3.1.79 \(\int x^3 (d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x)) \, dx\) [79]

Optimal. Leaf size=227 \[ \frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{7 c^4 d^2} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/c^4/d+1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/c^4/d^2+2/35*b*d*
x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/105*b*d*x^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-8/175*b*
c*d*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/49*b*c^3*d*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {272, 45, 4779, 12, 380} \begin {gather*} \frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{5 c^4 d}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}+\frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x
^2]) - (8*b*c*d*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[
1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c
*x]))/(7*c^4*d^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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 380

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

Rule 4779

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[Si
mplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p
- 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps

\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^3 \left (d-c^2 d x^2\right )^{3/2} \, dx\\ &=-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int x \left (d-c^2 d x\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int \left (\frac {\left (d-c^2 d x\right )^{3/2}}{c^2}-\frac {\left (d-c^2 d x\right )^{5/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4 d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 126, normalized size = 0.56 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (-105 a \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right )+b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )-105 b \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{3675 c^4 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*(-105*a*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 7
5*c^6*x^6) - 105*b*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2)*ArcSin[c*x]))/(3675*c^4*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 727, normalized size = 3.20

method	result	size
default	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}-64 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}+104 c^{4} x^{4}+112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-25 c^{2} x^{2}-56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+7 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+7 \arcsin \left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )+i\right ) d}{128 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) d}{128 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right ) d}{384 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (2 i+35 \arcsin \left (c x \right )\right ) \cos \left (6 \arcsin \left (c x \right )\right ) d}{39200 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i x^{2} c^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (37 i+35 \arcsin \left (c x \right )\right ) \sin \left (6 \arcsin \left (c x \right )\right ) d}{78400 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (7 i+15 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right ) d}{2400 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i x^{2} c^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+45 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right ) d}{4800 c^{4} \left (c^{2} x^{2}-1\right )}\right )\)	\(727\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-1/7*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+b*(-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*
c^8*x^8-144*c^6*x^6-64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56
*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+7*arcsin(c*x))*d/c^4/(c^2*x^2-1)-3/128*(-d*(c^2
*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)*d/c^4/(c^2*x^2-1)-3/128*(-d*(c^2*x^2-1))^(
1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)*d/c^4/(c^2*x^2-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(4*I*
(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d/c^4/(c^2*x^2
-1)+3/39200*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(2*I+35*arcsin(c*x))*cos(6*arcsin(c*x)
)*d/c^4/(c^2*x^2-1)+1/78400*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(37*I+35*arcsin(c*x))*
sin(6*arcsin(c*x))*d/c^4/(c^2*x^2-1)-1/2400*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(7*I+1
5*arcsin(c*x))*cos(4*arcsin(c*x))*d/c^4/(c^2*x^2-1)-1/4800*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^
(1/2)-I)*(11*I+45*arcsin(c*x))*sin(4*arcsin(c*x))*d/c^4/(c^2*x^2-1))

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 149, normalized size = 0.66 \begin {gather*} -\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a + \frac {{\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} - 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} + 210 \, d^{\frac {3}{2}} x\right )} b}{3675 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*b*arcsin(c*x) - 1/35*(5*(-c^2*
d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*a + 1/3675*(75*c^6*d^(3/2)*x^7 - 168*c^4*d^(3
/2)*x^5 + 35*c^2*d^(3/2)*x^3 + 210*d^(3/2)*x)*b/c^3

________________________________________________________________________________________

Fricas [A]
time = 4.09, size = 189, normalized size = 0.83 \begin {gather*} -\frac {{\left (75 \, b c^{7} d x^{7} - 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} + 210 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 105 \, {\left (5 \, a c^{8} d x^{8} - 13 \, a c^{6} d x^{6} + 9 \, a c^{4} d x^{4} + a c^{2} d x^{2} - 2 \, a d + {\left (5 \, b c^{8} d x^{8} - 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} + b c^{2} d x^{2} - 2 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3675 \, {\left (c^{6} x^{2} - c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3675*((75*b*c^7*d*x^7 - 168*b*c^5*d*x^5 + 35*b*c^3*d*x^3 + 210*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2
+ 1) + 105*(5*a*c^8*d*x^8 - 13*a*c^6*d*x^6 + 9*a*c^4*d*x^4 + a*c^2*d*x^2 - 2*a*d + (5*b*c^8*d*x^8 - 13*b*c^6*d
*x^6 + 9*b*c^4*d*x^4 + b*c^2*d*x^2 - 2*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________