3.6.82 \(\int x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2 \, dx\) [582]

Optimal. Leaf size=338 \[ \frac {16 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}{75 c^2}+\frac {8 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{225 c^2}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2}{125 c^2}+\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{5 c^2} \]

[Out]

16/75*b^2*d*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^2+8/225*b^2*d*e*(-c^2*x^2+1)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)
/c^2+2/125*b^2*d*e*(-c^2*x^2+1)^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^2-1/5*d*e*(-c^2*x^2+1)^2*(a+b*arcsin(c*x)
)^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c^2+2/5*b*d*e*x*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c/(-c^
2*x^2+1)^(1/2)-4/15*b*c*d*e*x^3*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+2/25*b*c
^3*d*e*x^5*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4823, 4767, 200, 4739, 12, 1261, 712} \begin {gather*} \frac {2 b d e x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{5 c \sqrt {1-c^2 x^2}}-\frac {d e \left (1-c^2 x^2\right )^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{5 c^2}-\frac {4 b c d e x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{25 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \left (1-c^2 x^2\right )^2 \sqrt {c d x+d} \sqrt {e-c e x}}{125 c^2}+\frac {8 b^2 d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{225 c^2}+\frac {16 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x}}{75 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(16*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(75*c^2) + (8*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^
2))/(225*c^2) + (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2)/(125*c^2) + (2*b*d*e*x*Sqrt[d + c*
d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*e*x^5*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*Arc
Sin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2
)/(5*c^2)

Rule 12

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4767

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] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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]

Rule 4823

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-d^2)*(g/e))^IntPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^Fr
acPart[q]), Int[(h*x)^m*(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, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]

Rubi steps

\begin {align*} \int x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (2 b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {16 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}{75 c^2}+\frac {8 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{225 c^2}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2}{125 c^2}+\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.38, size = 207, normalized size = 0.61 \begin {gather*} -\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (225 a^2 \left (-1+c^2 x^2\right )^3+30 a b c x \sqrt {1-c^2 x^2} \left (15-10 c^2 x^2+3 c^4 x^4\right )+2 b^2 \left (149-187 c^2 x^2+47 c^4 x^4-9 c^6 x^6\right )+30 b \left (15 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {1-c^2 x^2} \left (15-10 c^2 x^2+3 c^4 x^4\right )\right ) \text {ArcSin}(c x)+225 b^2 \left (-1+c^2 x^2\right )^3 \text {ArcSin}(c x)^2\right )}{1125 c^2 \left (-1+c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

-1/1125*(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(225*a^2*(-1 + c^2*x^2)^3 + 30*a*b*c*x*Sqrt[1 - c^2*x^2]*(15 - 10
*c^2*x^2 + 3*c^4*x^4) + 2*b^2*(149 - 187*c^2*x^2 + 47*c^4*x^4 - 9*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^3 + b*c
*x*Sqrt[1 - c^2*x^2]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcSin[c*x] + 225*b^2*(-1 + c^2*x^2)^3*ArcSin[c*x]^2))/(c^
2*(-1 + c^2*x^2))

________________________________________________________________________________________

Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int x \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x)

[Out]

int(x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x)

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 281, normalized size = 0.83 \begin {gather*} -\frac {{\left (-c^{2} d x^{2} e + d e\right )}^{\frac {5}{2}} b^{2} \arcsin \left (c x\right )^{2} e^{\left (-1\right )}}{5 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} e + d e\right )}^{\frac {5}{2}} a b \arcsin \left (c x\right ) e^{\left (-1\right )}}{5 \, c^{2} d} + \frac {2}{1125} \, b^{2} {\left (\frac {{\left (9 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {5}{2}} x^{4} e^{\frac {5}{2}} - 38 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}} x^{2} e^{\frac {5}{2}} + \frac {149 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}} e^{\frac {5}{2}}}{c^{2}}\right )} e^{\left (-1\right )}}{d} + \frac {15 \, {\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} e^{\frac {5}{2}} - 10 \, c^{2} d^{\frac {5}{2}} x^{3} e^{\frac {5}{2}} + 15 \, d^{\frac {5}{2}} x e^{\frac {5}{2}}\right )} \arcsin \left (c x\right ) e^{\left (-1\right )}}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} e + d e\right )}^{\frac {5}{2}} a^{2} e^{\left (-1\right )}}{5 \, c^{2} d} + \frac {2 \, {\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} e^{\frac {5}{2}} - 10 \, c^{2} d^{\frac {5}{2}} x^{3} e^{\frac {5}{2}} + 15 \, d^{\frac {5}{2}} x e^{\frac {5}{2}}\right )} a b e^{\left (-1\right )}}{75 \, c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/5*(-c^2*d*x^2*e + d*e)^(5/2)*b^2*arcsin(c*x)^2*e^(-1)/(c^2*d) - 2/5*(-c^2*d*x^2*e + d*e)^(5/2)*a*b*arcsin(c
*x)*e^(-1)/(c^2*d) + 2/1125*b^2*((9*sqrt(-c^2*x^2 + 1)*c^2*d^(5/2)*x^4*e^(5/2) - 38*sqrt(-c^2*x^2 + 1)*d^(5/2)
*x^2*e^(5/2) + 149*sqrt(-c^2*x^2 + 1)*d^(5/2)*e^(5/2)/c^2)*e^(-1)/d + 15*(3*c^4*d^(5/2)*x^5*e^(5/2) - 10*c^2*d
^(5/2)*x^3*e^(5/2) + 15*d^(5/2)*x*e^(5/2))*arcsin(c*x)*e^(-1)/(c*d)) - 1/5*(-c^2*d*x^2*e + d*e)^(5/2)*a^2*e^(-
1)/(c^2*d) + 2/75*(3*c^4*d^(5/2)*x^5*e^(5/2) - 10*c^2*d^(5/2)*x^3*e^(5/2) + 15*d^(5/2)*x*e^(5/2))*a*b*e^(-1)/(
c*d)

________________________________________________________________________________________

Fricas [A]
time = 1.99, size = 313, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {c d x + d} {\left (30 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (3 \, b^{2} c^{5} d x^{5} - 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \arcsin \left (c x\right ) e + {\left (3 \, a b c^{5} d x^{5} - 10 \, a b c^{3} d x^{3} + 15 \, a b c d x\right )} e\right )} \sqrt {-{\left (c x - 1\right )} e} + {\left (225 \, {\left (b^{2} c^{6} d x^{6} - 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} e + 450 \, {\left (a b c^{6} d x^{6} - 3 \, a b c^{4} d x^{4} + 3 \, a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right ) e + {\left (9 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} d x^{6} - {\left (675 \, a^{2} - 94 \, b^{2}\right )} c^{4} d x^{4} + {\left (675 \, a^{2} - 374 \, b^{2}\right )} c^{2} d x^{2} - {\left (225 \, a^{2} - 298 \, b^{2}\right )} d\right )} e\right )} \sqrt {-{\left (c x - 1\right )} e}\right )}}{1125 \, {\left (c^{4} x^{2} - c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

-1/1125*sqrt(c*d*x + d)*(30*sqrt(-c^2*x^2 + 1)*((3*b^2*c^5*d*x^5 - 10*b^2*c^3*d*x^3 + 15*b^2*c*d*x)*arcsin(c*x
)*e + (3*a*b*c^5*d*x^5 - 10*a*b*c^3*d*x^3 + 15*a*b*c*d*x)*e)*sqrt(-(c*x - 1)*e) + (225*(b^2*c^6*d*x^6 - 3*b^2*
c^4*d*x^4 + 3*b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2*e + 450*(a*b*c^6*d*x^6 - 3*a*b*c^4*d*x^4 + 3*a*b*c^2*d*x^2
- a*b*d)*arcsin(c*x)*e + (9*(25*a^2 - 2*b^2)*c^6*d*x^6 - (675*a^2 - 94*b^2)*c^4*d*x^4 + (675*a^2 - 374*b^2)*c^
2*d*x^2 - (225*a^2 - 298*b^2)*d)*e)*sqrt(-(c*x - 1)*e))/(c^4*x^2 - c^2)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)**(3/2)*(-c*e*x+e)**(3/2)*(a+b*asin(c*x))**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6188 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate((c*d*x + d)^(3/2)*(-c*e*x + e)^(3/2)*(b*arcsin(c*x) + a)^2*x, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c*x))^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2),x)

[Out]

int(x*(a + b*asin(c*x))^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2), x)

________________________________________________________________________________________