3.24.57 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\) [2357]

3.24.57.1 Optimal result

Integrand size = 26, antiderivative size = 178 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=-\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {100}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {1922677 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{762048 \sqrt {7}} \]

output

-1/12*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4-1922677/5334336*arctan(1/7*(1- 
2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+100/243*arcsin(1/11*22^(1/2)*(3+ 
5*x)^(1/2))*10^(1/2)-871/6048*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2+181/21 
6*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3-77269/254016*(1-2*x)^(1/2)*(3+5*x) 
^(1/2)/(2+3*x)
 

3.24.57.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (2583760+13434180 x+23185560 x^2+13290147 x^3\right )}{(2+3 x)^4}-2195200 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-1922677 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5334336} \]

input

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

output

((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2583760 + 13434180*x + 23185560*x^2 + 13 
290147*x^3))/(2 + 3*x)^4 - 2195200*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 
+ 5*x]] - 1922677*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/5 
334336
 

3.24.57.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 166, 27, 166, 27, 166, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{12} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^4}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{24} \left (\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac {1}{9} \int -\frac {(5 x+3)^{3/2} (960 x+1511)}{2 \sqrt {1-2 x} (3 x+2)^3}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \int \frac {(5 x+3)^{3/2} (960 x+1511)}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{42} \int \frac {3 \sqrt {5 x+3} (44800 x+55623)}{2 \sqrt {1-2 x} (3 x+2)^2}dx-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \int \frac {\sqrt {5 x+3} (44800 x+55623)}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{21} \int \frac {3136000 x+2731559}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \int \frac {3136000 x+2731559}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {3136000}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1922677}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {1922677}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1254400}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {3845354}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {1254400}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {1254400}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {3845354 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {627200}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {3845354 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\)

input

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

output

-1/12*((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^4 + ((181*Sqrt[1 - 2*x]* 
(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) + ((-871*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/( 
14*(2 + 3*x)^2) + ((-77269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)) + ( 
(627200*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 - (3845354*ArcTan[Sqr 
t[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/42)/28)/18)/24
 

3.24.57.3.1 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 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]
 

3.24.57.4 Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (13290147 x^{3}[phone] x^{2}[phone] x +2583760\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{254016 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {50 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{243}+\frac {1922677 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{10668672}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(142\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (155736837 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{10668672 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(315\)

input

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

output

-1/254016*(-1+2*x)*(3+5*x)^(1/2)*(13290147*x^3+23185560*x^2+13434180*x+258 
3760)/(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)+(50/243*10^(1/2)*arcsin(20/11*x+1/11)+1922677/10668672*7^(1/2)*arcta 
n(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)
 

3.24.57.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=-\frac {1922677 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 2195200 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (13290147 \, x^{3} + 23185560 \, x^{2} + 13434180 \, x + 2583760\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{10668672 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

input

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

output

-1/10668672*(1922677*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arct 
an(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) 
 + 2195200*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20*s 
qrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(13 
290147*x^3 + 23185560*x^2 + 13434180*x + 2583760)*sqrt(5*x + 3)*sqrt(-2*x 
+ 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

3.24.57.6 Sympy [F]

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

input

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

output

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

3.24.57.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {27065}{148176} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1176 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {5413 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32928 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {528205}{296352} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {50}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1922677}{10668672} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {802877}{1778112} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3667 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{197568 \, {\left (3 \, x + 2\right )}} \]

input

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

output

27065/148176*(-10*x^2 - x + 3)^(3/2) - 1/28*(-10*x^2 - x + 3)^(5/2)/(81*x^ 
4 + 216*x^3 + 216*x^2 + 96*x + 16) + 169/1176*(-10*x^2 - x + 3)^(5/2)/(27* 
x^3 + 54*x^2 + 36*x + 8) + 5413/32928*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12* 
x + 4) + 528205/296352*sqrt(-10*x^2 - x + 3)*x + 50/243*sqrt(10)*arcsin(20 
/11*x + 1/11) + 1922677/10668672*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/ 
11/abs(3*x + 2)) - 802877/1778112*sqrt(-10*x^2 - x + 3) + 3667/197568*(-10 
*x^2 - x + 3)^(3/2)/(3*x + 2)
 

3.24.57.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (136) = 272\).

Time = 0.65 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {1922677}{106686720} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {50}{243} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (77269 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 81002040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 31057924800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {8580356288000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {34321425152000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{127008 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

input

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

output

1922677/106686720*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)*sqr 
t(-10*x + 5) - sqrt(22)))) + 50/243*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)))) - 11/127008*sqrt(10)*(77269*((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 + 81002040*((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 + 3105792480 
0*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s 
qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 8580356288000*(sqrt(2)*sqrt(-10*x 
+ 5) - sqrt(22))/sqrt(5*x + 3) + 34321425152000*sqrt(5*x + 3)/(sqrt(2)*sqr 
t(-10*x + 5) - sqrt(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
 

3.24.57.9 Mupad [F(-1)]

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

input

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

output

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