\(\int \frac {(2+3 x+x^2)^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx\) [266]

Optimal result

Integrand size = 29, antiderivative size = 111 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=-\frac {8 \sqrt {5-2 x}}{2518569 (4+3 x)^{3/2}}+\frac {3969}{10580 (5-2 x)^{5/2} \sqrt {4+3 x}}-\frac {33621}{243340 (5-2 x)^{3/2} \sqrt {4+3 x}}+\frac {90023}{5596820 \sqrt {5-2 x} \sqrt {4+3 x}}-\frac {664267 \sqrt {5-2 x}}{1158541740 \sqrt {4+3 x}} \] Output:

-8/2518569*(5-2*x)^(1/2)/(4+3*x)^(3/2)+3969/10580/(5-2*x)^(5/2)/(4+3*x)^(1 
/2)-33621/243340/(5-2*x)^(3/2)/(4+3*x)^(1/2)+90023/5596820/(5-2*x)^(1/2)/( 
4+3*x)^(1/2)-664267/1158541740*(5-2*x)^(1/2)/(4+3*x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.38 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=\frac {2 \left (2831692+10008476 x+11656803 x^2+5222294 x^3+664267 x^4\right )}{96545145 (5-2 x)^{5/2} (4+3 x)^{3/2}} \] Input:

Integrate[(2 + 3*x + x^2)^2/((5 - 2*x)^(7/2)*(4 + 3*x)^(5/2)),x]
 

Output:

(2*(2831692 + 10008476*x + 11656803*x^2 + 5222294*x^3 + 664267*x^4))/(9654 
5145*(5 - 2*x)^(5/2)*(4 + 3*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.51 (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.

\(\displaystyle \int \frac {\left (x^2+3 x+2\right )^2}{(5-2 x)^{7/2} (3 x+4)^{5/2}} \, dx\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {2}{115} \int \frac {-920 x^3-7820 x^2-31510 x+5441}{32 (5-2 x)^{5/2} (3 x+4)^{5/2}}dx+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-920 x^3-7820 x^2-31510 x+5441}{(5-2 x)^{5/2} (3 x+4)^{5/2}}dx}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {\frac {2}{69} \int -\frac {3 \left (-10580 x^2-116380 x+166189\right )}{2 (5-2 x)^{3/2} (3 x+4)^{5/2}}dx-\frac {91056}{23 (5-2 x)^{3/2} (3 x+4)^{3/2}}}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{23} \int \frac {-10580 x^2-116380 x+166189}{(5-2 x)^{3/2} (3 x+4)^{5/2}}dx-\frac {91056}{23 (5-2 x)^{3/2} (3 x+4)^{3/2}}}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {\frac {1}{23} \left (\frac {381772}{23 \sqrt {5-2 x} (3 x+4)^{3/2}}-\frac {2}{23} \int -\frac {648087-121670 x}{2 \sqrt {5-2 x} (3 x+4)^{5/2}}dx\right )-\frac {91056}{23 (5-2 x)^{3/2} (3 x+4)^{3/2}}}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{23} \left (\frac {1}{23} \int \frac {648087-121670 x}{\sqrt {5-2 x} (3 x+4)^{5/2}}dx+\frac {381772}{23 \sqrt {5-2 x} (3 x+4)^{3/2}}\right )-\frac {91056}{23 (5-2 x)^{3/2} (3 x+4)^{3/2}}}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {\frac {1}{23} \left (\frac {1}{23} \left (\frac {1328534}{207} \int \frac {1}{\sqrt {5-2 x} (3 x+4)^{3/2}}dx-\frac {4861882 \sqrt {5-2 x}}{207 (3 x+4)^{3/2}}\right )+\frac {381772}{23 \sqrt {5-2 x} (3 x+4)^{3/2}}\right )-\frac {91056}{23 (5-2 x)^{3/2} (3 x+4)^{3/2}}}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\frac {1}{23} \left (\frac {1}{23} \left (-\frac {2657068 \sqrt {5-2 x}}{4761 \sqrt {3 x+4}}-\frac {4861882 \sqrt {5-2 x}}{207 (3 x+4)^{3/2}}\right )+\frac {381772}{23 \sqrt {5-2 x} (3 x+4)^{3/2}}\right )-\frac {91056}{23 (5-2 x)^{3/2} (3 x+4)^{3/2}}}{1840}+\frac {3969}{920 (5-2 x)^{5/2} (3 x+4)^{3/2}}\)

Input:

Int[(2 + 3*x + x^2)^2/((5 - 2*x)^(7/2)*(4 + 3*x)^(5/2)),x]
 

Output:

3969/(920*(5 - 2*x)^(5/2)*(4 + 3*x)^(3/2)) + (-91056/(23*(5 - 2*x)^(3/2)*( 
4 + 3*x)^(3/2)) + (381772/(23*Sqrt[5 - 2*x]*(4 + 3*x)^(3/2)) + ((-4861882* 
Sqrt[5 - 2*x])/(207*(4 + 3*x)^(3/2)) - (2657068*Sqrt[5 - 2*x])/(4761*Sqrt[ 
4 + 3*x]))/23)/23)/1840
 

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 = 1.71 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33

method	result	size
gosper	\(\frac {\frac {1328534}{96545145} x^{4}+\frac {10444588}{96545145} x^{3}+\frac {7771202}{32181715} x^{2}+\frac {20016952}{96545145} x +\frac {5663384}{96545145}}{\left (5-2 x \right )^{\frac {5}{2}} \left (3 x +4\right )^{\frac {3}{2}}}\)	\(37\)
default	\(\frac {\frac {1328534}{96545145} x^{4}+\frac {10444588}{96545145} x^{3}+\frac {7771202}{32181715} x^{2}+\frac {20016952}{96545145} x +\frac {5663384}{96545145}}{\left (5-2 x \right )^{\frac {5}{2}} \left (3 x +4\right )^{\frac {3}{2}}}\)	\(37\)
orering	\(-\frac {2 \left (664267 x^{4}+5222294 x^{3}[phone] x^{2}[phone] x +2831692\right ) \left (-5+2 x \right ) \left (x^{2}+3 x +2\right )^{2}}{96545145 \left (3 x +4\right )^{\frac {3}{2}} \left (2+x \right )^{2} \left (x +1\right )^{2} \left (5-2 x \right )^{\frac {7}{2}}}\)	\(62\)

Input:

int((x^2+3*x+2)^2/(5-2*x)^(7/2)/(3*x+4)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/96545145/(5-2*x)^(5/2)/(3*x+4)^(3/2)*(664267*x^4+5222294*x^3+11656803*x^ 
2+10008476*x+2831692)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.57 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=-\frac {2 \, {\left (664267 \, x^{4} + 5222294 \, x^{3} + 11656803 \, x^{2} + 10008476 \, x + 2831692\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{96545145 \, {\left (72 \, x^{5} - 348 \, x^{4} + 38 \, x^{3} + 1515 \, x^{2} - 600 \, x - 2000\right )}} \] Input:

integrate((x^2+3*x+2)^2/(5-2*x)^(7/2)/(4+3*x)^(5/2),x, algorithm="fricas")
 

Output:

-2/96545145*(664267*x^4 + 5222294*x^3 + 11656803*x^2 + 10008476*x + 283169 
2)*sqrt(3*x + 4)*sqrt(-2*x + 5)/(72*x^5 - 348*x^4 + 38*x^3 + 1515*x^2 - 60 
0*x - 2000)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=\int \frac {\left (x + 1\right )^{2} \left (x + 2\right )^{2}}{\left (5 - 2 x\right )^{\frac {7}{2}} \left (3 x + 4\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((x**2+3*x+2)**2/(5-2*x)**(7/2)/(4+3*x)**(5/2),x)
 

Output:

Integral((x + 1)**2*(x + 2)**2/((5 - 2*x)**(7/2)*(3*x + 4)**(5/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=\frac {664267 \, x}{579270870 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}} - \frac {x^{2}}{12 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}} - \frac {4649869}{6951250440 \, \sqrt {-6 \, x^{2} + 7 \, x + 20}} - \frac {4168427 \, x}{13140360 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}} - \frac {3969}{920 \, {\left (2 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x - 5 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}\right )}} - \frac {11007883}{13140360 \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}}} \] Input:

integrate((x^2+3*x+2)^2/(5-2*x)^(7/2)/(4+3*x)^(5/2),x, algorithm="maxima")
 

Output:

664267/579270870*x/sqrt(-6*x^2 + 7*x + 20) - 1/12*x^2/(-6*x^2 + 7*x + 20)^ 
(3/2) - 4649869/6951250440/sqrt(-6*x^2 + 7*x + 20) - 4168427/13140360*x/(- 
6*x^2 + 7*x + 20)^(3/2) - 3969/920/(2*(-6*x^2 + 7*x + 20)^(3/2)*x - 5*(-6* 
x^2 + 7*x + 20)^(3/2)) - 11007883/13140360/(-6*x^2 + 7*x + 20)^(3/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (81) = 162\).

Time = 0.37 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.56 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=-\frac {\sqrt {6} {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{3}}{695125044 \, {\left (3 \, x + 4\right )}^{\frac {3}{2}}} + \frac {31 \, \sqrt {6} {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}{57927087 \, \sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {6} {\left (3 \, x + 4\right )}^{\frac {3}{2}} {\left (\frac {93 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{173781261 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{3}} - \frac {2 \, {\left ({\left (24881 \, \sqrt {3} {\left (3 \, x + 4\right )} + 176870 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} - 600415 \, \sqrt {3}\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15}}{868906305 \, {\left (2 \, x - 5\right )}^{3}} \] Input:

integrate((x^2+3*x+2)^2/(5-2*x)^(7/2)/(4+3*x)^(5/2),x, algorithm="giac")
 

Output:

-1/695125044*sqrt(6)*(sqrt(2)*sqrt(-6*x + 15) - sqrt(46))^3/(3*x + 4)^(3/2 
) + 31/57927087*sqrt(6)*(sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt(3*x + 4) 
 - 4/173781261*sqrt(6)*(3*x + 4)^(3/2)*(93*(sqrt(2)*sqrt(-6*x + 15) - sqrt 
(46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46))^3 - 2/86890630 
5*((24881*sqrt(3)*(3*x + 4) + 176870*sqrt(3))*(3*x + 4) - 600415*sqrt(3))* 
sqrt(3*x + 4)*sqrt(-6*x + 15)/(2*x - 5)^3
 

Mupad [B] (verification not implemented)

Time = 13.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=-\frac {\sqrt {5-2\,x}\,\left (\frac {664267\,x^4}{1158541740}+\frac {2611147\,x^3}{579270870}+\frac {3885601\,x^2}{386180580}+\frac {2502119\,x}{289635435}+\frac {707923}{289635435}\right )}{\frac {75\,x\,\sqrt {3\,x+4}}{8}-\frac {125\,\sqrt {3\,x+4}}{6}+\frac {35\,x^2\,\sqrt {3\,x+4}}{4}-\frac {37\,x^3\,\sqrt {3\,x+4}}{6}+x^4\,\sqrt {3\,x+4}} \] Input:

int((3*x + x^2 + 2)^2/((5 - 2*x)^(7/2)*(3*x + 4)^(5/2)),x)
 

Output:

-((5 - 2*x)^(1/2)*((2502119*x)/289635435 + (3885601*x^2)/386180580 + (2611 
147*x^3)/579270870 + (664267*x^4)/1158541740 + 707923/289635435))/((75*x*( 
3*x + 4)^(1/2))/8 - (125*(3*x + 4)^(1/2))/6 + (35*x^2*(3*x + 4)^(1/2))/4 - 
 (37*x^3*(3*x + 4)^(1/2))/6 + 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 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x)^{7/2} (4+3 x)^{5/2}} \, dx=\frac {\frac {1328534}{96545145} x^{4}+\frac {10444588}{96545145} x^{3}+\frac {7771202}{32181715} x^{2}+\frac {20016952}{96545145} x +\frac {5663384}{96545145}}{\sqrt {3 x +4}\, \sqrt {-2 x +5}\, \left (12 x^{3}-44 x^{2}-5 x +100\right )} \] Input:

int((x^2+3*x+2)^2/(5-2*x)^(7/2)/(4+3*x)^(5/2),x)
 

Output:

(2*(664267*x**4 + 5222294*x**3 + 11656803*x**2 + 10008476*x + 2831692))/(9 
6545145*sqrt(3*x + 4)*sqrt( - 2*x + 5)*(12*x**3 - 44*x**2 - 5*x + 100))