\(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx\) [1026]

Optimal result

Integrand size = 26, antiderivative size = 238 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {103 (1-2 x)^{5/2} \sqrt {3+5 x}}{4536 (2+3 x)^5}+\frac {29075 (1-2 x)^{3/2} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {1028605 \sqrt {1-2 x} \sqrt {3+5 x}}{4572288 (2+3 x)^3}+\frac {200146505 \sqrt {1-2 x} \sqrt {3+5 x}}{128024064 (2+3 x)^2}+\frac {20886641735 \sqrt {1-2 x} \sqrt {3+5 x}}{1792336896 (2+3 x)}-\frac {185 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5292 (2+3 x)^6}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}-\frac {327738785 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2458624 \sqrt {7}} \] Output:

-103/4536*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^5+29075/254016*(1-2*x)^(3/2) 
*(3+5*x)^(1/2)/(2+3*x)^4+1028605/4572288*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3* 
x)^3+200146505/128024064*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+20886641735 
*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(3584673792+5377010688*x)-185/5292*(1-2*x)^(5 
/2)*(3+5*x)^(3/2)/(2+3*x)^6-1/21*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7-327 
738785/17210368*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.39 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (5897927808+52456780256 x+194338741616 x^2+384048502848 x^3+427105196104 x^4+253441751890 x^5+62659925205 x^6\right )}{(2+3 x)^7}-983216355 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{51631104} \] Input:

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]
 

Output:

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5897927808 + 52456780256*x + 194338741616 
*x^2 + 384048502848*x^3 + 427105196104*x^4 + 253441751890*x^5 + 6265992520 
5*x^6))/(2 + 3*x)^7 - 983216355*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt 
[3 + 5*x])])/51631104
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {107, 105, 105, 105, 105, 105, 105, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^8} \, dx\)

\(\Big \downarrow \) 107

\(\displaystyle \frac {37}{14} \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^7}dx+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{(3 x+2)^6}dx+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^5}dx+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \left (\frac {11}{8} \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \left (\frac {11}{8} \left (\frac {55}{42} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {37}{14} \left (\frac {55}{12} \left (\frac {33}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}\right )+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}\)

Input:

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]
 

Output:

(3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/(49*(2 + 3*x)^7) + (37*(((1 - 2*x)^(5/ 
2)*(3 + 5*x)^(7/2))/(6*(2 + 3*x)^6) + (55*(((1 - 2*x)^(3/2)*(3 + 5*x)^(7/2 
))/(5*(2 + 3*x)^5) + (33*((Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(4*(2 + 3*x)^4) 
+ (11*(-1/21*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^3 + (55*(-1/14*(Sqr 
t[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^2 + (33*(-1/7*(Sqrt[1 - 2*x]*Sqrt[3 
+ 5*x])/(2 + 3*x) - (11*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7* 
Sqrt[7])))/28))/42))/8))/10))/12))/14
 

Defintions of rubi rules used

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 105

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

rule 107

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[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

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 [A] (verified)

Time = 1.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (62659925205 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{7375872 \left (2+3 x \right )^{7} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {327738785 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{34420736 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(144\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (2150294168385 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{103262208 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\)	\(394\)

Input:

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x,method=_RETURNVERBOSE)
 

Output:

-1/7375872*(-1+2*x)*(3+5*x)^(1/2)*(62659925205*x^6+253441751890*x^5+427105 
196104*x^4+384048502848*x^3+194338741616*x^2+52456780256*x[phone])/(2+ 
3*x)^7/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+327 
738785/34420736*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+6 
7+111*x)^(1/2))*((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 = 161, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {983216355 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (62659925205 \, x^{6} + 253441751890 \, x^{5} + 427105196104 \, x^{4} + 384048502848 \, x^{3} + 194338741616 \, x^{2} + 52456780256 \, x + 5897927808\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{103262208 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="fricas")
 

Output:

-1/103262208*(983216355*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680* 
x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*arctan(1/14*sqrt(7)*(37*x + 20) 
*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(62659925205*x^6 + 25 
3441751890*x^5 + 427105196104*x^4 + 384048502848*x^3 + 194338741616*x^2 + 
52456780256*x + 5897927808)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2187*x^7 + 1020 
6*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
 

Sympy [F]

\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{8}}\, dx \] Input:

integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**8,x)
 

Output:

Integral((1 - 2*x)**(5/2)*(5*x + 3)**(5/2)/(3*x + 2)**8, x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.48 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {122277415}{271063296} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{49 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {1369 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{2744 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {162319 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{153664 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {3024121 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{2151296 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {24455483 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {2190708025}{180708864} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {4205402795}{361417728} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {4059472427 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1084253184 \, {\left (3 \, x + 2\right )}} + \frac {501088225}{8605184} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {327738785}{34420736} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {441499355}{17210368} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="maxima")
 

Output:

122277415/271063296*(-10*x^2 - x + 3)^(5/2) + 3/49*(-10*x^2 - x + 3)^(7/2) 
/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 13 
44*x + 128) + 37/196*(-10*x^2 - x + 3)^(7/2)/(729*x^6 + 2916*x^5 + 4860*x^ 
4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 1369/2744*(-10*x^2 - x + 3)^(7/2)/ 
(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 162319/153664*(-10 
*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 3024121/215 
1296*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 24455483/60236 
288*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 2190708025/180708864*(-10 
*x^2 - x + 3)^(3/2)*x + 4205402795/361417728*(-10*x^2 - x + 3)^(3/2) - 405 
9472427/1084253184*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 501088225/8605184*s 
qrt(-10*x^2 - x + 3)*x + 327738785/34420736*sqrt(7)*arcsin(37/11*x/abs(3*x 
 + 2) + 20/11/abs(3*x + 2)) - 441499355/17210368*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).

Time = 0.65 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.28 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {65547757}{68841472} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8857805 \, \sqrt {10} {\left (111 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{13} + 207200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 164185280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 63583027200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 12872125952000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 1273567232000000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {53489823744000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {213959294976000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3687936 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{7}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="giac")
 

Output:

65547757/68841472*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)*sqr 
t(-10*x + 5) - sqrt(22)))) - 8857805/3687936*sqrt(10)*(111*((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)))^13 + 207200*((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)))^11 + 1641 
85280*((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)))^9 - 63583027200*((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)))^7 - 12872125952000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s 
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 1 
273567232000000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq 
rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 53489823744000000*(s 
qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 213959294976000000*sqrt 
(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) 
 - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22)))^2 + 280)^7
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^8} \,d x \] Input:

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^8,x)
 

Output:

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 727, normalized size of antiderivative = 3.05 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx =\text {Too large to display} \] Input:

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x)
 

Output:

(2150294168385*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1 
)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**7 + 10034706119130*sqrt(7)*atan((sqrt 
(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) 
*x**6 + 20069412238260*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 
 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 + 22299346931400*sqrt(7)*at 
an((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/ 
sqrt(2))*x**4 + 14866231287600*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin( 
(sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 5946492515040*sqr 
t(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11 
))/2))/sqrt(2))*x**2 + 1321442781120*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan 
(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 125851693440*s 
qrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 
11))/2))/sqrt(2)) - 2150294168385*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(as 
in((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**7 - 10034706119130 
*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqr 
t(11))/2))/sqrt(2))*x**6 - 20069412238260*sqrt(7)*atan((sqrt(33) + sqrt(35 
)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 - 222993 
46931400*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt 
(5))/sqrt(11))/2))/sqrt(2))*x**4 - 14866231287600*sqrt(7)*atan((sqrt(33) + 
 sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x*...