\(\int \frac {(2+3 x+x^2)^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx\) [259]

Optimal result

Integrand size = 29, antiderivative size = 111 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=-\frac {12293}{2238728 \sqrt {5-2 x} \sqrt {4+3 x}}+\frac {36815 \sqrt {5-2 x}}{51490744 \sqrt {4+3 x}}+\frac {567 \sqrt {4+3 x}}{2116 (5-2 x)^{7/2}}-\frac {4851 \sqrt {4+3 x}}{48668 (5-2 x)^{5/2}}+\frac {40483 \sqrt {4+3 x}}{3358092 (5-2 x)^{3/2}} \] Output:

-12293/2238728/(5-2*x)^(1/2)/(4+3*x)^(1/2)+36815/51490744*(5-2*x)^(1/2)/(4 
+3*x)^(1/2)+567/2116*(4+3*x)^(1/2)/(5-2*x)^(7/2)-4851/48668*(4+3*x)^(1/2)/ 
(5-2*x)^(5/2)+40483/3358092*(4+3*x)^(1/2)/(5-2*x)^(3/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.38 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=\frac {2 \left (428116+1492084 x+1613677 x^2+716322 x^3+110445 x^4\right )}{19309029 (5-2 x)^{7/2} \sqrt {4+3 x}} \] Input:

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

Output:

(2*(428116 + 1492084*x + 1613677*x^2 + 716322*x^3 + 110445*x^4))/(19309029 
*(5 - 2*x)^(7/2)*Sqrt[4 + 3*x])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1193, 27, 2124, 27, 1193, 27, 87, 48}

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.

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

\(\Big \downarrow \) 1193

\(\displaystyle \frac {2}{161} \int -\frac {7 \left (184 x^3+1564 x^2+6302 x+4355\right )}{32 (5-2 x)^{7/2} (3 x+4)^{3/2}}dx+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}-\frac {1}{368} \int \frac {184 x^3+1564 x^2+6302 x+4355}{(5-2 x)^{7/2} (3 x+4)^{3/2}}dx\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {1}{368} \left (-\frac {2}{115} \int -\frac {5 \left (2116 x^2+23276 x+12727\right )}{2 (5-2 x)^{5/2} (3 x+4)^{3/2}}dx-\frac {13104}{23 (5-2 x)^{5/2} \sqrt {3 x+4}}\right )+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{368} \left (\frac {1}{23} \int \frac {2116 x^2+23276 x+12727}{(5-2 x)^{5/2} (3 x+4)^{3/2}}dx-\frac {13104}{23 (5-2 x)^{5/2} \sqrt {3 x+4}}\right )+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {1}{368} \left (\frac {1}{23} \left (\frac {2}{69} \int \frac {3 (8059-24334 x)}{2 (5-2 x)^{3/2} (3 x+4)^{3/2}}dx+\frac {168284}{69 (5-2 x)^{3/2} \sqrt {3 x+4}}\right )-\frac {13104}{23 (5-2 x)^{5/2} \sqrt {3 x+4}}\right )+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{368} \left (\frac {1}{23} \left (\frac {1}{23} \int \frac {8059-24334 x}{(5-2 x)^{3/2} (3 x+4)^{3/2}}dx+\frac {168284}{69 (5-2 x)^{3/2} \sqrt {3 x+4}}\right )-\frac {13104}{23 (5-2 x)^{5/2} \sqrt {3 x+4}}\right )+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{368} \left (\frac {1}{23} \left (\frac {1}{23} \left (-\frac {73630}{69} \int \frac {1}{(5-2 x)^{3/2} \sqrt {3 x+4}}dx-\frac {243026}{69 \sqrt {5-2 x} \sqrt {3 x+4}}\right )+\frac {168284}{69 (5-2 x)^{3/2} \sqrt {3 x+4}}\right )-\frac {13104}{23 (5-2 x)^{5/2} \sqrt {3 x+4}}\right )+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {1}{368} \left (\frac {1}{23} \left (\frac {1}{23} \left (-\frac {147260 \sqrt {3 x+4}}{1587 \sqrt {5-2 x}}-\frac {243026}{69 \sqrt {5-2 x} \sqrt {3 x+4}}\right )+\frac {168284}{69 (5-2 x)^{3/2} \sqrt {3 x+4}}\right )-\frac {13104}{23 (5-2 x)^{5/2} \sqrt {3 x+4}}\right )+\frac {567}{184 (5-2 x)^{7/2} \sqrt {3 x+4}}\)

Input:

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

Output:

567/(184*(5 - 2*x)^(7/2)*Sqrt[4 + 3*x]) + (-13104/(23*(5 - 2*x)^(5/2)*Sqrt 
[4 + 3*x]) + (168284/(69*(5 - 2*x)^(3/2)*Sqrt[4 + 3*x]) + (-243026/(69*Sqr 
t[5 - 2*x]*Sqrt[4 + 3*x]) - (147260*Sqrt[4 + 3*x])/(1587*Sqrt[5 - 2*x]))/2 
3)/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 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

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 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 = 1.67 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33

method	result	size
gosper	\(\frac {\frac {73630}{6436343} x^{4}+\frac {477548}{6436343} x^{3}+\frac {3227354}{19309029} x^{2}+\frac {2984168}{19309029} x +\frac {856232}{19309029}}{\left (5-2 x \right )^{\frac {7}{2}} \sqrt {3 x +4}}\)	\(37\)
default	\(\frac {\frac {73630}{6436343} x^{4}+\frac {477548}{6436343} x^{3}+\frac {3227354}{19309029} x^{2}+\frac {2984168}{19309029} x +\frac {856232}{19309029}}{\left (5-2 x \right )^{\frac {7}{2}} \sqrt {3 x +4}}\)	\(37\)
orering	\(-\frac {2 \left (110445 x^{4}+716322 x^{3}+1613677 x^{2}+1492084 x +428116\right ) \left (-5+2 x \right ) \left (x^{2}+3 x +2\right )^{2}}{19309029 \sqrt {3 x +4}\, \left (2+x \right )^{2} \left (x +1\right )^{2} \left (5-2 x \right )^{\frac {9}{2}}}\)	\(62\)

Input:

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

Output:

2/19309029/(5-2*x)^(7/2)/(3*x+4)^(1/2)*(110445*x^4+716322*x^3+1613677*x^2+ 
1492084*x+428116)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.57 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=\frac {2 \, {\left (110445 \, x^{4} + 716322 \, x^{3} + 1613677 \, x^{2} + 1492084 \, x + 428116\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{19309029 \, {\left (48 \, x^{5} - 416 \, x^{4} + 1160 \, x^{3} - 600 \, x^{2} - 2125 \, x + 2500\right )}} \] Input:

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

Output:

2/19309029*(110445*x^4 + 716322*x^3 + 1613677*x^2 + 1492084*x + 428116)*sq 
rt(3*x + 4)*sqrt(-2*x + 5)/(48*x^5 - 416*x^4 + 1160*x^3 - 600*x^2 - 2125*x 
 + 2500)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (81) = 162\).

Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.65 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=-\frac {36815 \, x}{25745372 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}} - \frac {1029773}{51490744 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}} - \frac {567}{184 \, {\left (8 \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x^{3} - 60 \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x^{2} + 150 \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x - 125 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}\right )}} - \frac {819}{529 \, {\left (4 \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x^{2} - 20 \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x + 25 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}\right )}} - \frac {42071}{146004 \, {\left (2 \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x - 5 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}\right )}} \] Input:

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

Output:

-36815/25745372*x/sqrt(-6*x^2 + 7*x + 20) - 1029773/51490744/sqrt(-6*x^2 + 
 7*x + 20) - 567/184/(8*sqrt(-6*x^2 + 7*x + 20)*x^3 - 60*sqrt(-6*x^2 + 7*x 
 + 20)*x^2 + 150*sqrt(-6*x^2 + 7*x + 20)*x - 125*sqrt(-6*x^2 + 7*x + 20)) 
- 819/529/(4*sqrt(-6*x^2 + 7*x + 20)*x^2 - 20*sqrt(-6*x^2 + 7*x + 20)*x + 
25*sqrt(-6*x^2 + 7*x + 20)) - 42071/146004/(2*sqrt(-6*x^2 + 7*x + 20)*x - 
5*sqrt(-6*x^2 + 7*x + 20))
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.12 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=-\frac {2 \, \sqrt {6} {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}{19309029 \, \sqrt {3 \, x + 4}} + \frac {8 \, \sqrt {6} \sqrt {3 \, x + 4}}{19309029 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}} + \frac {2 \, {\left ({\left ({\left (12293 \, \sqrt {3} {\left (3 \, x + 4\right )} + 41446 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} - 56603 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} - 243340 \, \sqrt {3}\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15}}{521343783 \, {\left (2 \, x - 5\right )}^{4}} \] Input:

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

Output:

-2/19309029*sqrt(6)*(sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt(3*x + 4) + 8 
/19309029*sqrt(6)*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)) + 2/5 
21343783*(((12293*sqrt(3)*(3*x + 4) + 41446*sqrt(3))*(3*x + 4) - 56603*sqr 
t(3))*(3*x + 4) - 243340*sqrt(3))*sqrt(3*x + 4)*sqrt(-6*x + 15)/(2*x - 5)^ 
4
 

Mupad [B] (verification not implemented)

Time = 13.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=\frac {\sqrt {5-2\,x}\,\left (\frac {36815\,x^4}{51490744}+\frac {119387\,x^3}{25745372}+\frac {1613677\,x^2}{154472232}+\frac {373021\,x}{38618058}+\frac {107029}{38618058}\right )}{\frac {625\,\sqrt {3\,x+4}}{16}-\frac {125\,x\,\sqrt {3\,x+4}}{2}+\frac {75\,x^2\,\sqrt {3\,x+4}}{2}-10\,x^3\,\sqrt {3\,x+4}+x^4\,\sqrt {3\,x+4}} \] Input:

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

Output:

((5 - 2*x)^(1/2)*((373021*x)/38618058 + (1613677*x^2)/154472232 + (119387* 
x^3)/25745372 + (36815*x^4)/51490744 + 107029/38618058))/((625*(3*x + 4)^( 
1/2))/16 - (125*x*(3*x + 4)^(1/2))/2 + (75*x^2*(3*x + 4)^(1/2))/2 - 10*x^3 
*(3*x + 4)^(1/2) + x^4*(3*x + 4)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.50 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{9/2} (4+3 x)^{3/2}} \, dx=\frac {-\frac {73630}{6436343} x^{4}-\frac {477548}{6436343} x^{3}-\frac {3227354}{19309029} x^{2}-\frac {2984168}{19309029} x -\frac {856232}{19309029}}{\sqrt {3 x +4}\, \sqrt {-2 x +5}\, \left (8 x^{3}-60 x^{2}+150 x -125\right )} \] Input:

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

Output:

(2*( - 110445*x**4 - 716322*x**3 - 1613677*x**2 - 1492084*x - 428116))/(19 
309029*sqrt(3*x + 4)*sqrt( - 2*x + 5)*(8*x**3 - 60*x**2 + 150*x - 125))