]

input

integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(13/2),x)
 

output

Timed out
 

3.26.93.7 Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {13}{2}}} \,d x } \]

input

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="maxima")
 

output

-integrate((3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(13/2), x)
 

3.26.93.8 Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {13}{2}}} \,d x } \]

input

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="giac")
 

output

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(13/2), x)
 

3.26.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{13/2}} \,d x \]

input

int(-((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^(13/2),x)
 

output

-int(((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^(13/2), x)