\(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx\) [980]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {2495 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 \sqrt {1-2 x}}{4 (2+3 x) \sqrt {3+5 x}}+\frac {5709 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \] Output:

-2495/12*(1-2*x)^(1/2)/(3+5*x)^(1/2)+7/6*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^( 
1/2)+55/4*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(1/2)+5709/28*7^(1/2)*arctan(1/7*( 
1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.22 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {1}{28} \left (-\frac {7 \sqrt {1-2 x} \left (3212+9815 x+7485 x^2\right )}{(2+3 x)^2 \sqrt {3+5 x}}-5709 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-5709 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \] Input:

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

Output:

((-7*Sqrt[1 - 2*x]*(3212 + 9815*x + 7485*x^2))/((2 + 3*x)^2*Sqrt[3 + 5*x]) 
 - 5709*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11 
] + Sqrt[5 - 10*x])] - 5709*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[ 
1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))])/28
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {107, 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.

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

\(\Big \downarrow \) 107

\(\displaystyle \frac {173}{28} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {173}{28} \left (\frac {33}{2} \int \frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}dx+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {173}{28} \left (\frac {33}{2} \left (-7 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {173}{28} \left (\frac {33}{2} \left (-14 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {173}{28} \left (\frac {33}{2} \left (2 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}\)

Input:

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

Output:

(3*(1 - 2*x)^(5/2))/(14*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (173*((1 - 2*x)^(3/2) 
/((2 + 3*x)*Sqrt[3 + 5*x]) + (33*((-2*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + 2*Sqr 
t[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/2))/28
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(88)=176\).

Time = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.76

method	result	size
default	\(-\frac {\left (256905 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+496683 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+319704 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +104790 x^{2} \sqrt {-10 x^{2}-x +3}+68508 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+137410 x \sqrt {-10 x^{2}-x +3}+44968 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{56 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(202\)

Input:

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

Output:

-1/56*(256905*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x 
^3+496683*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3 
19704*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+104790* 
x^2*(-10*x^2-x+3)^(1/2)+68508*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x 
^2-x+3)^(1/2))+137410*x*(-10*x^2-x+3)^(1/2)+44968*(-10*x^2-x+3)^(1/2))*(1- 
2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {5709 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (7485 \, x^{2} + 9815 \, x + 3212\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{56 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \] Input:

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

Output:

1/56*(5709*sqrt(7)*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37*x 
 + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(7485*x^2 + 981 
5*x + 3212)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {5709}{56} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2495 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2605}{12 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{18 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1127}{36 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:

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

Output:

-5709/56*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2495/ 
6*x/sqrt(-10*x^2 - x + 3) - 2605/12/sqrt(-10*x^2 - x + 3) + 49/18/(9*sqrt( 
-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3 
)) + 1127/36/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (88) = 176\).

Time = 0.22 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.70 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {5709}{560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {55 \, \sqrt {10} {\left (61 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {13384 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {53536 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \] Input:

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

Output:

-5709/560*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)))) - 11/2*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s 
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 55/ 
2*sqrt(10)*(61*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr 
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 13384*(sqrt(2)*sqrt(- 
10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 53536*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 
0*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)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.57 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {-51381 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-68508 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -22836 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+51381 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+68508 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +22836 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-52395 \sqrt {-2 x +1}\, x^{2}-68705 \sqrt {-2 x +1}\, x -22484 \sqrt {-2 x +1}}{28 \sqrt {5 x +3}\, \left (9 x^{2}+12 x +4\right )} \] Input:

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

Output:

( - 51381*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( 
- 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 68508*sqrt(5*x + 3)*sqrt 
(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11) 
)/2))/sqrt(2))*x - 22836*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*t 
an(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 51381*sqrt(5*x 
 + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) 
)/sqrt(11))/2))/sqrt(2))*x**2 + 68508*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) 
 + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 
 22836*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2 
*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 52395*sqrt( - 2*x + 1)*x**2 - 68 
705*sqrt( - 2*x + 1)*x - 22484*sqrt( - 2*x + 1))/(28*sqrt(5*x + 3)*(9*x**2 
 + 12*x + 4))