\(\int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\) [1207]

Optimal result

Integrand size = 26, antiderivative size = 182 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {4270537963 \sqrt {1-2 x} \sqrt {3+5 x}}{409600}-\frac {4270537963 \sqrt {1-2 x} (3+5 x)^{3/2}}{3379200}-\frac {22981 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}-\frac {481599 \sqrt {1-2 x} (3+5 x)^{5/2}}{12800}+\frac {8451 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {81}{160} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {2401 (3+5 x)^{7/2}}{264 (1-2 x)^{3/2}}+\frac {46975917593 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{409600 \sqrt {10}} \] Output:

-4270537963/409600*(1-2*x)^(1/2)*(3+5*x)^(1/2)-4270537963/3379200*(1-2*x)^ 
(1/2)*(3+5*x)^(3/2)-22981/66*(3+5*x)^(5/2)/(1-2*x)^(1/2)-481599/12800*(1-2 
*x)^(1/2)*(3+5*x)^(5/2)+8451/1280*(1-2*x)^(3/2)*(3+5*x)^(5/2)-81/160*(1-2* 
x)^(5/2)*(3+5*x)^(5/2)+2401/264*(3+5*x)^(7/2)/(1-2*x)^(3/2)[phone]/40 
96000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.51 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {-\frac {5 \sqrt {3+5 x} \left (21368105901-58600061024 x+18987469764 x^2+8217694800 x^3+4002203520 x^4+1423526400 x^5+248832000 x^6\right )}{(1-2 x)^{3/2}}-140927752779 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{6144000} \] Input:

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

Output:

((-5*Sqrt[3 + 5*x]*(21368105901 - 58600061024*x + 18987469764*x^2 + 821769 
4800*x^3 + 4002203520*x^4 + 1423526400*x^5 + 248832000*x^6))/(1 - 2*x)^(3/ 
2) - 140927752779*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x 
])])/6144000
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {108, 27, 167, 27, 170, 27, 164, 60, 60, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {(3 x+2)^4 (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(3 x+2)^3 (5 x+3)^{3/2} (195 x+122)}{2 (1-2 x)^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 x+2)^4 (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{6} \int \frac {(3 x+2)^3 (5 x+3)^{3/2} (195 x+122)}{(1-2 x)^{3/2}}dx\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{6} \left (-\frac {1}{11} \int -\frac {(3 x+2)^2 (5 x+3)^{3/2} (72285 x+45556)}{2 \sqrt {1-2 x}}dx-\frac {439 (5 x+3)^{5/2} (3 x+2)^3}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \int \frac {(3 x+2)^2 (5 x+3)^{3/2} (72285 x+45556)}{\sqrt {1-2 x}}dx-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (-\frac {1}{50} \int -\frac {5 (3 x+2) (5 x+3)^{3/2} (4540485 x+2892058)}{2 \sqrt {1-2 x}}dx-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {1}{20} \int \frac {(3 x+2) (5 x+3)^{3/2} (4540485 x+2892058)}{\sqrt {1-2 x}}dx-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {1}{20} \left (\frac {4270537963}{320} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx-\frac {3}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)\right )-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {1}{20} \left (\frac {4270537963}{320} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)\right )-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {1}{20} \left (\frac {4270537963}{320} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)\right )-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {1}{20} \left (\frac {4270537963}{320} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)\right )-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {1}{20} \left (\frac {4270537963}{320} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)\right )-\frac {14457}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {439 (3 x+2)^3 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}\)

Input:

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

Output:

((2 + 3*x)^4*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) + ((-439*(2 + 3*x)^3*(3 
+ 5*x)^(5/2))/(11*Sqrt[1 - 2*x]) + ((-14457*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 
 5*x)^(5/2))/10 + ((-3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(36714139 + 18161940* 
x))/160 + (4270537963*(-1/4*(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (33*(-1/2*(S 
qrt[1 - 2*x]*Sqrt[3 + 5*x]) + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqr 
t[10])))/8))/320)/20)/22)/6
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 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 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 = 0.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {\left (-4976640000 \sqrt {-10 x^{2}-x +3}\, x^{6}-28470528000 x^{5} \sqrt {-10 x^{2}-x +3}-80044070400 x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-164353896000 x^{3} \sqrt {-10 x^{2}-x +3}-563711011116 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -379749395280 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-427362118020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{24576000 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\)	\(181\)

Input:

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

Output:

1/24576000*(-4976640000*(-10*x^2-x+3)^(1/2)*x^6-28470528000*x^5*(-10*x^2-x 
+3)^(1/2)-80044070400*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin 
(20/11*x+1/11)*x^2-164353896000*x^3*(-10*x^2-x+3)^(1/2)-563711011116*10^(1 
/2)*arcsin(20/11*x+1/11)*x-379749395280*x^2*(-10*x^2-x+3)^(1/2)[phone] 
79*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-42736 
2118020*(-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 = 111, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {140927752779 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (248832000 \, x^{6} + 1423526400 \, x^{5} + 4002203520 \, x^{4} + 8217694800 \, x^{3} + 18987469764 \, x^{2} - 58600061024 \, x + 21368105901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24576000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:

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

Output:

-1/24576000*(140927752779*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*(248832000* 
x^6 + 1423526400*x^5 + 4002203520*x^4 + 8217694800*x^3 + 18987469764*x^2 - 
 58600061024*x + 21368105901)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 
 1)
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.95 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {81}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {891}{256} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11872553}{2048} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {514294407}{8192000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {139491}{5120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {2401 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {1029 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {189 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (2 \, x - 1\right )}} - \frac {4250367}{20480} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {89257707}{409600} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {800415}{512} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {132055 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{384 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {56595 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{64 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {24255 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (2 \, x - 1\right )}} + \frac {1452605 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {15827735 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (2 \, x - 1\right )}} \] Input:

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

Output:

-81/160*(-10*x^2 - x + 3)^(5/2) + 891/256*(-10*x^2 - x + 3)^(3/2)*x + 1187 
2553/2048*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 514294407/8192000*I*sqr 
t(5)*sqrt(2)*arcsin(20/11*x - 21/11) + 139491/5120*(-10*x^2 - x + 3)^(3/2) 
 - 2401/32*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 
1029/16*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 441/16*(-10*x 
^2 - x + 3)^(5/2)/(4*x^2 - 4*x + 1) - 189/32*(-10*x^2 - x + 3)^(5/2)/(2*x 
- 1) - 4250367/20480*sqrt(10*x^2 - 21*x + 8)*x + 89257707/409600*sqrt(10*x 
^2 - 21*x + 8) - 800415/512*sqrt(-10*x^2 - x + 3) - 132055/384*(-10*x^2 - 
x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 56595/64*(-10*x^2 - x + 3)^(3/2) 
/(4*x^2 - 4*x + 1) + 24255/128*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 1452605 
/768*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 15827735/768*sqrt(-10*x^2 - 
 x + 3)/(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {46975917593}{4096000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (12 \, {\left (72 \, {\left (4 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} + 509 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 20743 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 18487133 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4270537963 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 469759175930 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7751026402845 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{768000000 \, {\left (2 \, x - 1\right )}^{2}} \] Input:

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

Output:

46975917593/4096000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/76800 
0000*(4*(3*(12*(72*(4*(48*sqrt(5)*(5*x + 3) + 509*sqrt(5))*(5*x + 3) + 207 
43*sqrt(5))*(5*x + 3) + 18487133*sqrt(5))*(5*x + 3) + 4270537963*sqrt(5))* 
(5*x + 3) - 469759175930*sqrt(5))*(5*x + 3) + 7751026402845*sqrt(5))*sqrt( 
5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {-281855505558 \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x [phone] \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )[phone] \sqrt {5 x +3}\, x^{6}[phone] \sqrt {5 x +3}\, x^{5}[phone] \sqrt {5 x +3}\, x^{4}[phone] \sqrt {5 x +3}\, x^{3}[phone] \sqrt {5 x +3}\, x^{2}-586000610240 \sqrt {5 x +3}\, x [phone] \sqrt {5 x +3}}{12288000 \sqrt {-2 x +1}\, \left (2 x -1\right )} \] Input:

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

Output:

( - 281855505558*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5)) 
/sqrt(11))*x + 140927752779*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 
1)*sqrt(5))/sqrt(11)) + 2488320000*sqrt(5*x + 3)*x**6 + 14235264000*sqrt(5 
*x + 3)*x**5 + 40022035200*sqrt(5*x + 3)*x**4 + 82176948000*sqrt(5*x + 3)* 
x**3 + 189874697640*sqrt(5*x + 3)*x**2 - 586000610240*sqrt(5*x + 3)*x + 21 
3681059010*sqrt(5*x + 3))/(12288000*sqrt( - 2*x + 1)*(2*x - 1))