Integrand size = 32, antiderivative size = 362 \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \arcsin (c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}} \]
-1/32*b^2*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)-15/64*b^2*x*(c*d*x+d)^(3/2)*( -c*e*x+e)^(3/2)/(-c^2*x^2+1)+9/64*b^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*arc sin(c*x)/c/(-c^2*x^2+1)^(3/2)-3/8*b*c*x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2) *(a+b*arcsin(c*x))/(-c^2*x^2+1)^(3/2)+1/4*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/ 2)*(a+b*arcsin(c*x))^2+3/8*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin( c*x))^2/(-c^2*x^2+1)+1/8*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x) )^3/b/c/(-c^2*x^2+1)^(3/2)+1/8*b*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arc sin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 3.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.03 \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-96 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+8 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (12 a+8 b \sin (2 \arcsin (c x))+b \sin (4 \arcsin (c x)))+d e \sqrt {d+c d x} \sqrt {e-c e x} \left (160 a^2 c x \sqrt {1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+64 a b \cos (2 \arcsin (c x))+4 a b \cos (4 \arcsin (c x))-32 b^2 \sin (2 \arcsin (c x))-b^2 \sin (4 \arcsin (c x))\right )+4 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (16 b \cos (2 \arcsin (c x))+b \cos (4 \arcsin (c x))+4 a (8 \sin (2 \arcsin (c x))+\sin (4 \arcsin (c x))))}{256 c \sqrt {1-c^2 x^2}} \]
(32*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 96*a^2*d^(3/2) *e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(S qrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 8*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]* ArcSin[c*x]^2*(12*a + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]]) + d*e *Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64*a^2*c ^3*x^3*Sqrt[1 - c^2*x^2] + 64*a*b*Cos[2*ArcSin[c*x]] + 4*a*b*Cos[4*ArcSin[ c*x]] - 32*b^2*Sin[2*ArcSin[c*x]] - b^2*Sin[4*ArcSin[c*x]]) + 4*b*d*e*Sqrt [d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(16*b*Cos[2*ArcSin[c*x]] + b*Cos[4 *ArcSin[c*x]] + 4*a*(8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(256*c*S qrt[1 - c^2*x^2])
Time = 1.05 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.77, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {5178, 5158, 5156, 5138, 262, 223, 5152, 5182, 211, 211, 223}
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.
\(\displaystyle \int (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx\) |
\(\Big \downarrow \) 5178 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2dx}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 5158 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 5156 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-b c \int x (a+b \arcsin (c x))dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \left (\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\left (1-c^2 x^2\right )^{3/2}}\) |
((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*((x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin [c*x])^2)/4 + (3*((x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (a + b*A rcSin[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*ArcSin[c*x]))/2 - (b*c*(-1/2*(x*S qrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)))/4 - (b*c*(-1/4*((1 - c^ 2*x^2)^2*(a + b*ArcSin[c*x]))/c^2 + (b*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x *Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/2))/(1 - c^2*x^2)^ (3/2)
3.6.47.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 2)^q) Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
\[\int \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
\[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral(-(a^2*c^2*d*e*x^2 - a^2*d*e + (b^2*c^2*d*e*x^2 - b^2*d*e)*arcsin( c*x)^2 + 2*(a*b*c^2*d*e*x^2 - a*b*d*e)*arcsin(c*x))*sqrt(c*d*x + d)*sqrt(- c*e*x + e), x)
Timed out. \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
Exception generated. \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \]