\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{17/2}} \, dx\) [1062]

Optimal result

Integrand size = 29, antiderivative size = 261 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=\frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac {335723 \sqrt {2+5 x+3 x^2}}{80437500 \sqrt {3+2 x}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {335723 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{53625000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {594851 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{75075000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

594851/112612500*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)+335723/80437500*(3*x^2+ 
5*x+2)^(1/2)/(3+2*x)^(1/2)-1/7507500*(386846+328339*x)*(3*x^2+5*x+2)^(1/2) 
/(3+2*x)^(7/2)-1/64350*(8901+8399*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2)+1/ 
195*(94+119*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2)-335723/160875000*(-3*x^2 
-5*x-2)^(1/2)*EllipticE((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+ 
5*x+2)^(1/2)+594851/225225000*(-3*x^2-5*x-2)^(1/2)*EllipticF((1+x)^(1/2)*3 
^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 21.90 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=-\frac {-8 \left (2+5 x+3 x^2\right ) \left (4641518352+24502214271 x+55283449932 x^2+67557035830 x^3+46830142120 x^4+17742950508 x^5+3348834304 x^6+300807808 x^7\right )+2 (3+2 x)^7 \left (9400244 \left (2+5 x+3 x^2\right )+4700122 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-1131016 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{4504500000 (3+2 x)^{15/2} \sqrt {2+5 x+3 x^2}} \] Input:

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]
 

Output:

-1/4504500000*(-8*(2 + 5*x + 3*x^2)*(4641518352 + 24502214271*x + 55283449 
932*x^2 + 67557035830*x^3 + 46830142120*x^4 + 17742950508*x^5 + 3348834304 
*x^6 + 300807808*x^7) + 2*(3 + 2*x)^7*(9400244*(2 + 5*x + 3*x^2) + 4700122 
*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] - 1131016*Sqrt[5]*Sqrt[(1 
 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSi 
n[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/((3 + 2*x)^(15/2)*Sqrt[2 + 5*x + 3*x^2] 
)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {1229, 25, 1229, 27, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^{17/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}-\frac {1}{390} \int -\frac {(243 x+118) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{13/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{390} \int \frac {(243 x+118) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{13/2}}dx+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{390} \left (-\frac {1}{330} \int -\frac {3 (7449 x+6947) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{9/2}}dx-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \int \frac {(7449 x+6947) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{9/2}}dx-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (-\frac {1}{350} \int -\frac {477837 x+419330}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (328339 x+386846)}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \int \frac {477837 x+419330}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac {2}{15} \int -\frac {1784553 x+1501799}{2 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \int \frac {1784553 x+1501799}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {2}{5} \int \frac {3 (2350061 x+2037964)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \int \frac {2350061 x+2037964}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2350061}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {2974255}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2350061 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2974255 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2350061 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {2974255 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2350061 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {2974255 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{390} \left (\frac {1}{110} \left (\frac {1}{350} \left (\frac {1}{15} \left (\frac {4700122 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2350061 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2974255 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {1189702 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}\right )-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{165 (2 x+3)^{11/2}}\right )+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}\)

Input:

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]
 

Output:

((94 + 119*x)*(2 + 5*x + 3*x^2)^(5/2))/(195*(3 + 2*x)^(15/2)) + (-1/165*(( 
8901 + 8399*x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(11/2) + (-1/175*((38684 
6 + 328339*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(7/2) + ((1189702*Sqrt[2 + 
5*x + 3*x^2])/(15*(3 + 2*x)^(3/2)) + ((4700122*Sqrt[2 + 5*x + 3*x^2])/(5*S 
qrt[3 + 2*x]) - (3*((2350061*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]) - (2974255*Sqrt[-2 
 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqr 
t[2 + 5*x + 3*x^2])))/5)/15)/350)/110)/390
 

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]
 

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.40

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{24576 \left (x +\frac {3}{2}\right )^{8}}+\frac {3299 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{159744 \left (x +\frac {3}{2}\right )^{7}}-\frac {11439 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{183040 \left (x +\frac {3}{2}\right )^{6}}+\frac {149093 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1647360 \left (x +\frac {3}{2}\right )^{5}}-\frac {636491 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{10483200 \left (x +\frac {3}{2}\right )^{4}}+\frac {6055369 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{480480000 \left (x +\frac {3}{2}\right )^{3}}+\frac {594851 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{450450000 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {335723}{26812500} x^{2}+\frac {335723}{16087500} x +\frac {335723}{40218750}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {509491 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{1407656250 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {335723 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{804375000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(366\)
default	\(-\frac {-278491101120-2166360609060 x -14551138923990 x^{3}-7077085425240 x^{6}-2117239159800 x^{7}-17918874395580 x^{4}-7410075788250 x^{2}-346516258560 x^{8}-27072702720 x^{9}-14169231573180 x^{5}[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-35532922320 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{6} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-5139583407 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-47969445132 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-14213168928 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-23984722566 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{7} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-300807808 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{7} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-3158481984 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{6} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}-53299383480 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}}{16891875000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {15}{2}}}\)	\(761\)

Input:

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^(17/2),x,method=_RETURNVERBOSE)
 

Output:

((3*x^2+5*x+2)*(2*x+3))^(1/2)/(2*x+3)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-65/24576 
*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^8+3299/159744*(6*x^3+19*x^2+19*x+6)^( 
1/2)/(x+3/2)^7-11439/183040*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^6+149093/1 
647360*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^5-636491/10483200*(6*x^3+19*x^2 
+19*x+6)^(1/2)/(x+3/2)^4+6055369/480480000*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+ 
3/2)^3+594851/450450000*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2+335723/16087 
5000*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)+509491/1407656250*(-30* 
x-20)^(1/2)*(3+3*x)^(1/2)*(30*x+45)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*Elli 
pticF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))+335723/804375000*(-30*x-20)^(1/2) 
*(3+3*x)^(1/2)*(30*x+45)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*(1/3*EllipticE( 
1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-EllipticF(1/5*(-30*x-20)^(1/2),1/2*10^( 
1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.79 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=\frac {7967807 \, \sqrt {6} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 42301098 \, \sqrt {6} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (300807808 \, x^{7} + 3348834304 \, x^{6} + 17742950508 \, x^{5} + 46830142120 \, x^{4} + 67557035830 \, x^{3} + 55283449932 \, x^{2} + 24502214271 \, x + 4641518352\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{20270250000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(17/2),x, algorithm="fricas")
 

Output:

1/20270250000*(7967807*sqrt(6)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 
 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*weierstrassPInvers 
e(19/27, -28/729, x + 19/18) + 42301098*sqrt(6)*(256*x^8 + 3072*x^7 + 1612 
8*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*w 
eierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/ 
18)) + 36*(300807808*x^7 + 3348834304*x^6 + 17742950508*x^5 + 46830142120* 
x^4 + 67557035830*x^3 + 55283449932*x^2 + 24502214271*x + 4641518352)*sqrt 
(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x 
^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(17/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {17}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(17/2),x, algorithm="maxima")
 

Output:

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)
 

Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {17}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(17/2),x, algorithm="giac")
 

Output:

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{17/2}} \,d x \] Input:

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(17/2),x)
                                                                                    
                                                                                    
 

Output:

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(17/2), x)
 

Reduce [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx=\text {too large to display} \] Input:

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(17/2),x)
 

Output:

(91309680*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**5 + 206968608*sqrt(2*x + 
 3)*sqrt(3*x**2 + 5*x + 2)*x**4 - 110876040*sqrt(2*x + 3)*sqrt(3*x**2 + 5* 
x + 2)*x**3 - 700749456*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 - 711978 
580*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x - 242697442*sqrt(2*x + 3)*sqrt( 
3*x**2 + 5*x + 2) - 7963676160*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x 
**2)/(1536*x**11 + 23296*x**10 + 160000*x**9 + 656640*x**8 + 1788480*x**7 
+ 3392928*x**6 + 4572288*x**5 + 4374000*x**4 + 2908710*x**3 + 1279395*x**2 
 + 334611*x + 39366),x)*x**8 - 95564113920*int((sqrt(2*x + 3)*sqrt(3*x**2 
+ 5*x + 2)*x**2)/(1536*x**11 + 23296*x**10 + 160000*x**9 + 656640*x**8 + 1 
788480*x**7 + 3392928*x**6 + 4572288*x**5 + 4374000*x**4 + 2908710*x**3 + 
1279395*x**2 + 334611*x + 39366),x)*x**7 - 501711598080*int((sqrt(2*x + 3) 
*sqrt(3*x**2 + 5*x + 2)*x**2)/(1536*x**11 + 23296*x**10 + 160000*x**9 + 65 
6640*x**8 + 1788480*x**7 + 3392928*x**6 + 4572288*x**5 + 4374000*x**4 + 29 
08710*x**3 + 1279395*x**2 + 334611*x + 39366),x)*x**6 - 1505134794240*int( 
(sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(1536*x**11 + 23296*x**10 + 16 
0000*x**9 + 656640*x**8 + 1788480*x**7 + 3392928*x**6 + 4572288*x**5 + 437 
4000*x**4 + 2908710*x**3 + 1279395*x**2 + 334611*x + 39366),x)*x**5 - 2822 
127739200*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(1536*x**11 + 23 
296*x**10 + 160000*x**9 + 656640*x**8 + 1788480*x**7 + 3392928*x**6 + 4572 
288*x**5 + 4374000*x**4 + 2908710*x**3 + 1279395*x**2 + 334611*x + 3936...