\(\int (5-x) (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2} \, dx\) [1033]

Optimal result

Integrand size = 29, antiderivative size = 224 \[ \int (5-x) (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {33629 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{56133}-\frac {2297 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}{6237}-\frac {145}{567} (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}+\frac {1}{297} (178-27 x) (3+2 x)^{7/2} \sqrt {2+5 x+3 x^2}-\frac {32567 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{16038 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {168145 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{112266 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

-33629/56133*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)-2297/6237*(3+2*x)^(3/2)*(3* 
x^2+5*x+2)^(1/2)-145/567*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(1/2)+1/297*(178-27*x 
)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^(1/2)-32567/48114*(-3*x^2-5*x-2)^(1/2)*Ellip 
ticE((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/2)+168145 
/336798*(-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 = 31.45 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.93 \[ \int (5-x) (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-11846900-64194200 x-137602437 x^2-147414969 x^3-80563032 x^4-18348768 x^5+789264 x^6+734832 x^7\right )+227969 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-127082 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{336798 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/336798*(2*Sqrt[3 + 2*x]*(-11846900 - 64194200*x - 137602437*x^2 - 14741 
4969*x^3 - 80563032*x^4 - 18348768*x^5 + 789264*x^6 + 734832*x^7) + 227969 
*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Ell 
ipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 127082*Sqrt[5]*Sqrt[(1 + x) 
/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/ 
3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1236, 27, 1236, 27, 1236, 27, 1231, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{33} \int \frac {5}{2} (2 x+3)^{3/2} (73 x+112) \sqrt {3 x^2+5 x+2}dx-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{33} \int (2 x+3)^{3/2} (73 x+112) \sqrt {3 x^2+5 x+2}dx-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {5}{33} \left (\frac {2}{27} \int \frac {3}{2} \sqrt {2 x+3} (1213 x+1637) \sqrt {3 x^2+5 x+2}dx+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \int \sqrt {2 x+3} (1213 x+1637) \sqrt {3 x^2+5 x+2}dx+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {2}{21} \int \frac {(31151 x+43694) \sqrt {3 x^2+5 x+2}}{2 \sqrt {2 x+3}}dx+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \int \frac {(31151 x+43694) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \left (\frac {1}{45} \sqrt {2 x+3} (280359 x+250447) \sqrt {3 x^2+5 x+2}-\frac {1}{90} \int \frac {227969 x+257881}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \left (\frac {1}{90} \left (\frac {168145}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {227969}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (280359 x+250447)\right )+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \left (\frac {1}{90} \left (\frac {168145 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {227969 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (280359 x+250447)\right )+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \left (\frac {1}{90} \left (\frac {168145 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {227969 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (280359 x+250447)\right )+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \left (\frac {1}{90} \left (\frac {168145 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {227969 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (280359 x+250447)\right )+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {5}{33} \left (\frac {1}{9} \left (\frac {1}{21} \left (\frac {1}{90} \left (\frac {168145 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {227969 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (280359 x+250447)\right )+\frac {2426}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {146}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2}{33} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}\)

Input:

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

Output:

(-2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2))/33 + (5*((146*(3 + 2*x)^(3/2) 
*(2 + 5*x + 3*x^2)^(3/2))/27 + ((2426*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2 
))/21 + ((Sqrt[3 + 2*x]*(250447 + 280359*x)*Sqrt[2 + 5*x + 3*x^2])/45 + (( 
-227969*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]) + (168145*Sqrt[-2 - 5*x - 3*x^2]*Ellipt 
icF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))/9 
0)/21)/9))/33
 

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 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 1231

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

rule 1236

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

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 = 1.86 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-22044960 x^{7}-23677920 x^{6}[phone] x^{5}+89736 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )+227969 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{30311820 x^{3}[phone] x^{2}[phone] x [phone]}\)	\(156\)
risch	\(-\frac {\left (40824 x^{4}-85428 x^{3}-878130 x^{2}-1465281 x -683491\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{56133}-\frac {\left (\frac {257881 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{1683990 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {32567 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{240570 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(213\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {8 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{11}+\frac {452 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{297}+\frac {8870 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{567}+\frac {162809 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{6237}+\frac {683491 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{56133}+\frac {257881 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{1683990 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {32567 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{240570 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(275\)

Input:

int((5-x)*(2*x+3)^(5/2)*(3*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/5051970*(2*x+3)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-22044960*x^7-23677920*x^6+55 
0463040*x^5+89736*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*El 
lipticF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))+227969*(-30*x-20)^(1/2)*(3+3*x) 
^(1/2)*15^(1/2)*(2*x+3)^(1/2)*EllipticE(1/5*(-30*x-20)^(1/2),1/2*10^(1/2)) 
[phone]*x^4+4422449070*x^3+4148590320*x^2+1960021350*x[phone])/(6*x 
^3+19*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.30 \[ \int (5-x) (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{56133} \, {\left (40824 \, x^{4} - 85428 \, x^{3} - 878130 \, x^{2} - 1465281 \, x - 683491\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} - \frac {310447}{6062364} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {32567}{48114} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \] Input:

integrate((5-x)*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")
 

Output:

-1/56133*(40824*x^4 - 85428*x^3 - 878130*x^2 - 1465281*x - 683491)*sqrt(3* 
x^2 + 5*x + 2)*sqrt(2*x + 3) - 310447/6062364*sqrt(6)*weierstrassPInverse( 
19/27, -28/729, x + 19/18) + 32567/48114*sqrt(6)*weierstrassZeta(19/27, -2 
8/729, weierstrassPInverse(19/27, -28/729, x + 19/18))
 

Sympy [F]

\[ \int (5-x) (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2} \, dx=- \int \left (- 45 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 51 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 8 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 4 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \] Input:

integrate((5-x)*(3+2*x)**(5/2)*(3*x**2+5*x+2)**(1/2),x)
 

Output:

-Integral(-45*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-51*x*sq 
rt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2*sqrt(2*x + 3)*sq 
rt(3*x**2 + 5*x + 2), x) - Integral(4*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x 
 + 2), x)
 

Maxima [F]

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

integrate((5-x)*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((5-x)*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(-(2*x + 3)^(5/2)*(x - 5)*(5*x + 3*x^2 + 2)^(1/2),x)
 

Output:

-int((2*x + 3)^(5/2)*(x - 5)*(5*x + 3*x^2 + 2)^(1/2), x)
 

Reduce [F]

\[ \int (5-x) (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {8 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{4}}{11}+\frac {452 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{297}+\frac {8870 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{567}+\frac {162809 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{6237}+\frac {953507 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{79002}+\frac {32567 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{33858}-\frac {323 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{252} \] Input:

int((5-x)*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(1/2),x)
 

Output:

( - 344736*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**4 + 721392*sqrt(2*x + 3 
)*sqrt(3*x**2 + 5*x + 2)*x**3 + 7415320*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 
2)*x**2 + 12373484*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x + 5721042*sqrt(2 
*x + 3)*sqrt(3*x**2 + 5*x + 2) + 455938*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5 
*x + 2)*x**2)/(6*x**3 + 19*x**2 + 19*x + 6),x) - 607563*int((sqrt(2*x + 3) 
*sqrt(3*x**2 + 5*x + 2))/(6*x**3 + 19*x**2 + 19*x + 6),x))/474012