\(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx\) [1007]

Optimal result

Integrand size = 26, antiderivative size = 150 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {871 (1-2 x)^{3/2} \sqrt {3+5 x}}{8640}-\frac {181}{432} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {1922677 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \] Output:

77269/259200*(1-2*x)^(1/2)*(3+5*x)^(1/2)+871/8640*(1-2*x)^(3/2)*(3+5*x)^(1 
/2)-181/432*(1-2*x)^(5/2)*(3+5*x)^(1/2)+1/12*(1-2*x)^(5/2)*(3+5*x)^(3/2)+1 
922677/7776000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-98/243*7^(1/2) 
*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {30 \sqrt {1-2 x} \left (178797+990815 x-666900 x^2-1740000 x^3+2160000 x^4\right )-1922677 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-3136000 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7776000 \sqrt {3+5 x}} \] Input:

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

Output:

(30*Sqrt[1 - 2*x]*(178797 + 990815*x - 666900*x^2 - 1740000*x^3 + 2160000* 
x^4) - 1922677*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 313 
6000*Sqrt[7]*Sqrt[3 + 5*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/ 
(7776000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {112, 27, 171, 27, 171, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {1}{12} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac {1}{12} \int -\frac {(1-2 x)^{3/2} \sqrt {5 x+3} (181 x+102)}{2 (3 x+2)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (181 x+102)}{3 x+2}dx+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{24} \left (\frac {1}{45} \int \frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (7093 x+3422)}{2 (3 x+2)}dx+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (7093 x+3422)}{3 x+2}dx+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{30} \int \frac {\sqrt {5 x+3} (390869 x+77646)}{2 \sqrt {1-2 x} (3 x+2)}dx+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \int \frac {\sqrt {5 x+3} (390869 x+77646)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (-\frac {1}{6} \int -\frac {1922677 x+2013518}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (\frac {1}{12} \int \frac {1922677 x+2013518}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (\frac {1}{12} \left (\frac {1922677}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {2195200}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (\frac {1}{12} \left (\frac {2195200}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3845354}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (\frac {1}{12} \left (\frac {4390400}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {3845354}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (\frac {1}{12} \left (\frac {3845354}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {627200}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{24} \left (\frac {1}{30} \left (\frac {1}{60} \left (\frac {1}{12} \left (\frac {1922677}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {627200}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {390869}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {7093}{30} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {181}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}\)

Input:

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

Output:

((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/12 + ((181*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/ 
2))/45 + ((7093*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/30 + ((-390869*Sqrt[1 - 2*x 
]*Sqrt[3 + 5*x])/6 + ((1922677*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]) 
/3 - (627200*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/12) 
/60)/30)/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 112

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

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 = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (25920000 x^{3} \sqrt {-10 x^{2}-x +3}-36432000 x^{2} \sqrt {-10 x^{2}-x +3}+1922677 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3136000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}+3575940 \sqrt {-10 x^{2}-x +3}\right )}{15552000 \sqrt {-10 x^{2}-x +3}}\)	\(132\)
risch	\(-\frac {\left (432000 x^{3}-607200 x^{2}+230940 x +59599\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{259200 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1922677 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{15552000}-\frac {49 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{243}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(136\)

Input:

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

Output:

1/15552000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(25920000*x^3*(-10*x^2-x+3)^(1/2)-3 
6432000*x^2*(-10*x^2-x+3)^(1/2)+1922677*10^(1/2)*arcsin(20/11*x+1/11)+3136 
000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x* 
(-10*x^2-x+3)^(1/2)+3575940*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {1}{259200} \, {\left (432000 \, x^{3} - 607200 \, x^{2} + 230940 \, x + 59599\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {49}{243} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {1922677}{15552000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

1/259200*(432000*x^3 - 607200*x^2 + 230940*x + 59599)*sqrt(5*x + 3)*sqrt(- 
2*x + 1) - 49/243*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sq 
rt(-2*x + 1)/(10*x^2 + x - 3)) - 1922677/15552000*sqrt(10)*arctan(1/20*sqr 
t(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx=-\frac {1}{6} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {271}{1080} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7093}{4320} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1922677}{15552000} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {49}{243} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {135521}{259200} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

-1/6*(-10*x^2 - x + 3)^(3/2)*x + 271/1080*(-10*x^2 - x + 3)^(3/2) + 7093/4 
320*sqrt(-10*x^2 - x + 3)*x + 1922677/15552000*sqrt(10)*arcsin(20/11*x + 1 
/11) + 49/243*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 
135521/259200*sqrt(-10*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.33 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {49}{2430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{1296000} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 577 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 23769 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 390869 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1922677}{15552000} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \] Input:

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

Output:

49/2430*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s 
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 
5) - sqrt(22)))) + 1/1296000*(12*(8*(36*sqrt(5)*(5*x + 3) - 577*sqrt(5))*( 
5*x + 3) + 23769*sqrt(5))*(5*x + 3) - 390869*sqrt(5))*sqrt(5*x + 3)*sqrt(- 
10*x + 5) + 1922677/15552000*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))))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx=-\frac {1922677 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{7776000}+\frac {98 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{243}-\frac {98 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{243}+\frac {5 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{3}-\frac {253 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{108}+\frac {1283 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{1440}+\frac {59599 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{259200} \] Input:

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

Output:

( - 1922677*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 3136000*s 
qrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 
11))/2))/sqrt(2)) - 3136000*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sq 
rt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 12960000*sqrt(5*x + 3)*sq 
rt( - 2*x + 1)*x**3 - 18216000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 69282 
00*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 1787970*sqrt(5*x + 3)*sqrt( - 2*x + 
1))/7776000