\(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx\) [874]

Optimal result

Integrand size = 26, antiderivative size = 151 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {6005 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {625115 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {794365 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \] Output:

-1/12*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+37/504*(1-2*x)^(1/2)*(3+5*x)^( 
1/2)/(2+3*x)^3+6005/14112*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+625115*(1- 
2*x)^(1/2)*(3+5*x)^(1/2)/(395136+592704*x)-794365/153664*7^(1/2)*arctan(1/ 
7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {121 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (594416+2617388 x+3834760 x^2+1875345 x^3\right )}{121 (2+3 x)^4}-6565 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{153664} \] Input:

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

Output:

(121*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(594416 + 2617388*x + 3834760*x^2 + 1 
875345*x^3))/(121*(2 + 3*x)^4) - 6565*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7 
]*Sqrt[3 + 5*x])]))/153664
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {108, 27, 168, 27, 168, 27, 168, 27, 104, 217}

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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{24} \left (\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}-\frac {1}{21} \int -\frac {5 (203-296 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \int \frac {203-296 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{14} \int \frac {25661-24020 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {1201 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \int \frac {25661-24020 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {1201 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {1429857}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {125023 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {1201 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {1429857}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {125023 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {1201 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {1429857}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {125023 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {1201 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {125023 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {1429857 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {1201 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\)

Input:

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

Output:

-1/12*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^4 + ((37*Sqrt[1 - 2*x]*Sqrt[ 
3 + 5*x])/(21*(2 + 3*x)^3) + (5*((1201*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 
 + 3*x)^2) + ((125023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (142985 
7*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/42)/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 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 168

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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])
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1875345 x^{3}+3834760 x^{2}+2617388 x +594416\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {794365 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(129\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (64343565 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}+8321824 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(250\)

Input:

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

Output:

-1/21952*(-1+2*x)*(3+5*x)^(1/2)*(1875345*x^3+3834760*x^2+2617388*x+594416) 
/(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) 
+794365/307328*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 = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {794365 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1875345 \, x^{3} + 3834760 \, x^{2} + 2617388 \, x + 594416\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:

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

Output:

-1/307328*(794365*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)) - 
14*(1875345*x^3 + 3834760*x^2 + 2617388*x + 594416)*sqrt(5*x + 3)*sqrt(-2* 
x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {794365}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {32825}{16464} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {185 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {19695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {242905 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

794365/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 
32825/16464*sqrt(-10*x^2 - x + 3) + 3/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 
 216*x^3 + 216*x^2 + 96*x + 16) + 185/392*(-10*x^2 - x + 3)^(3/2)/(27*x^3 
+ 54*x^2 + 36*x + 8) + 19695/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 
 4) - 242905/65856*sqrt(-10*x^2 - x + 3)/(3*x + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).

Time = 0.28 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {158873}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {605 \, \sqrt {10} {\left (1313 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 1578920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 374767680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {28822976000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {115291904000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:

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

Output:

158873/614656*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(-1 
0*x + 5) - sqrt(22)))) - 605/10976*sqrt(10)*(1313*((sqrt(2)*sqrt(-10*x + 5 
) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s 
qrt(22)))^7 - 1578920*((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 - 374767680*((sq 
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2) 
*sqrt(-10*x + 5) - sqrt(22)))^3 - 28822976000*(sqrt(2)*sqrt(-10*x + 5) - s 
qrt(22))/sqrt(5*x + 3) + 115291904000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 
5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s 
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4
 

Mupad [B] (verification not implemented)

Time = 11.17 (sec) , antiderivative size = 1509, normalized size of antiderivative = 9.99 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

((2430615287*((1 - 2*x)^(1/2) - 1)^7)/(76562500*(3^(1/2) - (5*x + 3)^(1/2) 
)^7) - (4227392*((1 - 2*x)^(1/2) - 1)^3)/(2734375*(3^(1/2) - (5*x + 3)^(1/ 
2))^3) - (29304073*((1 - 2*x)^(1/2) - 1)^5)/(7656250*(3^(1/2) - (5*x + 3)^ 
(1/2))^5) - (1572266*((1 - 2*x)^(1/2) - 1))/(133984375*(3^(1/2) - (5*x + 3 
)^(1/2))) - (2430615287*((1 - 2*x)^(1/2) - 1)^9)/(30625000*(3^(1/2) - (5*x 
 + 3)^(1/2))^9) + (29304073*((1 - 2*x)^(1/2) - 1)^11)/(490000*(3^(1/2) - ( 
5*x + 3)^(1/2))^11) + (132106*((1 - 2*x)^(1/2) - 1)^13)/(875*(3^(1/2) - (5 
*x + 3)^(1/2))^13) + (786133*((1 - 2*x)^(1/2) - 1)^15)/(109760*(3^(1/2) - 
(5*x + 3)^(1/2))^15) + (474659*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3828125*( 
3^(1/2) - (5*x + 3)^(1/2))^2) + (7936034*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/ 
(2734375*(3^(1/2) - (5*x + 3)^(1/2))^4) - (402724691*3^(1/2)*((1 - 2*x)^(1 
/2) - 1)^6)/(38281250*(3^(1/2) - (5*x + 3)^(1/2))^6) + (6732597583*3^(1/2) 
*((1 - 2*x)^(1/2) - 1)^8)/(267968750*(3^(1/2) - (5*x + 3)^(1/2))^8) - (402 
724691*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(6125000*(3^(1/2) - (5*x + 3)^(1/ 
2))^10) + (3968017*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(35000*(3^(1/2) - (5* 
x + 3)^(1/2))^12) + (474659*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(15680*(3^(1 
/2) - (5*x + 3)^(1/2))^14))/((45056*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1 
/2) - (5*x + 3)^(1/2))^2) + (294784*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1 
/2) - (5*x + 3)^(1/2))^4) - (1921024*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^( 
1/2) - (5*x + 3)^(1/2))^6) + (5828656*((1 - 2*x)^(1/2) - 1)^8)/(390625*...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.93 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(64343565*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqr 
t(5))/sqrt(11))/2))/sqrt(2))*x**4 + 171582840*sqrt(7)*atan((sqrt(33) - sqr 
t(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 17 
1582840*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt( 
5))/sqrt(11))/2))/sqrt(2))*x**2 + 76259040*sqrt(7)*atan((sqrt(33) - sqrt(3 
5)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 12709840 
*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqr 
t(11))/2))/sqrt(2)) - 64343565*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin( 
(sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 171582840*sqrt(7) 
*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2 
))/sqrt(2))*x**3 - 171582840*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((s 
qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 76259040*sqrt(7)*at 
an((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/ 
sqrt(2))*x - 12709840*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 
2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 13127415*sqrt(5*x + 3)*sqrt( - 
2*x + 1)*x**3 + 26843320*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 18321716*sq 
rt(5*x + 3)*sqrt( - 2*x + 1)*x + 4160912*sqrt(5*x + 3)*sqrt( - 2*x + 1))/( 
153664*(81*x**4 + 216*x**3 + 216*x**2 + 96*x + 16))