\(\int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\) [913]

Optimal result

Integrand size = 26, antiderivative size = 116 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2}}{1375 \sqrt {3+5 x}}+\frac {10409 \sqrt {1-2 x} \sqrt {3+5 x}}{44000}-\frac {801 (1-2 x)^{3/2} \sqrt {3+5 x}}{2000}+\frac {9}{100} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {10409 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4000 \sqrt {10}} \] Output:

-2/1375*(1-2*x)^(3/2)/(3+5*x)^(1/2)+10409/44000*(1-2*x)^(1/2)*(3+5*x)^(1/2 
)-801/2000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+9/100*(1-2*x)^(5/2)*(3+5*x)^(1/2)+1 
0409/40000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (-893+3825 x+13140 x^2+7200 x^3\right )-10409 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{40000 \sqrt {3+5 x}} \] Input:

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

Output:

(10*Sqrt[1 - 2*x]*(-893 + 3825*x + 13140*x^2 + 7200*x^3) - 10409*Sqrt[30 + 
 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(40000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {108, 27, 170, 27, 164, 64, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

\(\displaystyle \frac {14}{5} \left (\frac {1}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{30} \int -\frac {3 (6-5 x) (3 x+2)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {14}{5} \left (\frac {1}{20} \int \frac {(6-5 x) (3 x+2)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {14}{5} \left (\frac {1}{20} \left (\frac {1487}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} (73-60 x) \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {14}{5} \left (\frac {1}{20} \left (\frac {1487}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} (73-60 x) \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {14}{5} \left (\frac {1}{20} \left (\frac {1487 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} (73-60 x) \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

Input:

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

Output:

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (14*((Sqrt[1 - 2*x]*(2 
+ 3*x)^2*Sqrt[3 + 5*x])/10 + (-1/80*((73 - 60*x)*Sqrt[1 - 2*x]*Sqrt[3 + 5* 
x]) + (1487*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10]))/20))/5
 

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 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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

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] 
 && IntegerQ[m]
 

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 = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {\left (144000 x^{3} \sqrt {-10 x^{2}-x +3}+52045 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +262800 x^{2} \sqrt {-10 x^{2}-x +3}+31227 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+76500 x \sqrt {-10 x^{2}-x +3}-17860 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{80000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(116\)

Input:

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

Output:

1/80000*(144000*x^3*(-10*x^2-x+3)^(1/2)+52045*10^(1/2)*arcsin(20/11*x+1/11 
)*x+262800*x^2*(-10*x^2-x+3)^(1/2)+31227*10^(1/2)*arcsin(20/11*x+1/11)+765 
00*x*(-10*x^2-x+3)^(1/2)-17860*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2 
-x+3)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {10409 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (7200 \, x^{3} + 13140 \, x^{2} + 3825 \, x - 893\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{80000 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

-1/80000*(10409*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5* 
x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(7200*x^3 + 13140*x^2 + 3825* 
x - 893)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {10409}{80000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{250} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {81}{200} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {693}{20000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{625 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

10409/80000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/250*(-10*x^2 - x + 
3)^(3/2) + 81/200*sqrt(-10*x^2 - x + 3)*x + 693/20000*sqrt(-10*x^2 - x + 3 
) - 2/625*sqrt(-10*x^2 - x + 3)/(5*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {9}{100000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 463 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {10409}{40000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{3125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \] Input:

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

Output:

9/100000*(4*(8*sqrt(5)*(5*x + 3) + sqrt(5))*(5*x + 3) - 463*sqrt(5))*sqrt( 
5*x + 3)*sqrt(-10*x + 5) + 10409/40000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt( 
5*x + 3)) - 1/6250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x 
+ 3) + 2/3125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {-10409 \sqrt {5 x +3}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )+72000 \sqrt {-2 x +1}\, x^{3}+131400 \sqrt {-2 x +1}\, x^{2}+38250 \sqrt {-2 x +1}\, x -8930 \sqrt {-2 x +1}}{40000 \sqrt {5 x +3}} \] Input:

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

Output:

( - 10409*sqrt(5*x + 3)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) 
 + 72000*sqrt( - 2*x + 1)*x**3 + 131400*sqrt( - 2*x + 1)*x**2 + 38250*sqrt 
( - 2*x + 1)*x - 8930*sqrt( - 2*x + 1))/(40000*sqrt(5*x + 3))