3.25.9 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\) [2409]

3.25.9.1 Optimal result

Integrand size = 26, antiderivative size = 209 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {3733455 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {113135 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {10285 \sqrt {1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}-\frac {41068005 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]

output

1/14*(1-2*x)^(7/2)*(3+5*x)^(5/2)/(2+3*x)^6+17/28*(1-2*x)^(5/2)*(3+5*x)^(5/ 
2)/(2+3*x)^5+935/224*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4-41068005/122931 
2*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-113135/12544*(3+ 
5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2+10285/448*(3+5*x)^(5/2)*(1-2*x)^(1/2)/( 
2+3*x)^3-3733455/175616*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
 

3.25.9.2 Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (123208128+910641904 x+2692519968 x^2+3982356144 x^3+2946673460 x^4+872316385 x^5\right )}{(2+3 x)^6}-41068005 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312} \]

input

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

output

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(123208128 + 910641904*x + 2692519968*x^2 
+ 3982356144*x^3 + 2946673460*x^4 + 872316385*x^5))/(2 + 3*x)^6 - 41068005 
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1229312
 

3.25.9.3 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 = {107, 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.

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

\(\Big \downarrow \) 107

\(\displaystyle \frac {85}{28} \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^6}dx+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}dx+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \left (\frac {33}{8} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^4}dx+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \left (\frac {33}{8} \left (\frac {11}{6} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {85}{28} \left (\frac {11}{2} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\right )+\frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}\)

input

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

output

((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/(14*(2 + 3*x)^6) + (85*(((1 - 2*x)^(5/2) 
*(3 + 5*x)^(5/2))/(5*(2 + 3*x)^5) + (11*(((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)) 
/(4*(2 + 3*x)^4) + (33*((Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)^3) + 
(11*(-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))/6))/8))/2))/28
 

3.25.9.3.1 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 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])
 

3.25.9.4 Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (872316385 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{175616 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {41068005 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(139\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (29938575645 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{2458624 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\)	\(346\)

input

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

output

-1/175616*(-1+2*x)*(3+5*x)^(1/2)*(872316385*x^5+2946673460*x^4+3982356144* 
x^3+2692519968*x^2+910641904*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)[phone]/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)
 

3.25.9.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {41068005 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (872316385 \, x^{5} + 2946673460 \, x^{4} + 3982356144 \, x^{3} + 2692519968 \, x^{2} + 910641904 \, x + 123208128\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2458624 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

input

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

output

-1/2458624*(41068005*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*(872316385*x^5 + 2946673460*x^4 + 39823561 
44*x^3 + 2692519968*x^2 + 910641904*x + 123208128)*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)
 

3.25.9.6 Sympy [F]

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

input

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

output

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

3.25.9.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.31 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {7709075}{921984} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{6 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {47 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{84 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {2805 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {103785 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {4625445 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {62789925}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {41068005}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {55323015}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {18300755 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3687936 \, {\left (3 \, x + 2\right )}} \]

input

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

output

7709075/921984*(-10*x^2 - x + 3)^(3/2) + 1/6*(-10*x^2 - x + 3)^(5/2)/(729* 
x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 47/84*(-10 
*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) 
+ 2805/1568*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 1 
6) + 103785/21952*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4 
625445/614656*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 62789925/614656 
*sqrt(-10*x^2 - x + 3)*x + 41068005/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x 
 + 2) + 20/11/abs(3*x + 2)) - 55323015/1229312*sqrt(-10*x^2 - x + 3) + 183 
00755/3687936*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
 

3.25.9.8 Giac [B] (verification not implemented)

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

Time = 0.75 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {8213601}{4917248} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {805255 \, \sqrt {10} {\left (51 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 80920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 59615360 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 14778086400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 1776355840000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {87772876800000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {351091507200000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]

input

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

output

8213601/4917248*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)))) - 805255/87808*sqrt(10)*(51*((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 + 80920*((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 - 59615360*((s 
qrt(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 - 14778086400*((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)))^5 - 1776355840000*((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 - 87772876800 
000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 351091507200000*s 
qrt(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
 

3.25.9.9 Mupad [F(-1)]

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

input

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

output

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