3.25.21 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\) [2421]

3.25.21.1 Optimal result

Integrand size = 26, antiderivative size = 209 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {805255 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {73205 \sqrt {1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}-\frac {8857805 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]

output

1/6*(1-2*x)^(5/2)*(3+5*x)^(7/2)/(2+3*x)^6+11/12*(1-2*x)^(3/2)*(3+5*x)^(7/2 
)/(2+3*x)^5-8857805/1229312*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2) 
)*7^(1/2)-73205/37632*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-1331/1344*(3+5 
*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3+121/32*(3+5*x)^(7/2)*(1-2*x)^(1/2)/(2+3* 
x)^4-805255/175616*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
 

3.25.21.2 Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (79536960+589734736 x+1743189856 x^2+2573967504 x^3+1905431420 x^4+568572155 x^5\right )}{(2+3 x)^6}-26573415 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3687936} \]

input

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]
 

output

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(79536960 + 589734736*x + 1743189856*x^2 + 
 2573967504*x^3 + 1905431420*x^4 + 568572155*x^5))/(2 + 3*x)^6 - 26573415* 
Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3687936
 

3.25.21.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {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.

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 217

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

input

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]
 

output

((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*(Sqrt[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
 

3.25.21.3.1 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 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])
 

3.25.21.4 Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (568572155 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{526848 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {8857805 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(139\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (19372019535 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\)	\(346\)

input

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x,method=_RETURNVERBOSE)
 

output

-1/526848*(-1+2*x)*(3+5*x)^(1/2)*(568572155*x^5+1905431420*x^4+2573967504* 
x^3+1743189856*x^2+589734736*x[phone])/(2+3*x)^6/(-(-1+2*x)*(3+5*x))^(1/ 
2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+8857805/2458624*7^(1/2)*arctan(9/ 
14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((1-2*x)*(3+5*x)) 
^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

3.25.21.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {26573415 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (568572155 \, x^{5} + 1905431420 \, x^{4} + 2573967504 \, x^{3} + 1743189856 \, x^{2} + 589734736 \, x + 79536960\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="fricas")
 

output

-1/7375872*(26573415*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2 
160*x^2 + 576*x + 64)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(- 
2*x + 1)/(10*x^2 + x - 3)) - 14*(568572155*x^5 + 1905431420*x^4 + 25739675 
04*x^3 + 1743189856*x^2 + 589734736*x + 79536960)*sqrt(5*x + 3)*sqrt(-2*x 
+ 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
 

3.25.21.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\text {Timed out} \]

input

integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**7,x)
 

output

Timed out
 

3.25.21.7 Maxima [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.44 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {3304795}{19361664} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{14 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {4387 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{10976 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {81733 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{153664 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {660959 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{4302592 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {59208325}{12907776} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {113659535}{25815552} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {109715471 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{77446656 \, {\left (3 \, x + 2\right )}} + \frac {13542925}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {8857805}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {11932415}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="maxima")
 

output

3304795/19361664*(-10*x^2 - x + 3)^(5/2) + 1/14*(-10*x^2 - x + 3)^(7/2)/(7 
29*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 37/196* 
(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 
32) + 4387/10976*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96* 
x + 16) + 81733/153664*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8 
) + 660959/4302592*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 59208325/1 
2907776*(-10*x^2 - x + 3)^(3/2)*x + 113659535/25815552*(-10*x^2 - x + 3)^( 
3/2) - 109715471/77446656*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 13542925/614 
656*sqrt(-10*x^2 - x + 3)*x + 8857805/2458624*sqrt(7)*arcsin(37/11*x/abs(3 
*x + 2) + 20/11/abs(3*x + 2)) - 11932415/1229312*sqrt(-10*x^2 - x + 3)
 

3.25.21.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).

Time = 0.68 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {1771561}{4917248} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8857805 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 4760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 3104640 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 869299200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 104491520000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {5163110400000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {20652441600000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="giac")
 

output

1771561/4917248*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)*sqrt( 
-10*x + 5) - sqrt(22)))) - 8857805/263424*sqrt(10)*(3*((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 + 4760*((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 + 3104640*((sq 
rt(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 - 869299200*((sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 
)))^5 - 104491520000*((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)))^3 - 5163110400000*( 
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 20652441600000*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) - sqrt( 
22)))^2 + 280)^6
 

3.25.21.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \]

input

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^7,x)
 

output

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^7, x)