\(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\) [1023]

Optimal result

Integrand size = 26, antiderivative size = 200 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {778885 \sqrt {1-2 x} \sqrt {3+5 x}}{381024}-\frac {23255 (1-2 x)^{5/2} \sqrt {3+5 x}}{127008 (2+3 x)^2}+\frac {517345 (1-2 x)^{3/2} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {185 (1-2 x)^{5/2} (3+5 x)^{3/2}}{1512 (2+3 x)^3}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {1850}{729} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {3304795 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{326592 \sqrt {7}} \] Output:

778885/381024*(1-2*x)^(1/2)*(3+5*x)^(1/2)-23255/127008*(1-2*x)^(5/2)*(3+5* 
x)^(1/2)/(2+3*x)^2+517345*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(508032+762048*x)-18 
5/1512*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^3-1/12*(1-2*x)^(5/2)*(3+5*x)^(5 
/2)/(2+3*x)^4+1850/729*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+330479 
5/2286144*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (5093072+25998852 x+45563928 x^2+29475315 x^3+3628800 x^4\right )}{(2+3 x)^4}-5801600 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+3304795 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2286144} \] Input:

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^5,x]
 

Output:

((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5093072 + 25998852*x + 45563928*x^2 + 29 
475315*x^3 + 3628800*x^4))/(2 + 3*x)^4 - 5801600*Sqrt[10]*ArcTan[Sqrt[5/2 
- 5*x]/Sqrt[3 + 5*x]] + 3304795*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt 
[3 + 5*x])])/2286144
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 171, 27, 175, 64, 104, 217, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^5} \, dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{12} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^4}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{24} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^4}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {5}{24} \left (-\frac {1}{9} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (980 x+731)}{2 (3 x+2)^3}dx-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{24} \left (-\frac {1}{18} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (980 x+731)}{(3 x+2)^3}dx-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{6} \int -\frac {(4153-3180 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^2}dx-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (-\frac {1}{12} \int \frac {(4153-3180 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}-\frac {1}{21} \int \frac {3 (87213-199660 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)}dx\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}-\frac {1}{14} \int \frac {(87213-199660 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 171

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{6} \int -\frac {2 (1657600 x+884747)}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (-\frac {1}{3} \int \frac {1657600 x+884747}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 175

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {660959}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {1657600}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 64

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {660959}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {663040}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {1321918}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {663040}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (-\frac {663040}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1321918 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {5}{24} \left (\frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (-\frac {331520}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {1321918 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {99830}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {6273 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )-\frac {233 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

Input:

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^5,x]
 

Output:

-1/12*((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^4 - (5*((-37*(1 - 2*x)^( 
3/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) + ((-233*Sqrt[1 - 2*x]*(3 + 5*x)^(5/ 
2))/(6*(2 + 3*x)^2) + ((6273*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(7*(2 + 3*x)) 
+ ((-99830*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3 + ((-331520*Sqrt[10]*ArcSin[Sqrt 
[2/11]*Sqrt[3 + 5*x]])/3 - (1321918*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 
 5*x])])/(3*Sqrt[7]))/3)/14)/12)/18))/24
 

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 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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 108

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

rule 166

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

rule 171

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

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

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]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (3628800 x^{4}[phone] x^{3}[phone] x^{2}[phone] x +5093072\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{108864 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {925 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{729}+\frac {3304795 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{4572288}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(148\)
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (267688395 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-469929600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-1253145600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-152409600 x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-1253145600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-1237963230 x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -556953600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1913684976 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-92825600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1091951784 x \sqrt {-10 x^{2}-x +3}-213909024 \sqrt {-10 x^{2}-x +3}\right )}{4572288 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(332\)

Input:

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

Output:

-1/108864*(-1+2*x)*(3+5*x)^(1/2)*(3628800*x^4+29475315*x^3+45563928*x^2+25 
998852*x+5093072)/(2+3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1 
/2)/(1-2*x)^(1/2)-(-925/729*10^(1/2)*arcsin(20/11*x+1/11)+3304795/4572288* 
7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))) 
*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {3304795 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5801600 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (3628800 \, x^{4} + 29475315 \, x^{3} + 45563928 \, x^{2} + 25998852 \, x + 5093072\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4572288 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:

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

Output:

1/4572288*(3304795*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan 
(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 
 5801600*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20*sqr 
t(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(3628 
800*x^4 + 29475315*x^3 + 45563928*x^2 + 25998852*x + 5093072)*sqrt(5*x + 3 
)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {5755}{49392} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1151 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {182225}{98784} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1488395}{1778112} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {44881 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{197568 \, {\left (3 \, x + 2\right )}} - \frac {28675}{127008} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {925}{729} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {3304795}{4572288} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {1643795}{762048} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

5755/49392*(-10*x^2 - x + 3)^(5/2) + 3/28*(-10*x^2 - x + 3)^(7/2)/(81*x^4 
+ 216*x^3 + 216*x^2 + 96*x + 16) + 37/392*(-10*x^2 - x + 3)^(7/2)/(27*x^3 
+ 54*x^2 + 36*x + 8) + 1151/10976*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 
4) + 182225/98784*(-10*x^2 - x + 3)^(3/2)*x - 1488395/1778112*(-10*x^2 - x 
 + 3)^(3/2) + 44881/197568*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) - 28675/12700 
8*sqrt(-10*x^2 - x + 3)*x + 925/729*sqrt(10)*arcsin(20/11*x + 1/11) - 3304 
795/4572288*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 16 
43795/762048*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (152) = 304\).

Time = 0.50 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=-\frac {660959}{9144576} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {925}{729} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {20}{243} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {55 \, {\left (8191 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 7386792 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2164545600 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 2731201984000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{54432 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:

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

Output:

-660959/9144576*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x 
+ 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt( 
-10*x + 5) - sqrt(22)))) + 925/729*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 
 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(- 
10*x + 5) - sqrt(22)))) + 20/243*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 5 
5/54432*(8191*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) 
 - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 7386792*sqrt( 
10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/ 
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 2164545600*sqrt(10)*((sqrt(2)*sq 
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 
0*x + 5) - sqrt(22)))^3 + 2731201984000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) 
 - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5* 
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.80 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 469929600*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**4 - 12 
53145600*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**3 - 1253145 
600*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**2 - 556953600*sq 
rt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x - 92825600*sqrt(10)*asi 
n((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) - 267688395*sqrt(7)*atan((sqrt(33) 
- sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 
 - 713835720*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)* 
sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 713835720*sqrt(7)*atan((sqrt(33) - 
sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 
 317260320*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sq 
rt(5))/sqrt(11))/2))/sqrt(2))*x - 52876720*sqrt(7)*atan((sqrt(33) - sqrt(3 
5)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 267688395* 
sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt 
(11))/2))/sqrt(2))*x**4 + 713835720*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan( 
asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 713835720*sq 
rt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(1 
1))/2))/sqrt(2))*x**2 + 317260320*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(as 
in((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 52876720*sqrt(7)* 
atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2) 
)/sqrt(2)) + 76204800*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 + 618981615*s...