\(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\) [982]

Optimal result

Integrand size = 26, antiderivative size = 173 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}+\frac {145708761 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \] Output:

-63678595/9408*(1-2*x)^(1/2)/(3+5*x)^(1/2)+7/12*(1-2*x)^(1/2)/(2+3*x)^4/(3 
+5*x)^(1/2)+33/8*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^(1/2)+8063/224*(1-2*x)^(1 
/2)/(2+3*x)^2/(3+5*x)^(1/2)+1403963/3136*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(1/ 
2)[phone]/21952*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 3.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {-\frac {7 \sqrt {1-2 x} \left (327908240+1985778980 x+4508028900 x^2+4546951839 x^3+1719322065 x^4\right )}{(2+3 x)^4 \sqrt {3+5 x}}-145708761 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-145708761 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )}{21952} \] Input:

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

Output:

((-7*Sqrt[1 - 2*x]*(327908240 + 1985778980*x + 4508028900*x^2 + 4546951839 
*x^3 + 1719322065*x^4))/((2 + 3*x)^4*Sqrt[3 + 5*x]) - 145708761*Sqrt[7]*Ar 
cTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x] 
)] - 145708761*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt 
[11] + Sqrt[5 - 10*x]))])/21952
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {109, 27, 168, 27, 168, 27, 168, 27, 169, 27, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{12} \int \frac {11 (31-48 x)}{2 \sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{3/2}}dx+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{24} \int \frac {31-48 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{3/2}}dx+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {11}{24} \left (\frac {1}{21} \int \frac {21 (373-540 x)}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \int \frac {373-540 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{14} \int \frac {68993-87960 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \int \frac {68993-87960 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {8140931-7657980 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {382899 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \left (\frac {1}{14} \int \frac {8140931-7657980 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {382899 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (-\frac {2}{11} \int \frac {437126283}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {127357190 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {382899 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (-39738753 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {127357190 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {382899 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (-79477506 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {127357190 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {382899 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {11}{24} \left (\frac {1}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (\frac {79477506 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {127357190 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {382899 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {2199 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {9 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\)

Input:

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

Output:

(7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (11*((9*Sqrt[1 - 2*x])/ 
((2 + 3*x)^3*Sqrt[3 + 5*x]) + ((2199*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*Sqrt[3 
 + 5*x]) + ((382899*Sqrt[1 - 2*x])/(7*(2 + 3*x)*Sqrt[3 + 5*x]) + ((-127357 
190*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (79477506*ArcTan[Sqrt[1 - 2*x]/(Sq 
rt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/14)/28)/2))/24
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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. \(297\) vs. \(2(134)=268\).

Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.72

method	result	size
default	\(-\frac {\left (59012048205 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{43904 \left (2+3 x \right )^{4} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(298\)

Input:

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

Output:

-1/43904*(59012048205*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ 
(1/2))*x^5+192772690803*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3 
)^(1/2))*x^4+251784739008*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x 
+3)^(1/2))*x^3+24070508910*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar 
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+63657325746*x^3*(-10* 
x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2- 
x+3)^(1/2))*x[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arcta 
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^ 
(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^4/(-10*x^2-x+3 
)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {145708761 \, \sqrt {7} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1719322065 \, x^{4} + 4546951839 \, x^{3} + 4508028900 \, x^{2} + 1985778980 \, x + 327908240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \] Input:

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

Output:

1/43904*(145708761*sqrt(7)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368 
*x + 48)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10* 
x^2 + x - 3)) - 14*(1719322065*x^4 + 4546951839*x^3 + 4508028900*x^2 + 198 
5778980*x + 327908240)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(405*x^5 + 1323*x^4 + 
 1728*x^3 + 1128*x^2 + 368*x + 48)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (134) = 268\).

Time = 0.13 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.71 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {145708761}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {63678595 \, x}{4704 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {66486521}{9408 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{36 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {665}{72 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {7799}{96 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {457237}{448 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:

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

Output:

-145708761/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) 
 + 63678595/4704*x/sqrt(-10*x^2 - x + 3) - 66486521/9408/sqrt(-10*x^2 - x 
+ 3) + 49/36/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 
 + 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-1 
0*x^2 - x + 3)) + 665/72/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - 
 x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 7799 
/96/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10 
*x^2 - x + 3)) + 457237/448/(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. 427 vs. \(2 (134) = 268\).

Time = 0.35 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {145708761}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {275}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {11 \, \sqrt {10} {\left (13252949 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 8830442040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2086818820800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {170309125952000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {681236503808000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1568 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:

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

Output:

-145708761/439040*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)*sqr 
t(-10*x + 5) - sqrt(22)))) - 275/2*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 
))) - 11/1568*sqrt(10)*(13252949*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr 
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 883 
0442040*((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 + 2086818820800*((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 + 170309125952000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 
))/sqrt(5*x + 3) - 681236503808000*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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.88 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 11802409641*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**4 - 31473092376*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**3 - 31473092376*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 - 13988041056*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*ta 
n(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 2331340176*sq 
rt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s 
qrt(5))/sqrt(11))/2))/sqrt(2)) + 11802409641*sqrt(5*x + 3)*sqrt(7)*atan((s 
qrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt( 
2))*x**4 + 31473092376*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**3 + 31473092376 
*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 + 13988041056*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 + 2331340176*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35) 
*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 12035254455* 
sqrt( - 2*x + 1)*x**4 - 31828662873*sqrt( - 2*x + 1)*x**3 - 31556202300*sq 
rt( - 2*x + 1)*x**2 - 13900452860*sqrt( - 2*x + 1)*x - 2295357680*sqrt( - 
2*x + 1))/(21952*sqrt(5*x + 3)*(81*x**4 + 216*x**3 + 216*x**2 + 96*x + ...