\(\int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\) [1218]

Optimal result

Integrand size = 26, antiderivative size = 116 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2401 \sqrt {3+5 x}}{264 (1-2 x)^{3/2}}-\frac {55909 \sqrt {3+5 x}}{1452 \sqrt {1-2 x}}-\frac {12447 \sqrt {1-2 x} \sqrt {3+5 x}}{1600}+\frac {81}{160} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {392283 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}} \] Output:

2401/264*(3+5*x)^(1/2)/(1-2*x)^(3/2)-55909/1452*(3+5*x)^(1/2)/(1-2*x)^(1/2 
)-12447/1600*(1-2*x)^(1/2)*(3+5*x)^(1/2)+81/160*(1-2*x)^(3/2)*(3+5*x)^(1/2 
)+392283/16000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {-10 \sqrt {3+5 x} \left (21305631-61036064 x+14544684 x^2+2352240 x^3\right )[phone] \sqrt {10-20 x} (-1+2 x) \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{5808000 (1-2 x)^{3/2}} \] Input:

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

Output:

(-10*Sqrt[3 + 5*x]*(21305631 - 61036064*x + 14544684*x^2 + 2352240*x^3) + 
142398729*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] 
)/(5808000*(1 - 2*x)^(3/2))
 

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 = {109, 27, 167, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {7 (3 x+2)^3 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(3 x+2)^2 (717 x+436)}{2 (1-2 x)^{3/2} \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 (3 x+2)^3 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(3 x+2)^2 (717 x+436)}{(1-2 x)^{3/2} \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{66} \left (-\frac {1}{11} \int -\frac {3 (3 x+2) (39667 x+24326)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1589 \sqrt {5 x+3} (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{66} \left (\frac {3}{22} \int \frac {(3 x+2) (39667 x+24326)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1589 (3 x+2)^2 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{66} \left (\frac {3}{22} \left (\frac {47466243}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{400} \sqrt {1-2 x} \sqrt {5 x+3} (2380020 x+5735477)\right )-\frac {1589 (3 x+2)^2 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{66} \left (\frac {3}{22} \left (\frac {47466243 \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}}{2000}-\frac {1}{400} \sqrt {1-2 x} \sqrt {5 x+3} (2380020 x+5735477)\right )-\frac {1589 (3 x+2)^2 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{66} \left (\frac {3}{22} \left (\frac {47466243 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}}-\frac {1}{400} \sqrt {1-2 x} \sqrt {5 x+3} (2380020 x+5735477)\right )-\frac {1589 (3 x+2)^2 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}\)

Input:

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

Output:

(7*(2 + 3*x)^3*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + ((-1589*(2 + 3*x)^2*S 
qrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*(-1/400*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x] 
*(5735477 + 2380020*x)) + (47466243*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400 
*Sqrt[10])))/22)/66
 

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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 167

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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

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 = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {\left (569594916 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-47044800 x^{3} \sqrt {-10 x^{2}-x +3}-569594916 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -290893680 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-426112620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{11616000 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\)	\(130\)

Input:

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

Output:

1/11616000*(569594916*10^(1/2)*arcsin(20/11*x+1/11)*x^2-47044800*x^3*(-10* 
x^2-x+3)^(1/2)-569594916*10^(1/2)*arcsin(20/11*x+1/11)*x-290893680*x^2*(-1 
0*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10 
*x^2-x+3)^(1/2)-426112620*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/(1-2*x)^(3/2) 
/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {142398729 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2352240 \, x^{3} + 14544684 \, x^{2} - 61036064 \, x + 21305631\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11616000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:

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

Output:

-1/11616000*(142398729*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20 
*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(2352240*x^3 + 
 14544684*x^2 - 61036064*x + 21305631)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^ 
2 - 4*x + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {392283}{32000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {81}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {11637}{1600} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2401 \, \sqrt {-10 \, x^{2} - x + 3}}{264 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {55909 \, \sqrt {-10 \, x^{2} - x + 3}}{1452 \, {\left (2 \, x - 1\right )}} \] Input:

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

Output:

392283/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 81/80*sqrt(-10*x^2 - 
 x + 3)*x - 11637/1600*sqrt(-10*x^2 - x + 3) + 2401/264*sqrt(-10*x^2 - x + 
 3)/(4*x^2 - 4*x + 1) + 55909/1452*sqrt(-10*x^2 - x + 3)/(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {392283}{16000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9801 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 263 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 94936582 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1566381795 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{72600000 \, {\left (2 \, x - 1\right )}^{2}} \] Input:

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

Output:

392283/16000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/72600000*(4* 
(9801*(12*sqrt(5)*(5*x + 3) + 263*sqrt(5))*(5*x + 3) - 94936582*sqrt(5))*( 
5*x + 3) + 1566381795*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {-284797458 \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x [phone] \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )[phone] \sqrt {5 x +3}\, x^{3}[phone] \sqrt {5 x +3}\, x^{2}-610360640 \sqrt {5 x +3}\, x [phone] \sqrt {5 x +3}}{5808000 \sqrt {-2 x +1}\, \left (2 x -1\right )} \] Input:

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

Output:

( - 284797458*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sq 
rt(11))*x + 142398729*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqr 
t(5))/sqrt(11)) + 23522400*sqrt(5*x + 3)*x**3 + 145446840*sqrt(5*x + 3)*x* 
*2 - 610360640*sqrt(5*x + 3)*x + 213056310*sqrt(5*x + 3))/(5808000*sqrt( - 
 2*x + 1)*(2*x - 1))