\(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\) [2379]

Optimal result

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

[Out]

145708761/21952*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-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)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {145708761 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}}-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {5 x+3}}+\frac {1403963 \sqrt {1-2 x}}{3136 (3 x+2) \sqrt {5 x+3}}+\frac {8063 \sqrt {1-2 x}}{224 (3 x+2)^2 \sqrt {5 x+3}}+\frac {33 \sqrt {1-2 x}}{8 (3 x+2)^3 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}} \]

[In]

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

[Out]

(-63678595*Sqrt[1 - 2*x])/(9408*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (33*Sqrt[1
- 2*x])/(8*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (8063*Sqrt[1 - 2*x])/(224*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (1403963*Sqrt[
1 - 2*x])/(3136*(2 + 3*x)*Sqrt[3 + 5*x]) + (145708761*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqr
t[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 210

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])

Rubi steps \begin{align*} \text {integral}& = \frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {1}{12} \int \frac {\frac {341}{2}-264 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx \\ & = \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 {1}{252} \int \frac {\frac {86163}{4}-31185 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx \\ & = \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 {\int \frac {\frac {15937383}{8}-2539845 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{3528} \\ & = \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 {\int \frac {\frac {1880555061}{16}-\frac {442248345 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{24696} \\ & = -\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 {\int \frac {100976171373}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{135828} \\ & = -\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 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272} \\ & = -\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 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136} \\ & = -\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 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

((-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]*ArcTan[(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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(134)=268\).

Time = 1.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.72

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

[In]

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

[Out]

-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)*arct
an(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)*arctan(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+63112404600*x^2*(-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))[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)

none

Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {145708761 \, \sqrt {7} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1719322065 \, x^{4} + 4546951839 \, x^{3} + 4508028900 \, x^{2} + 1985778980 \, x + 327908240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

[In]

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

[Out]

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 + 1
985778980*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]

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

[In]

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

[Out]

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 = 0.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.71 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {145708761}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {63678595 \, x}{4704 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {66486521}{9408 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{36 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {665}{72 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {7799}{96 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {457237}{448 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]

[In]

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

[Out]

-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(-10*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*s
qrt(-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 = 0.55 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {145708761}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {275}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {11 \, \sqrt {10} {\left (13252949 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 8830442040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2086818820800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {170309125952000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {681236503808000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1568 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

[In]

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

[Out]

-145708761/439040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 275/2*sqrt(10)*((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))) - 11/1568*sqrt(10)*(13252949*(
(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 +
8830442040*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))^5 + 2086818820800*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^3 + 170309125952000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 68123650380
8000*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. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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