\(\int \frac {(5-2 x)^{3/2} (4+3 x)^3}{\sqrt {2+3 x+x^2}} \, dx\) [560]

Optimal result

Integrand size = 29, antiderivative size = 185 \[ \int \frac {(5-2 x)^{3/2} (4+3 x)^3}{\sqrt {2+3 x+x^2}} \, dx=\frac {15763}{105} \sqrt {5-2 x} \sqrt {2+3 x+x^2}+\frac {657}{35} (5-2 x)^{3/2} \sqrt {2+3 x+x^2}-\frac {237}{14} (5-2 x)^{5/2} \sqrt {2+3 x+x^2}+\frac {3}{2} (5-2 x)^{7/2} \sqrt {2+3 x+x^2}-\frac {131897 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{70 \sqrt {2+3 x+x^2}}+\frac {15763 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{10 \sqrt {2+3 x+x^2}} \] Output:

15763/105*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)+657/35*(5-2*x)^(3/2)*(x^2+3*x+2) 
^(1/2)-237/14*(5-2*x)^(5/2)*(x^2+3*x+2)^(1/2)+3/2*(5-2*x)^(7/2)*(x^2+3*x+2 
)^(1/2)-131897/70*(-x^2-3*x-2)^(1/2)*EllipticE((2+x)^(1/2),1/3*2^(1/2))/(x 
^2+3*x+2)^(1/2)+15763/10*(-x^2-3*x-2)^(1/2)*EllipticF((2+x)^(1/2),1/3*2^(1 
/2))/(x^2+3*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 32.87 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.95 \[ \int \frac {(5-2 x)^{3/2} (4+3 x)^3}{\sqrt {2+3 x+x^2}} \, dx=\frac {-2 \sqrt {5-2 x} \left (272474+485069 x+242242 x^2+3421 x^3-32166 x^4-3420 x^5+2520 x^6\right )-395691 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )+64668 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )}{210 (-5+2 x) \sqrt {2+3 x+x^2}} \] Input:

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

Output:

(-2*Sqrt[5 - 2*x]*(272474 + 485069*x + 242242*x^2 + 3421*x^3 - 32166*x^4 - 
 3420*x^5 + 2520*x^6) - 395691*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[( 
2 + x)/(-5 + 2*x)]*EllipticE[ArcSin[3/Sqrt[5 - 2*x]], 7/9] + 64668*(5 - 2* 
x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticF[ArcSin[3/ 
Sqrt[5 - 2*x]], 7/9])/(210*(-5 + 2*x)*Sqrt[2 + 3*x + x^2])
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1291, 27, 2184, 27, 1236, 27, 1236, 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.

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

\(\Big \downarrow \) 1291

\(\displaystyle \frac {3}{2} (5-2 x)^{7/2} \sqrt {x^2+3 x+2}-\frac {1}{36} \int -\frac {9 (5-2 x)^{3/2} \left (474 x^2+1068 x+451\right )}{\sqrt {x^2+3 x+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {(5-2 x)^{3/2} \left (474 x^2+1068 x+451\right )}{\sqrt {x^2+3 x+2}}dx+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{4} \left (\frac {1}{14} \int \frac {4 (5-2 x)^{3/2} (657 x+986)}{\sqrt {x^2+3 x+2}}dx-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \int \frac {(5-2 x)^{3/2} (657 x+986)}{\sqrt {x^2+3 x+2}}dx-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {2}{5} \int \frac {\sqrt {5-2 x} (15763 x+22679)}{2 \sqrt {x^2+3 x+2}}dx+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \int \frac {\sqrt {5-2 x} (15763 x+22679)}{\sqrt {x^2+3 x+2}}dx+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {131897 x+166792}{2 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {131897 x+166792}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {993069}{2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {131897}{2} \int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx\right )+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {331023 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {395691 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {993069 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {131897 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {331023 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {131897 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{4} \left (\frac {2}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {331023 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {395691 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {31526}{3} \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )+\frac {1314}{5} \sqrt {x^2+3 x+2} (5-2 x)^{3/2}\right )-\frac {474}{7} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )+\frac {3}{2} \sqrt {x^2+3 x+2} (5-2 x)^{7/2}\)

Input:

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

Output:

(3*(5 - 2*x)^(7/2)*Sqrt[2 + 3*x + x^2])/2 + ((-474*(5 - 2*x)^(5/2)*Sqrt[2 
+ 3*x + x^2])/7 + (2*((1314*(5 - 2*x)^(3/2)*Sqrt[2 + 3*x + x^2])/5 + ((315 
26*Sqrt[5 - 2*x]*Sqrt[2 + 3*x + x^2])/3 + ((-395691*Sqrt[-2 - 3*x - x^2]*E 
llipticE[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] + (331023*Sqrt[-2 
- 3*x - x^2]*EllipticF[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2])/3)/ 
5))/7)/4
 

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 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]
 

rule 1291

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && NeQ[m + n + 2*p + 1, 0]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (-10080 x^{6}+13680 x^{5}+9970 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+131897 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+128664 x^{4}-13684 x^{3}-441380 x^{2}-357512 x -34720\right )}{840 x^{3}+420 x^{2}-4620 x -4200}\)	\(141\)
risch	\(\frac {\left (1260 x^{3}-2340 x^{2}-7983 x -868\right ) \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{105 \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}+\frac {\left (-\frac {83396 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2205 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {131897 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{4410 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(239\)
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (\frac {2661 x \sqrt {-2 x^{3}-x^{2}+11 x +10}}{35}+\frac {124 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{15}-\frac {83396 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{735 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {131897 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{1470 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {156 x^{2} \sqrt {-2 x^{3}-x^{2}+11 x +10}}{7}-12 x^{3} \sqrt {-2 x^{3}-x^{2}+11 x +10}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(250\)

Input:

int((5-2*x)^(3/2)*(3*x+4)^3/(x^2+3*x+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/420*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)*(-10080*x^6+13680*x^5+9970*(5-2*x)^( 
1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)*EllipticF(1/3*(5-2*x)^(1/2),3/7*7^(1/2) 
)+131897*(5-2*x)^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)*EllipticE(1/3*(5-2*x) 
^(1/2),3/7*7^(1/2))+128664*x^4-13684*x^3-441380*x^2-357512*x-34720)/(2*x^3 
+x^2-11*x-10)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.32 \[ \int \frac {(5-2 x)^{3/2} (4+3 x)^3}{\sqrt {2+3 x+x^2}} \, dx=-\frac {1}{105} \, {\left (1260 \, x^{3} - 2340 \, x^{2} - 7983 \, x - 868\right )} \sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5} - \frac {173771}{252} \, \sqrt {-2} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) + \frac {131897}{210} \, \sqrt {-2} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) \] Input:

integrate((5-2*x)^(3/2)*(4+3*x)^3/(x^2+3*x+2)^(1/2),x, algorithm="fricas")
 

Output:

-1/105*(1260*x^3 - 2340*x^2 - 7983*x - 868)*sqrt(x^2 + 3*x + 2)*sqrt(-2*x 
+ 5) - 173771/252*sqrt(-2)*weierstrassPInverse(67/3, 440/27, x + 1/6) + 13 
1897/210*sqrt(-2)*weierstrassZeta(67/3, 440/27, weierstrassPInverse(67/3, 
440/27, x + 1/6))
 

Sympy [F]

\[ \int \frac {(5-2 x)^{3/2} (4+3 x)^3}{\sqrt {2+3 x+x^2}} \, dx=\int \frac {\left (5 - 2 x\right )^{\frac {3}{2}} \left (3 x + 4\right )^{3}}{\sqrt {\left (x + 1\right ) \left (x + 2\right )}}\, dx \] Input:

integrate((5-2*x)**(3/2)*(4+3*x)**3/(x**2+3*x+2)**(1/2),x)
 

Output:

Integral((5 - 2*x)**(3/2)*(3*x + 4)**3/sqrt((x + 1)*(x + 2)), x)
 

Maxima [F]

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

integrate((5-2*x)^(3/2)*(4+3*x)^3/(x^2+3*x+2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((5-2*x)^(3/2)*(4+3*x)^3/(x^2+3*x+2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((5 - 2*x)^(3/2)*(3*x + 4)^3)/(3*x + x^2 + 2)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(((5 - 2*x)^(3/2)*(3*x + 4)^3)/(3*x + x^2 + 2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(5-2 x)^{3/2} (4+3 x)^3}{\sqrt {2+3 x+x^2}} \, dx=-12 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{3}+\frac {156 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{7}+\frac {2661 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{35}-\frac {43387 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{70}+\frac {131897 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{70}-\frac {594817 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{140} \] Input:

int((5-2*x)^(3/2)*(4+3*x)^3/(x^2+3*x+2)^(1/2),x)
 

Output:

( - 1680*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**3 + 3120*sqrt( - 2*x + 5 
)*sqrt(x**2 + 3*x + 2)*x**2 + 10644*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)* 
x - 86774*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2) + 263794*int((sqrt( - 2*x 
+ 5)*sqrt(x**2 + 3*x + 2)*x**2)/(2*x**3 + x**2 - 11*x - 10),x) - 594817*in 
t((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(2*x**3 + x**2 - 11*x - 10),x))/ 
140