\(\int \frac {(4+3 x)^{3/2} (2+3 x+x^2)^2}{(5-2 x)^{9/2}} \, dx\) [231]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=-\frac {23113 \sqrt {4+3 x}}{384 \sqrt {5-2 x}}-\frac {407}{256} \sqrt {5-2 x} \sqrt {4+3 x}+\frac {191 (4+3 x)^{3/2}}{24 (5-2 x)^{3/2}}+\frac {567 (4+3 x)^{5/2}}{184 (5-2 x)^{7/2}}-\frac {21483 (4+3 x)^{5/2}}{10580 (5-2 x)^{5/2}}+\frac {(4+3 x)^{5/2}}{96 \sqrt {5-2 x}}+\frac {18529}{256} \sqrt {\frac {3}{2}} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right ) \] Output:

-23113/384*(4+3*x)^(1/2)/(5-2*x)^(1/2)-407/256*(5-2*x)^(1/2)*(4+3*x)^(1/2) 
+191/24*(4+3*x)^(3/2)/(5-2*x)^(3/2)+567/184*(4+3*x)^(5/2)/(5-2*x)^(7/2)-21 
483/10580*(4+3*x)^(5/2)/(5-2*x)^(5/2)+1/96*(4+3*x)^(5/2)/(5-2*x)^(1/2)+185 
29/512*arcsin(1/23*46^(1/2)*(4+3*x)^(1/2))*6^(1/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.50 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=-\frac {\sqrt {4+3 x} \left (15872816875-21288830852 x+9948453832 x^2-1762305408 x^3+44309040 x^4+1523520 x^5\right )}{2031360 (5-2 x)^{7/2}}-\frac {18529}{256} \sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {15}{2}-3 x}}{\sqrt {4+3 x}}\right ) \] Input:

Integrate[((4 + 3*x)^(3/2)*(2 + 3*x + x^2)^2)/(5 - 2*x)^(9/2),x]
 

Output:

-1/2031360*(Sqrt[4 + 3*x]*(15872816875 - 21288830852*x + 9948453832*x^2 - 
1762305408*x^3 + 44309040*x^4 + 1523520*x^5))/(5 - 2*x)^(7/2) - (18529*Sqr 
t[3/2]*ArcTan[Sqrt[15/2 - 3*x]/Sqrt[4 + 3*x]])/256
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1193, 27, 2124, 27, 1193, 27, 87, 60, 60, 64, 223}

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.

\(\displaystyle \int \frac {(3 x+4)^{3/2} \left (x^2+3 x+2\right )^2}{(5-2 x)^{9/2}} \, dx\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {2}{161} \int -\frac {7 (3 x+4)^{3/2} \left (184 x^3+1564 x^2+6302 x+14561\right )}{32 (5-2 x)^{7/2}}dx+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}-\frac {1}{368} \int \frac {(3 x+4)^{3/2} \left (184 x^3+1564 x^2+6302 x+14561\right )}{(5-2 x)^{7/2}}dx\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {1}{368} \left (-\frac {2}{115} \int -\frac {2645 (3 x+4)^{3/2} \left (4 x^2+44 x+247\right )}{2 (5-2 x)^{5/2}}dx-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{368} \left (23 \int \frac {(3 x+4)^{3/2} \left (4 x^2+44 x+247\right )}{(5-2 x)^{5/2}}dx-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {2}{69} \int -\frac {3 (3 x+4)^{3/2} (46 x+1385)}{2 (5-2 x)^{3/2}}dx+\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}-\frac {1}{23} \int \frac {(3 x+4)^{3/2} (46 x+1385)}{(5-2 x)^{3/2}}dx\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {1}{23} \left (\frac {18529}{23} \int \frac {(3 x+4)^{3/2}}{\sqrt {5-2 x}}dx-\frac {3000 (3 x+4)^{5/2}}{23 \sqrt {5-2 x}}\right )+\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {1}{23} \left (\frac {18529}{23} \left (\frac {69}{8} \int \frac {\sqrt {3 x+4}}{\sqrt {5-2 x}}dx-\frac {1}{4} \sqrt {5-2 x} (3 x+4)^{3/2}\right )-\frac {3000 (3 x+4)^{5/2}}{23 \sqrt {5-2 x}}\right )+\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {1}{23} \left (\frac {18529}{23} \left (\frac {69}{8} \left (\frac {23}{4} \int \frac {1}{\sqrt {5-2 x} \sqrt {3 x+4}}dx-\frac {1}{2} \sqrt {5-2 x} \sqrt {3 x+4}\right )-\frac {1}{4} \sqrt {5-2 x} (3 x+4)^{3/2}\right )-\frac {3000 (3 x+4)^{5/2}}{23 \sqrt {5-2 x}}\right )+\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {1}{23} \left (\frac {18529}{23} \left (\frac {69}{8} \left (\frac {23}{6} \int \frac {1}{\sqrt {\frac {23}{3}-\frac {2}{3} (3 x+4)}}d\sqrt {3 x+4}-\frac {1}{2} \sqrt {5-2 x} \sqrt {3 x+4}\right )-\frac {1}{4} \sqrt {5-2 x} (3 x+4)^{3/2}\right )-\frac {3000 (3 x+4)^{5/2}}{23 \sqrt {5-2 x}}\right )+\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{368} \left (23 \left (\frac {1}{23} \left (\frac {18529}{23} \left (\frac {69}{8} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{2 \sqrt {6}}-\frac {1}{2} \sqrt {5-2 x} \sqrt {3 x+4}\right )-\frac {1}{4} \sqrt {5-2 x} (3 x+4)^{3/2}\right )-\frac {3000 (3 x+4)^{5/2}}{23 \sqrt {5-2 x}}\right )+\frac {764 (3 x+4)^{5/2}}{69 (5-2 x)^{3/2}}\right )-\frac {85932 (3 x+4)^{5/2}}{115 (5-2 x)^{5/2}}\right )+\frac {567 (3 x+4)^{5/2}}{184 (5-2 x)^{7/2}}\)

Input:

Int[((4 + 3*x)^(3/2)*(2 + 3*x + x^2)^2)/(5 - 2*x)^(9/2),x]
 

Output:

(567*(4 + 3*x)^(5/2))/(184*(5 - 2*x)^(7/2)) + ((-85932*(4 + 3*x)^(5/2))/(1 
15*(5 - 2*x)^(5/2)) + 23*((764*(4 + 3*x)^(5/2))/(69*(5 - 2*x)^(3/2)) + ((- 
3000*(4 + 3*x)^(5/2))/(23*Sqrt[5 - 2*x]) + (18529*(-1/4*(Sqrt[5 - 2*x]*(4 
+ 3*x)^(3/2)) + (69*(-1/2*(Sqrt[5 - 2*x]*Sqrt[4 + 3*x]) + (23*ArcSin[Sqrt[ 
2/23]*Sqrt[4 + 3*x]])/(2*Sqrt[6])))/8))/23)/23))/368
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\left (2352441840 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right ) x^{4}-6094080 x^{5} \sqrt {-6 x^{2}+7 x +20}-23524418400 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right ) x^{3}-177236160 x^{4} \sqrt {-6 x^{2}+7 x +20}[phone] \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right ) x^{2}[phone] x^{3} \sqrt {-6 x^{2}+7 x +20}-147027615000 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right ) x -39793815328 x^{2} \sqrt {-6 x^{2}+7 x +20}[phone] \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )[phone] x \sqrt {-6 x^{2}+7 x +20}-63491267500 \sqrt {-6 x^{2}+7 x +20}\right ) \sqrt {3 x +4}}{8125440 \left (5-2 x \right )^{\frac {7}{2}} \sqrt {-6 x^{2}+7 x +20}}\)	\(192\)

Input:

int((3*x+4)^(3/2)*(x^2+3*x+2)^2/(5-2*x)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

1/8125440*(2352441840*6^(1/2)*arcsin(12/23*x-7/23)*x^4-6094080*x^5*(-6*x^2 
+7*x+20)^(1/2)-23524418400*6^(1/2)*arcsin(12/23*x-7/23)*x^3-177236160*x^4* 
(-6*x^2+7*x+20)^(1/2)[phone]*6^(1/2)*arcsin(12/23*x-7/23)*x^2+7049221 
632*x^3*(-6*x^2+7*x+20)^(1/2)-147027615000*6^(1/2)*arcsin(12/23*x-7/23)*x- 
39793815328*x^2*(-6*x^2+7*x+20)^(1/2)[phone]*6^(1/2)*arcsin(12/23*x-7 
/23)[phone]*x*(-6*x^2+7*x+20)^(1/2)-63491267500*(-6*x^2+7*x+20)^(1/2) 
)*(3*x+4)^(1/2)/(5-2*x)^(7/2)/(-6*x^2+7*x+20)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.79 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=-\frac {147027615 \, \sqrt {\frac {3}{2}} {\left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )} \arctan \left (\frac {\sqrt {\frac {3}{2}} {\left (12 \, x - 7\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{6 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + 2 \, {\left (1523520 \, x^{5} + 44309040 \, x^{4} - 1762305408 \, x^{3} + 9948453832 \, x^{2} - 21288830852 \, x + 15872816875\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{4062720 \, {\left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )}} \] Input:

integrate((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x)^(9/2),x, algorithm="fricas")
 

Output:

-1/4062720*(147027615*sqrt(3/2)*(16*x^4 - 160*x^3 + 600*x^2 - 1000*x + 625 
)*arctan(1/6*sqrt(3/2)*(12*x - 7)*sqrt(3*x + 4)*sqrt(-2*x + 5)/(6*x^2 - 7* 
x - 20)) + 2*(1523520*x^5 + 44309040*x^4 - 1762305408*x^3 + 9948453832*x^2 
 - 21288830852*x + 15872816875)*sqrt(3*x + 4)*sqrt(-2*x + 5))/(16*x^4 - 16 
0*x^3 + 600*x^2 - 1000*x + 625)
 

Sympy [F]

\[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=\int \frac {\left (x + 1\right )^{2} \left (x + 2\right )^{2} \left (3 x + 4\right )^{\frac {3}{2}}}{\left (5 - 2 x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((4+3*x)**(3/2)*(x**2+3*x+2)**2/(5-2*x)**(9/2),x)
 

Output:

Integral((x + 1)**2*(x + 2)**2*(3*x + 4)**(3/2)/(5 - 2*x)**(9/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (115) = 230\).

Time = 0.13 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.83 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=\frac {18529}{1024} \, \sqrt {3} \sqrt {2} \arcsin \left (\frac {12}{23} \, x - \frac {7}{23}\right ) - \frac {69}{256} \, \sqrt {-6 \, x^{2} + 7 \, x + 20} - \frac {3969 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}}{64 \, {\left (32 \, x^{5} - 400 \, x^{4} + 2000 \, x^{3} - 5000 \, x^{2} + 6250 \, x - 3125\right )}} - \frac {63 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}}{16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625} - \frac {191 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}}{48 \, {\left (8 \, x^{3} - 60 \, x^{2} + 150 \, x - 125\right )}} + \frac {{\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}}{4 \, x^{2} - 20 \, x + 25} + \frac {{\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}}{64 \, {\left (2 \, x - 5\right )}} - \frac {39123 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}}{128 \, {\left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )}} - \frac {283311 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}}{640 \, {\left (8 \, x^{3} - 60 \, x^{2} + 150 \, x - 125\right )}} + \frac {38981 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}}{1380 \, {\left (4 \, x^{2} - 20 \, x + 25\right )}} + \frac {6843139 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}}{84640 \, {\left (2 \, x - 5\right )}} \] Input:

integrate((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x)^(9/2),x, algorithm="maxima")
 

Output:

18529/1024*sqrt(3)*sqrt(2)*arcsin(12/23*x - 7/23) - 69/256*sqrt(-6*x^2 + 7 
*x + 20) - 3969/64*(-6*x^2 + 7*x + 20)^(3/2)/(32*x^5 - 400*x^4 + 2000*x^3 
- 5000*x^2 + 6250*x - 3125) - 63*(-6*x^2 + 7*x + 20)^(3/2)/(16*x^4 - 160*x 
^3 + 600*x^2 - 1000*x + 625) - 191/48*(-6*x^2 + 7*x + 20)^(3/2)/(8*x^3 - 6 
0*x^2 + 150*x - 125) + (-6*x^2 + 7*x + 20)^(3/2)/(4*x^2 - 20*x + 25) + 1/6 
4*(-6*x^2 + 7*x + 20)^(3/2)/(2*x - 5) - 39123/128*sqrt(-6*x^2 + 7*x + 20)/ 
(16*x^4 - 160*x^3 + 600*x^2 - 1000*x + 625) - 283311/640*sqrt(-6*x^2 + 7*x 
 + 20)/(8*x^3 - 60*x^2 + 150*x - 125) + 38981/1380*sqrt(-6*x^2 + 7*x + 20) 
/(4*x^2 - 20*x + 25) + 6843139/84640*sqrt(-6*x^2 + 7*x + 20)/(2*x - 5)
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.68 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=\frac {18529}{512} \, \sqrt {6} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right ) - \frac {{\left (4 \, {\left (2 \, {\left (2 \, {\left (2645 \, {\left (4 \, \sqrt {3} {\left (3 \, x + 4\right )} + 269 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} - 123220968 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} + 6537822059 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} - 129629347225 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} + 1788884991705 \, \sqrt {3}\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15}}{164540160 \, {\left (2 \, x - 5\right )}^{4}} \] Input:

integrate((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x)^(9/2),x, algorithm="giac")
 

Output:

18529/512*sqrt(6)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4)) - 1/164540160*(4*(2* 
(2*(2645*(4*sqrt(3)*(3*x + 4) + 269*sqrt(3))*(3*x + 4) - 123220968*sqrt(3) 
)*(3*x + 4) + 6537822059*sqrt(3))*(3*x + 4) - 129629347225*sqrt(3))*(3*x + 
 4) + 1788884991705*sqrt(3))*sqrt(3*x + 4)*sqrt(-6*x + 15)/(2*x - 5)^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=\int \frac {{\left (3\,x+4\right )}^{3/2}\,{\left (x^2+3\,x+2\right )}^2}{{\left (5-2\,x\right )}^{9/2}} \,d x \] Input:

int(((3*x + 4)^(3/2)*(3*x + x^2 + 2)^2)/(5 - 2*x)^(9/2),x)
 

Output:

int(((3*x + 4)^(3/2)*(3*x + x^2 + 2)^2)/(5 - 2*x)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2}} \, dx=\frac {-1176220920 \sqrt {-2 x +5}\, \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right ) x^{3}[phone] \sqrt {-2 x +5}\, \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right ) x^{2}-22054142250 \sqrt {-2 x +5}\, \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right ) x [phone] \sqrt {-2 x +5}\, \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )+3047040 \sqrt {3 x +4}\, x^{5}[phone] \sqrt {3 x +4}\, x^{4}-3524610816 \sqrt {3 x +4}\, x^{3}[phone] \sqrt {3 x +4}\, x^{2}-42577661704 \sqrt {3 x +4}\, x [phone] \sqrt {3 x +4}}{4062720 \sqrt {-2 x +5}\, \left (8 x^{3}-60 x^{2}+150 x -125\right )} \] Input:

int((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x)^(9/2),x)
 

Output:

( - 1176220920*sqrt( - 2*x + 5)*sqrt(6)*asin((sqrt( - 2*x + 5)*sqrt(3))/sq 
rt(23))*x**3 + 8821656900*sqrt( - 2*x + 5)*sqrt(6)*asin((sqrt( - 2*x + 5)* 
sqrt(3))/sqrt(23))*x**2 - 22054142250*sqrt( - 2*x + 5)*sqrt(6)*asin((sqrt( 
 - 2*x + 5)*sqrt(3))/sqrt(23))*x + 18378451875*sqrt( - 2*x + 5)*sqrt(6)*as 
in((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23)) + 3047040*sqrt(3*x + 4)*x**5 + 886 
18080*sqrt(3*x + 4)*x**4 - 3524610816*sqrt(3*x + 4)*x**3 + 19896907664*sqr 
t(3*x + 4)*x**2 - 42577661704*sqrt(3*x + 4)*x + 31745633750*sqrt(3*x + 4)) 
/(4062720*sqrt( - 2*x + 5)*(8*x**3 - 60*x**2 + 150*x - 125))