\(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\) [949]

Optimal result

Integrand size = 26, antiderivative size = 180 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=-\frac {37 (1-2 x)^{3/2} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {409 \sqrt {1-2 x} \sqrt {3+5 x}}{3024 (2+3 x)^3}+\frac {36149 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {3831323 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^5}-\frac {1625151 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \] Output:

-37/840*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^4+409/3024*(1-2*x)^(1/2)*(3+5* 
x)^(1/2)/(2+3*x)^3+36149/84672*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+38313 
23*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2370816+3556224*x)-1/15*(1-2*x)^(3/2)*(3+5 
*x)^(3/2)/(2+3*x)^5-1625151/307328*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2 
)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (12157344+71866904 x+158785356 x^2+155783350 x^3+57469845 x^4\right )}{(2+3 x)^5}-8125755 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1536640} \] Input:

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

Output:

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12157344 + 71866904*x + 158785356*x^2 + 1 
55783350*x^3 + 57469845*x^4))/(2 + 3*x)^5 - 8125755*Sqrt[7]*ArcTan[Sqrt[1 
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1536640
 

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 = {107, 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.

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

\(\Big \downarrow \) 107

\(\displaystyle \frac {37}{14} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}dx+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {33}{8} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^4}dx+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {33}{8} \left (\frac {11}{6} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {37}{14} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {37}{14} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {37}{14} \left (\frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}\)

Input:

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

Output:

(3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(35*(2 + 3*x)^5) + (37*(((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))/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 = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (57469845 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{219520 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1625151 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(134\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1974558465 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\)	\(298\)

Input:

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

Output:

-1/219520*(-1+2*x)*(3+5*x)^(1/2)*(57469845*x^4+155783350*x^3+158785356*x^2 
[phone]*x[phone])/(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)+1625151/614656*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 = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=-\frac {8125755 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (57469845 \, x^{4} + 155783350 \, x^{3} + 158785356 \, x^{2} + 71866904 \, x + 12157344\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \] Input:

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

Output:

-1/3073280*(8125755*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*(57469845*x^4 + 155783350*x^3 + 158785356*x^2 + 71866904 
*x + 12157344)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 
 + 720*x^2 + 240*x + 32)
 

Sympy [F]

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

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

Output:

Integral((1 - 2*x)**(3/2)*(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 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {305065}{230496} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{392 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {4107 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5488 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {183039 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{153664 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {2484735}{153664} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1625151}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2189253}{307328} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {724201 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{921984 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

305065/230496*(-10*x^2 - x + 3)^(3/2) + 3/35*(-10*x^2 - x + 3)^(5/2)/(243* 
x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 111/392*(-10*x^2 - x + 
3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 4107/5488*(-10*x^2 - x 
 + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 183039/153664*(-10*x^2 - x + 3) 
^(5/2)/(9*x^2 + 12*x + 4) + 2484735/153664*sqrt(-10*x^2 - x + 3)*x + 16251 
51/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2189 
253/307328*sqrt(-10*x^2 - x + 3) + 724201/921984*(-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 = 0.39 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {1625151}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (111 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 145040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 66232320 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 11371136000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {682268160000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2729072640000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \] Input:

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

Output:

1625151/6146560*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)))) - 14641/21952*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)))^9 + 145040*((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 - 66232320*((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)))^5 - 11371136000*((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 - 682268160000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) 
 + 2729072640000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((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)))^2 + 280)^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.98 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx =\text {Too large to display} \] Input:

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

Output:

(1974558465*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s 
qrt(5))/sqrt(11))/2))/sqrt(2))*x**5 + 6581861550*sqrt(7)*atan((sqrt(33) - 
sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 
 8775815400*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s 
qrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 5850543600*sqrt(7)*atan((sqrt(33) - 
sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 
 1950181200*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s 
qrt(5))/sqrt(11))/2))/sqrt(2))*x + 260024160*sqrt(7)*atan((sqrt(33) - sqrt 
(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 19745584 
65*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s 
qrt(11))/2))/sqrt(2))*x**5 - 6581861550*sqrt(7)*atan((sqrt(33) + sqrt(35)* 
tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 87758154 
00*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s 
qrt(11))/2))/sqrt(2))*x**3 - 5850543600*sqrt(7)*atan((sqrt(33) + sqrt(35)* 
tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 19501812 
00*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s 
qrt(11))/2))/sqrt(2))*x - 260024160*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan( 
asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 402288915*sqrt(5* 
x + 3)*sqrt( - 2*x + 1)*x**4 + 1090483450*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x 
**3 + 1111497492*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 503068328*sqrt(5...