\(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx\) [2456]

Optimal result

Integrand size = 26, antiderivative size = 195 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=-\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}-\frac {519421265 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}} \]

[Out]

7/12*(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2)-519421265/3136*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1
/2)-25024175/1344*(1-2*x)^(1/2)/(3+5*x)^(3/2)+847/72*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^(3/2)+36817/288*(1-2*x)^(
1/2)/(2+3*x)^2/(3+5*x)^(3/2)+2992825/1344*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)[phone]/1344*(1-2*x)^(1/2)/(3+
5*x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 154, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=-\frac {519421265 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}}+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {5 x+3}}+\frac {2992825 \sqrt {1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {25024175 \sqrt {1-2 x}}{1344 (5 x+3)^{3/2}} \]

[In]

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

[Out]

(-25024175*Sqrt[1 - 2*x])/(1344*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)^(3/2)) + (847
*Sqrt[1 - 2*x])/(72*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (36817*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (
2992825*Sqrt[1 - 2*x])/(1344*(2 + 3*x)*(3 + 5*x)^(3/2)) + (227000875*Sqrt[1 - 2*x])/(1344*Sqrt[3 + 5*x]) - (51
9421265*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[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 154

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*((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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {1}{12} \int \frac {\left (\frac {495}{2}-264 x\right ) \sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {1}{108} \int \frac {-\frac {126423}{4}+49236 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {31923045}{8}+\frac {11597355 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx}{1512} \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {5879900565}{16}+\frac {942739875 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{10584} \\ & = -\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {-\frac {663674731335}{32}+\frac {156075779475 x}{8}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{174636} \\ & = -\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}-\frac {\int -\frac {35635934727855}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{960498} \\ & = -\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}+\frac {519421265}{896} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}+\frac {519421265}{448} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}-\frac {519421265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.90 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=\frac {7056 (1-2 x)^{7/2}+65016 (1-2 x)^{7/2} (2+3 x)+(2+3 x)^2 \left (716706 (1-2 x)^{7/2}+9444023 (2+3 x) \left (3 (1-2 x)^{5/2}-55 (2+3 x) \left (-\sqrt {1-2 x} (62+107 x)+21 \sqrt {7} (3+5 x)^{3/2} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )\right )\right )}{65856 (2+3 x)^4 (3+5 x)^{3/2}} \]

[In]

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

[Out]

(7056*(1 - 2*x)^(7/2) + 65016*(1 - 2*x)^(7/2)*(2 + 3*x) + (2 + 3*x)^2*(716706*(1 - 2*x)^(7/2) + 9444023*(2 + 3
*x)*(3*(1 - 2*x)^(5/2) - 55*(2 + 3*x)*(-(Sqrt[1 - 2*x]*(62 + 107*x)) + 21*Sqrt[7]*(3 + 5*x)^(3/2)*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))))/(65856*(2 + 3*x)^4*(3 + 5*x)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(150)=300\).

Time = 1.18 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {\left (3155484184875 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{18816 \left (2+3 x \right )^{4} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(346\)

[In]

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

[Out]

1/18816*(3155484184875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+12201205514850*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+19648148191155*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^4+1287094961250*x^5*(-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^3+4176132792300*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+5417063350650*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+3511408936896*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)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=-\frac {1558263795 \, \sqrt {7} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (91935354375 \, x^{5} + 298295199450 \, x^{4} + 386933096475 \, x^{3} + 250814924064 \, x^{2} + 81243850516 \, x + 10520317456\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{18816 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

[In]

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

[Out]

-1/18816*(1558263795*sqrt(7)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*arctan(1/
14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(91935354375*x^5 + 298295199450*x^4
+ 386933096475*x^3 + 250814924064*x^2 + 81243850516*x + 10520317456)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2025*x^6
+ 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (150) = 300\).

Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.67 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=\frac {519421265}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {227000875 \, x}{672 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {79003515}{448 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {24449315 \, x}{288 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{324 \, {\left (81 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 96 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 16 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {37387}{648 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {571291}{864 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {60813781}{5184 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {237706249}{5184 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

519421265/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 227000875/672*x/sqrt(-10*x^2 - x +
3) + 79003515/448/sqrt(-10*x^2 - x + 3) + 24449315/288*x/(-10*x^2 - x + 3)^(3/2) + 2401/324/(81*(-10*x^2 - x +
3)^(3/2)*x^4 + 216*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 - x + 3)^(3/2)
*x + 16*(-10*x^2 - x + 3)^(3/2)) + 37387/648/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2
+ 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 571291/864/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(
-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 60813781/5184/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^
2 - x + 3)^(3/2)) - 237706249/5184/(-10*x^2 - x + 3)^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (150) = 300\).

Time = 0.64 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.50 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=-\frac {55}{48} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {103884253}{12544} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {9295}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {55 \, \sqrt {10} {\left (6089929 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 4375094808 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1081495934400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {90973105216000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {363892420864000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{224 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

[In]

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

[Out]

-55/48*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)))^3 + 103884253/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 9295/2*sqrt(10)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 55/224*sqrt
(10)*(6089929*((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 + 4375094808*((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 + 1081495934400*((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 + 90973105216000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)
- 363892420864000*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. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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