\(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\) [1012]

Optimal result

Integrand size = 26, antiderivative size = 180 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=-\frac {33 (1-2 x)^{7/2} \sqrt {3+5 x}}{1960 (2+3 x)^4}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{3920 (2+3 x)^3}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{3136 (2+3 x)^2}+\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}-\frac {(1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}-\frac {483153 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \] Output:

-33/1960*(1-2*x)^(7/2)*(3+5*x)^(1/2)/(2+3*x)^4+121/3920*(1-2*x)^(5/2)*(3+5 
*x)^(1/2)/(2+3*x)^3+1331/3136*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^2+43923* 
(1-2*x)^(1/2)*(3+5*x)^(1/2)/(12544+18816*x)-1/35*(1-2*x)^(7/2)*(3+5*x)^(3/ 
2)/(2+3*x)^5-483153/43904*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x) 
^(1/2))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (3620448+21361768 x+47166452 x^2+46327530 x^3+17153435 x^4\right )}{(2+3 x)^5}-2415765 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{219520} \] Input:

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]
 

Output:

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3620448 + 21361768*x + 47166452*x^2 + 463 
27530*x^3 + 17153435*x^4))/(2 + 3*x)^5 - 2415765*Sqrt[7]*ArcTan[Sqrt[1 - 2 
*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/219520
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {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.

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

\(\Big \downarrow \) 105

\(\displaystyle \frac {11}{2} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}dx+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 217

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

Input:

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]
 

Output:

((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
 

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])
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (17153435 x^{4}[phone] x^{3}[phone] x^{2}[phone] x +3620448\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{31360 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {483153 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{87808 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(134\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (587030895 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{439040 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\)	\(298\)

Input:

int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x,method=_RETURNVERBOSE)
 

Output:

-1/31360*(-1+2*x)*(3+5*x)^(1/2)*(17153435*x^4+46327530*x^3+47166452*x^2+21 
361768*x+3620448)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1 
/2)/(1-2*x)^(1/2)+483153/87808*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)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=-\frac {2415765 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (17153435 \, x^{4} + 46327530 \, x^{3} + 47166452 \, x^{2} + 21361768 \, x + 3620448\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{439040 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x, algorithm="fricas")
 

Output:

-1/439040*(2415765*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x 
 + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^ 
2 + x - 3)) - 14*(17153435*x^4 + 46327530*x^3 + 47166452*x^2 + 21361768*x 
+ 3620448)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 7 
20*x^2 + 240*x + 32)
 

Sympy [F]

\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{6}}\, dx \] Input:

integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**6,x)
 

Output:

Integral((1 - 2*x)**(5/2)*(5*x + 3)**(3/2)/(3*x + 2)**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.26 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {90695}{32928} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{56 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1221 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {54417 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {738705}{21952} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {483153}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {650859}{43904} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {215303 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{131712 \, {\left (3 \, x + 2\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x, algorithm="maxima")
 

Output:

90695/32928*(-10*x^2 - x + 3)^(3/2) + 1/5*(-10*x^2 - x + 3)^(5/2)/(243*x^5 
 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 33/56*(-10*x^2 - x + 3)^(5 
/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1221/784*(-10*x^2 - x + 3)^ 
(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 54417/21952*(-10*x^2 - x + 3)^(5/2)/( 
9*x^2 + 12*x + 4) + 738705/21952*sqrt(-10*x^2 - x + 3)*x + 483153/87808*sq 
rt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 650859/43904*sqr 
t(-10*x^2 - x + 3) + 215303/131712*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).

Time = 0.42 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {483153}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 3920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 2007040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 307328000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {18439680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {73758720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3136 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x, algorithm="giac")
 

Output:

483153/878080*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(-1 
0*x + 5) - sqrt(22)))) - 161051/3136*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) - sq 
rt(22)))^9 + 3920*((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 - 2007040*((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 - 307328000*((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 
- 18439680000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 7375872 
0000*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)^5
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^6} \,d x \] Input:

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^6,x)
 

Output:

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.98 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx =\text {Too large to display} \] Input:

int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x)
 

Output:

(587030895*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sq 
rt(5))/sqrt(11))/2))/sqrt(2))*x**5 + 1956769650*sqrt(7)*atan((sqrt(33) - s 
qrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 
2609026200*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sq 
rt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 1739350800*sqrt(7)*atan((sqrt(33) - s 
qrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 
579783600*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqr 
t(5))/sqrt(11))/2))/sqrt(2))*x + 77304480*sqrt(7)*atan((sqrt(33) - sqrt(35 
)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 587030895*s 
qrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 
11))/2))/sqrt(2))*x**5 - 1956769650*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan( 
asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 2609026200*s 
qrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 
11))/2))/sqrt(2))*x**3 - 1739350800*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan( 
asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 579783600*sq 
rt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(1 
1))/2))/sqrt(2))*x - 77304480*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin(( 
sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 120074045*sqrt(5*x + 3) 
*sqrt( - 2*x + 1)*x**4 + 324292710*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 3 
30165164*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 149532376*sqrt(5*x + 3)*...