3.25.33 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\) [2433]

3.25.33.1 Optimal result

Integrand size = 26, antiderivative size = 151 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247 (1-2 x)^{5/2} \sqrt {3+5 x}}{168 (2+3 x)^3}+\frac {13585 (1-2 x)^{3/2} \sqrt {3+5 x}}{672 (2+3 x)^2}+\frac {149435 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}-\frac {1643785 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}} \]

output

-1643785/3136*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+3/28 
*(1-2*x)^(7/2)*(3+5*x)^(1/2)/(2+3*x)^4+247/168*(1-2*x)^(5/2)*(3+5*x)^(1/2) 
/(2+3*x)^3+13585/672*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^2+149435/448*(1-2 
*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
 

3.25.33.2 Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (3699216+16236916 x+23794744 x^2+11637735 x^3\right )}{(2+3 x)^4}-4931355 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9408} \]

input

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

output

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3699216 + 16236916*x + 23794744*x^2 + 116 
37735*x^3))/(2 + 3*x)^4 - 4931355*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq 
rt[3 + 5*x])])/9408
 

3.25.33.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {107, 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.

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

\(\Big \downarrow \) 107

\(\displaystyle \frac {247}{56} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4 \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {247}{56} \left (\frac {55}{6} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {247}{56} \left (\frac {55}{6} \left (\frac {33}{4} \int \frac {\sqrt {1-2 x}}{(3 x+2)^2 \sqrt {5 x+3}}dx+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {247}{56} \left (\frac {55}{6} \left (\frac {33}{4} \left (\frac {11}{2} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {247}{56} \left (\frac {55}{6} \left (\frac {33}{4} \left (11 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {247}{56} \left (\frac {55}{6} \left (\frac {33}{4} \left (\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}\right )+\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}\)

input

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

output

(3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (247*(((1 - 2*x)^(5/2 
)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (55*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 
*(2 + 3*x)^2) + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) - (11*ArcTan[ 
Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7]))/4))/6))/56
 

3.25.33.3.1 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])
 

3.25.33.4 Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (11637735 x^{3}[phone] x^{2}[phone] x +3699216\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1344 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1643785 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6272 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(129\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (399439755 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{18816 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(250\)

input

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

output

-1/1344*(-1+2*x)*(3+5*x)^(1/2)*(11637735*x^3+23794744*x^2+16236916*x+36992 
16)/(2+3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1 
/2)+1643785/6272*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)
 

3.25.33.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=-\frac {4931355 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (11637735 \, x^{3} + 23794744 \, x^{2} + 16236916 \, x + 3699216\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{18816 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

input

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

output

-1/18816*(4931355*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan( 
1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 
14*(11637735*x^3 + 23794744*x^2 + 16236916*x + 3699216)*sqrt(5*x + 3)*sqrt 
(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

3.25.33.6 Sympy [F]

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

input

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

output

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

3.25.33.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {1643785}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{36 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1477 \, \sqrt {-10 \, x^{2} - x + 3}}{216 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {37091 \, \sqrt {-10 \, x^{2} - x + 3}}{864 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {3879245 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (3 \, x + 2\right )}} \]

input

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

output

1643785/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4 
9/36*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1477 
/216*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 37091/864*sqrt(- 
10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 3879245/12096*sqrt(-10*x^2 - x + 3)/( 
3*x + 2)
 

3.25.33.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).

Time = 0.48 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {328757}{12544} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {6655 \, \sqrt {10} {\left (1947 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 1009736 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 213012800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {16266432000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {65065728000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{672 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

input

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

output

328757/12544*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)))) + 6655/672*sqrt(10)*(1947*((sqrt(2)*sqrt(-10*x + 5) 
- sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr 
t(22)))^7 + 1009736*((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 + 213012800*((sqrt 
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s 
qrt(-10*x + 5) - sqrt(22)))^3 + 16266432000*(sqrt(2)*sqrt(-10*x + 5) - sqr 
t(22))/sqrt(5*x + 3) - 65065728000*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)^4
 

3.25.33.9 Mupad [F(-1)]

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

input

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

output

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