\(\int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx\) [992]

Optimal result

Integrand size = 26, antiderivative size = 182 \[ \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {288621 (1-2 x)^{7/2} (3+5 x)^{3/2}}{280000}+\frac {3321 (1-2 x)^{9/2} (3+5 x)^{3/2}}{5600}-\frac {27}{280} (1-2 x)^{11/2} (3+5 x)^{3/2}+\frac {3735929329 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000000 \sqrt {10}} \] Output:

339629939/256000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)[phone]/76800000*(1-2*x)^ 
(3/2)*(3+5*x)^(1/2)+2806859/19200000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-255169/64 
0000*(1-2*x)^(7/2)*(3+5*x)^(1/2)-288621/280000*(1-2*x)^(7/2)*(3+5*x)^(3/2) 
+3321/5600*(1-2*x)^(9/2)*(3+5*x)^(3/2)-27/280*(1-2*x)^(11/2)*(3+5*x)^(3/2) 
[phone]/2560000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (-2037835593+86060710125 x+178815054340 x^2-205753776800 x^3-582885168000 x^4+30032640000 x^5+736128000000 x^6+414720000000 x^7\right )-78454515909 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{53760000000 \sqrt {3+5 x}} \] Input:

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

Output:

(10*Sqrt[1 - 2*x]*(-2037835593 + 86060710125*x + 178815054340*x^2 - 205753 
776800*x^3 - 582885168000*x^4 + 30032640000*x^5 + 736128000000*x^6 + 41472 
0000000*x^7) - 78454515909*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 
 5*x]])/(53760000000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {111, 27, 164, 60, 60, 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.

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

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{70} \int -\frac {1}{2} (1-2 x)^{5/2} (3 x+2) \sqrt {5 x+3} (801 x+506)dx-\frac {3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{140} \int (1-2 x)^{5/2} (3 x+2) \sqrt {5 x+3} (801 x+506)dx-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \int (1-2 x)^{5/2} \sqrt {5 x+3}dx}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{140} \left (\frac {1786183 \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )}{4000}-\frac {3 (1-2 x)^{7/2} (5 x+3)^{3/2} (26700 x+33857)}{2000}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{3/2}\)

Input:

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

Output:

(-3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/70 + ((-3*(1 - 2*x)^(7/2) 
*(3 + 5*x)^(3/2)*(33857 + 26700*x))/2000 + (1786183*(-1/8*((1 - 2*x)^(7/2) 
*Sqrt[3 + 5*x]) + (11*(((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + (11*(((1 - 2*x 
)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*Arc 
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/16))/4000)/140
 

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 111

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

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 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 = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.65

method	result	size
risch	\(-\frac {\left (82944000000 x^{6}[phone] x^{5}-52468992000 x^{4}-85095638400 x^{3}[phone] x^{2}[phone] x -679278531\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5376000000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3735929329 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5120000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(118\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1658880000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}-1049379840000 x^{4} \sqrt {-10 x^{2}-x +3}-1701912768000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-13585570620 \sqrt {-10 x^{2}-x +3}\right )}{107520000000 \sqrt {-10 x^{2}-x +3}}\)	\(155\)

Input:

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

Output:

-1/5376000000*(82944000000*x^6+97459200000*x^5-52468992000*x^4-85095638400 
*x^3+9906627680*x^2+29819034260*x-679278531)*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+ 
2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)[phone]/51200 
00000*10^(1/2)*arcsin(20/11*x+1/11)*((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 = 87, normalized size of antiderivative = 0.48 \[ \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {1}{5376000000} \, {\left (82944000000 \, x^{6} + 97459200000 \, x^{5} - 52468992000 \, x^{4} - 85095638400 \, x^{3} + 9906627680 \, x^{2} + 29819034260 \, x - 679278531\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3735929329}{5120000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

1/5376000000*(82944000000*x^6 + 97459200000*x^5 - 52468992000*x^4 - 850956 
38400*x^3 + 9906627680*x^2 + 29819034260*x - 679278531)*sqrt(5*x + 3)*sqrt 
(-2*x + 1) - 3735929329/5120000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1 
)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.66 \[ \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=-\frac {54}{35} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {1161}{700} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {47529}{70000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {5697497}{5600000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {5531929}{67200000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30875449}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3735929329}{5120000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30875449}{256000000} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

-54/35*(-10*x^2 - x + 3)^(3/2)*x^4 - 1161/700*(-10*x^2 - x + 3)^(3/2)*x^3 
+ 47529/70000*(-10*x^2 - x + 3)^(3/2)*x^2 + 5697497/5600000*(-10*x^2 - x + 
 3)^(3/2)*x - 5531929/67200000*(-10*x^2 - x + 3)^(3/2) + 30875449/12800000 
*sqrt(-10*x^2 - x + 3)*x - 3735929329/5120000000*sqrt(10)*arcsin(-20/11*x 
- 1/11) + 30875449/256000000*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (131) = 262\).

Time = 0.19 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.45 \[ \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx =\text {Too large to display} \] Input:

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

Output:

9/89600000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933) 
*(5*x + 3) - 7838433)*(5*x + 3) + 98794353)*(5*x + 3) - 1568443065)*(5*x + 
 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*arcs 
in(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/400000000*sqrt(5)*(2*(4*(8*(4*(16*(10 
0*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695)*(5 
*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcs 
in(1/11*sqrt(22)*sqrt(5*x + 3))) + 33/320000000*sqrt(5)*(2*(4*(8*(12*(80*x 
 - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5 
*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 
 3))) - 17/384000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3 
) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt 
(22)*sqrt(5*x + 3))) - 77/60000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293 
)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5 
*x + 3))) + 13/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 
143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 12/25*sqrt(5)*(11*sqrt( 
2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=-\frac {3735929329 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2560000000}+\frac {108 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{6}}{7}+\frac {1269 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{70}-\frac {68319 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{7000}-\frac {8864129 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{560000}+\frac {61916423 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{33600000}+\frac {1490951713 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{268800000}-\frac {226426177 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1792000000} \] Input:

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

Output:

( - 78454515909*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 82944 
0000000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**6 + 974592000000*sqrt(5*x + 3)*s 
qrt( - 2*x + 1)*x**5 - 524689920000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 
850956384000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 99066276800*sqrt(5*x + 
3)*sqrt( - 2*x + 1)*x**2 + 298190342600*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 
 6792785310*sqrt(5*x + 3)*sqrt( - 2*x + 1))/53760000000