\(\int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\) [999]

Optimal result

Integrand size = 26, antiderivative size = 149 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^2}+\frac {925 \sqrt {1-2 x} \sqrt {3+5 x}}{216 (2+3 x)}-\frac {8}{81} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {32765 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{648 \sqrt {7}} \] Output:

-1/9*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^3+5/12*(1-2*x)^(3/2)*(3+5*x)^(1/2 
)/(2+3*x)^2+925*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(432+648*x)-8/81*arcsin(1/11*2 
2^(1/2)*(3+5*x)^(1/2))*10^(1/2)-32765/4536*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 = 103, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (3856+11106 x+7689 x^2\right )}{(2+3 x)^3}+448 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-32765 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4536} \] Input:

Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^4,x]
 

Output:

((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3856 + 11106*x + 7689*x^2))/(2 + 3*x)^3 
+ 448*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 32765*Sqrt[7]*ArcTa 
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4536
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {108, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{9} \int -\frac {5 (1-2 x)^{3/2} (12 x+5)}{2 (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{18} \int \frac {(1-2 x)^{3/2} (12 x+5)}{(3 x+2)^3 \sqrt {5 x+3}}dx-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {5}{18} \left (-\frac {1}{6} \int \frac {3 \sqrt {1-2 x} (32 x+83)}{2 (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{18} \left (-\frac {1}{4} \int \frac {\sqrt {1-2 x} (32 x+83)}{(3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{3} \int -\frac {2099-128 x}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (-\frac {1}{6} \int \frac {2099-128 x}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 175

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {128}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {6553}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 64

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {256}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {6553}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {256}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {13106}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {256}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {13106 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {128}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {13106 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{2 (3 x+2)^2}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}\)

Input:

Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^4,x]
 

Output:

-1/9*((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^3 - (5*((-3*(1 - 2*x)^(3/2) 
*Sqrt[3 + 5*x])/(2*(2 + 3*x)^2) + ((-185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*( 
2 + 3*x)) + ((128*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (13106*A 
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/6)/4))/18
 

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]
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (7689 x^{2}+11106 x +3856\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{216 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {4 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{81}-\frac {32765 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{9072}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(138\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (884655 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-12096 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+1769310 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-24192 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+1179540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -16128 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +322938 x^{2} \sqrt {-10 x^{2}-x +3}+262120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3584 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+466452 x \sqrt {-10 x^{2}-x +3}+161952 \sqrt {-10 x^{2}-x +3}\right )}{9072 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\)	\(253\)

Input:

int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/216*(-1+2*x)*(3+5*x)^(1/2)*(7689*x^2+11106*x+3856)/(2+3*x)^3/(-(-1+2*x) 
*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)-(4/81*10^(1/2)*arcsi 
n(20/11*x+1/11)-32765/9072*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 = 156, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=-\frac {32765 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 448 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (7689 \, x^{2} + 11106 \, x + 3856\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9072 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^4,x, algorithm="fricas")
 

Output:

-1/9072*(32765*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(3 
7*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 448*sqrt(10)*(2 
7*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*s 
qrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(7689*x^2 + 11106*x + 3856)*sqrt(5*x 
+ 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=-\frac {4}{81} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {32765}{9072} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {145}{54} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{9 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {29 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1105 \, \sqrt {-10 \, x^{2} - x + 3}}{216 \, {\left (3 \, x + 2\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^4,x, algorithm="maxima")
 

Output:

-4/81*sqrt(10)*arcsin(20/11*x + 1/11) + 32765/9072*sqrt(7)*arcsin(37/11*x/ 
abs(3*x + 2) + 20/11/abs(3*x + 2)) + 145/54*sqrt(-10*x^2 - x + 3) + 7/9*(- 
10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 29/12*(-10*x^2 - x + 
3)^(3/2)/(9*x^2 + 12*x + 4) - 1105/216*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. 377 vs. \(2 (113) = 226\).

Time = 0.34 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.53 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\frac {6553}{18144} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4}{81} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (989 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 795200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {72520000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {290080000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{108 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^4,x, algorithm="giac")
 

Output:

6553/18144*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)))) - 4/81*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sq 
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 
) - sqrt(22)))) - 11/108*sqrt(10)*(989*((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 
 - 795200*((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)))^3 - 72520000*(sqrt(2)*sqrt(-10 
*x + 5) - sqrt(22))/sqrt(5*x + 3) + 290080000*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*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3
 

Mupad [F(-1)]

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

int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^4,x)
 

Output:

int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.86 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx =\text {Too large to display} \] Input:

int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^4,x)
 

Output:

(12096*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**3 + 24192*sqr 
t(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**2 + 16128*sqrt(10)*asin 
((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x + 3584*sqrt(10)*asin((sqrt( - 2*x 
+ 1)*sqrt(5))/sqrt(11)) + 884655*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asi 
n((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 1769310*sqrt(7) 
*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2 
))/sqrt(2))*x**2 + 1179540*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqr 
t( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 262120*sqrt(7)*atan((sqr 
t(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2) 
) - 884655*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sq 
rt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 1769310*sqrt(7)*atan((sqrt(33) + sqrt 
(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 117 
9540*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5)) 
/sqrt(11))/2))/sqrt(2))*x - 262120*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(a 
sin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 161469*sqrt(5*x + 
3)*sqrt( - 2*x + 1)*x**2 + 233226*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 80976 
*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(4536*(27*x**3 + 54*x**2 + 36*x + 8))