\(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx\) [1226]

Optimal result

Integrand size = 26, antiderivative size = 166 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=-\frac {101485 \sqrt {3+5 x}}{45276 (1-2 x)^{3/2}}-\frac {3471145 \sqrt {3+5 x}}{3486252 \sqrt {1-2 x}}+\frac {\sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {193 \sqrt {3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac {423 \sqrt {3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}-\frac {330255 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}} \] Output:

-101485/45276*(3+5*x)^(1/2)/(1-2*x)^(3/2)-3471145/3486252*(3+5*x)^(1/2)/(1 
-2*x)^(1/2)+1/7*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^3+193/196*(3+5*x)^(1/2 
)/(1-2*x)^(3/2)/(2+3*x)^2+423/56*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)-33025 
5/134456*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 = 84, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {\frac {7 \sqrt {3+5 x} \left (44829024-48873610 x-244982277 x^2+140350860 x^3+374883660 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^3}-119882565 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{48807528} \] Input:

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

Output:

((7*Sqrt[3 + 5*x]*(44829024 - 48873610*x - 244982277*x^2 + 140350860*x^3 + 
 374883660*x^4))/((1 - 2*x)^(3/2)*(2 + 3*x)^3) - 119882565*Sqrt[7]*ArcTan[ 
Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/48807528
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {114, 27, 168, 27, 168, 27, 169, 27, 169, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{21} \int \frac {3 (11-80 x)}{2 (1-2 x)^{5/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \int \frac {11-80 x}{(1-2 x)^{5/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (\frac {1}{14} \int -\frac {11580 x+811}{2 (1-2 x)^{5/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac {1}{28} \int \frac {11580 x+811}{(1-2 x)^{5/2} (3 x+2)^2 \sqrt {5 x+3}}dx\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {1}{7} \int \frac {35 (23688 x+8453)}{2 (1-2 x)^{5/2} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \int \frac {23688 x+8453}{(1-2 x)^{5/2} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \left (\frac {81188 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int -\frac {1217820 x+85319}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx\right )\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \left (\frac {1}{231} \int \frac {1217820 x+85319}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {81188 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \left (\frac {1}{231} \left (\frac {2776916 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int \frac {23976513}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {81188 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \left (\frac {1}{231} \left (\frac {2776916 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2179683}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {81188 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \left (\frac {1}{231} \left (\frac {2776916 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {4359366}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {81188 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{14} \left (\frac {1}{28} \left (\frac {2961 \sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}-\frac {5}{2} \left (\frac {1}{231} \left (\frac {4359366 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}+\frac {2776916 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {81188 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\right )+\frac {193 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}\)

Input:

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

Output:

Sqrt[3 + 5*x]/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + ((193*Sqrt[3 + 5*x])/(14*( 
1 - 2*x)^(3/2)*(2 + 3*x)^2) + ((2961*Sqrt[3 + 5*x])/((1 - 2*x)^(3/2)*(2 + 
3*x)) - (5*((81188*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)) + ((2776916*Sqrt[3 
 + 5*x])/(77*Sqrt[1 - 2*x]) + (4359366*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[ 
3 + 5*x])])/(7*Sqrt[7]))/231))/2)/28)/14
 

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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(127)=254\).

Time = 0.27 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.80

method	result	size
default	\(\frac {\left (12947317020 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-5394715425 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}-6953188770 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -3429751878 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-684230540 x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{97615056 \left (2+3 x \right )^{3} \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\)	\(298\)

Input:

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

Output:

1/97615056*(12947317020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3 
)^(1/2))*x^5+12947317020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 
3)^(1/2))*x^4-5394715425*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 
3)^(1/2))*x^3+5248371240*x^4*(-10*x^2-x+3)^(1/2)-6953188770*7^(1/2)*arctan 
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1964912040*x^3*(-10*x^2-x 
+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 
/2))*x-3429751878*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(3 
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-684230540*x*(-10*x^2-x+3)^(1/2)+62760 
6336*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(3/2)/(-10*x^2-x 
+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=-\frac {119882565 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (374883660 \, x^{4} + 140350860 \, x^{3} - 244982277 \, x^{2} - 48873610 \, x + 44829024\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{97615056 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \] Input:

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

Output:

-1/97615056*(119882565*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x 
+ 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 
+ x - 3)) - 14*(374883660*x^4 + 140350860*x^3 - 244982277*x^2 - 48873610*x 
 + 44829024)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 5 
8*x^2 + 4*x + 8)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (127) = 254\).

Time = 0.38 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {66051}{537824} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {32 \, {\left (932 \, \sqrt {5} {\left (5 \, x + 3\right )} - 5511 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{152523525 \, {\left (2 \, x - 1\right )}^{2}} + \frac {297 \, \sqrt {10} {\left (15599 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 5723200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {607208000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {2428832000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \] Input:

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

Output:

66051/537824*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)))) - 32/152523525*(932*sqrt(5)*(5*x + 3) - 5511*sqrt(5) 
)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 297/67228*sqrt(10)*(15599*(( 
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 + 5723200*((sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 
)))^3 + 607208000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 242 
8832000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqr 
t(-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. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4\,\sqrt {5\,x+3}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.00 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(6473658510*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((s 
qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 9710487765*sqrt( - 
2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt 
(5))/sqrt(11))/2))/sqrt(2))*x**3 + 2157886170*sqrt( - 2*x + 1)*sqrt(7)*ata 
n((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/s 
qrt(2))*x**2 - 2397651300*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(3 
5)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 95906052 
0*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x 
 + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 6473658510*sqrt( - 2*x + 1)*sqrt(7 
)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/ 
2))/sqrt(2))*x**4 - 9710487765*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + s 
qrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 
2157886170*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sq 
rt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 2397651300*sqrt( - 2 
*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt( 
5))/sqrt(11))/2))/sqrt(2))*x + 959060520*sqrt( - 2*x + 1)*sqrt(7)*atan((sq 
rt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2 
)) - 2624185620*sqrt(5*x + 3)*x**4 - 982456020*sqrt(5*x + 3)*x**3 + 171487 
5939*sqrt(5*x + 3)*x**2 + 342115270*sqrt(5*x + 3)*x - 313803168*sqrt(5*x + 
 3))/(48807528*sqrt( - 2*x + 1)*(54*x**4 + 81*x**3 + 18*x**2 - 20*x - 8...