3.94 \(\int x^3 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=278 \[ \frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/c^4/d+1/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arcsin(c*x))/c^4/d^2+2/63*b*d^
2*x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/189*b*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/21
*b*c*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+19/441*b*c^3*d^2*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1
/2)-1/81*b*c^5*d^2*x^9*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {266, 43, 4691, 12, 373} \[ \frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4 d}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]

[Out]

(2*b*d^2*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[1 - c
^2*x^2]) - (b*c*d^2*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[1 - c^2*x^2]) + (19*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2])/(
441*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(
a + b*ArcSin[c*x]))/(7*c^4*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*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 43

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 266

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 373

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 4691

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d +
e*x^2])/Sqrt[1 - c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (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 )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 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 )^{5/2} \, dx\\ &=-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname {Subst}\left (\int x \left (d-c^2 d x\right )^{5/2} \, dx,x,x^2\right )\\ &=-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname {Subst}\left (\int \left (\frac {\left (d-c^2 d x\right )^{5/2}}{c^2}-\frac {\left (d-c^2 d x\right )^{7/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^4 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 137, normalized size = 0.49 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (-63 a \left (7 c^2 x^2+2\right ) \left (1-c^2 x^2\right )^{7/2}-63 b \left (7 c^2 x^2+2\right ) \left (1-c^2 x^2\right )^{7/2} \sin ^{-1}(c x)+b \left (-49 c^9 x^9+171 c^7 x^7-189 c^5 x^5+21 c^3 x^3+126 c x\right )\right )}{3969 c^4 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(-63*a*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2) + b*(126*c*x + 21*c^3*x^3 - 189*c^5*x^5 +
171*c^7*x^7 - 49*c^9*x^9) - 63*b*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2)*ArcSin[c*x]))/(3969*c^4*Sqrt[1 - c^2*x^2]
)

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fricas [A]  time = 0.53, size = 255, normalized size = 0.92 \[ \frac {{\left (49 \, b c^{9} d^{2} x^{9} - 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} - 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 63 \, {\left (7 \, a c^{10} d^{2} x^{10} - 26 \, a c^{8} d^{2} x^{8} + 34 \, a c^{6} d^{2} x^{6} - 16 \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + 2 \, a d^{2} + {\left (7 \, b c^{10} d^{2} x^{10} - 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} - 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/3969*((49*b*c^9*d^2*x^9 - 171*b*c^7*d^2*x^7 + 189*b*c^5*d^2*x^5 - 21*b*c^3*d^2*x^3 - 126*b*c*d^2*x)*sqrt(-c^
2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 63*(7*a*c^10*d^2*x^10 - 26*a*c^8*d^2*x^8 + 34*a*c^6*d^2*x^6 - 16*a*c^4*d^2*x
^4 - a*c^2*d^2*x^2 + 2*a*d^2 + (7*b*c^10*d^2*x^10 - 26*b*c^8*d^2*x^8 + 34*b*c^6*d^2*x^6 - 16*b*c^4*d^2*x^4 - b
*c^2*d^2*x^2 + 2*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [C]  time = 0.38, size = 1063, normalized size = 3.82 \[ a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (256 c^{10} x^{10}-704 c^{8} x^{8}-256 i \sqrt {-c^{2} x^{2}+1}\, x^{9} c^{9}+688 c^{6} x^{6}+576 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}-280 c^{4} x^{4}-432 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+41 c^{2} x^{2}+120 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-9 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+9 \arcsin \left (c x \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{2}}{25088 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+3 \arcsin \left (c x \right )\right ) d^{2}}{576 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 (i+\arcsin \left (c x \right )\right ) d^{2}}{256 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^{2}}{256 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^{2}}{576 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}+64 c^{8} x^{8}-112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-144 c^{6} x^{6}+56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+104 c^{4} x^{4}-7 i \sqrt {-c^{2} x^{2}+1}\, x c -25 c^{2} x^{2}+1\right ) \left (-i+7 \arcsin \left (c x \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (256 i \sqrt {-c^{2} x^{2}+1}\, x^{9} c^{9}+256 c^{10} x^{10}-576 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}-704 c^{8} x^{8}+432 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+688 c^{6} x^{6}-120 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-280 c^{4} x^{4}+9 i \sqrt {-c^{2} x^{2}+1}\, x c +41 c^{2} x^{2}-1\right ) \left (-i+9 \arcsin \left (c x \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}-1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x)

[Out]

a*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d)^(7/2))+b*(1/41472*(-d*(c^2*x^2-1))^(1/2)*(256
*c^10*x^10-704*c^8*x^8-256*I*(-c^2*x^2+1)^(1/2)*x^9*c^9+688*c^6*x^6+576*I*(-c^2*x^2+1)^(1/2)*x^7*c^7-280*c^4*x
^4-432*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+41*c^2*x^2+120*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-9*I*(-c^2*x^2+1)^(1/2)*x*c-1)*
(I+9*arcsin(c*x))*d^2/c^4/(c^2*x^2-1)-3/25088*(-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^2/c^4/(c^2*x^2-1)+1/576*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2
-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))*d^2/c^4/(c^2*x^2-1)-3/256*(-d*
(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+arcsin(c*x))*d^2/c^4/(c^2*x^2-1)-3/256*(-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^2/c^4/(c^2*x^2-1)+1/576*(-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^2/c^
4/(c^2*x^2-1)-3/25088*(-d*(c^2*x^2-1))^(1/2)*(64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1
/2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*(
-I+7*arcsin(c*x))*d^2/c^4/(c^2*x^2-1)+1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*I*(-c^2*x^2+1)^(1/2)*x^9*c^9+256*c^1
0*x^10-576*I*(-c^2*x^2+1)^(1/2)*x^7*c^7-704*c^8*x^8+432*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+688*c^6*x^6-120*I*(-c^2*x
^2+1)^(1/2)*x^3*c^3-280*c^4*x^4+9*I*(-c^2*x^2+1)^(1/2)*x*c+41*c^2*x^2-1)*(-I+9*arcsin(c*x))*d^2/c^4/(c^2*x^2-1
))

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maxima [A]  time = 0.42, size = 160, normalized size = 0.58 \[ -\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} - 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} - 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

-1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*b*arcsin(c*x) - 1/63*(7*(-c^2*
d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a - 1/3969*(49*c^8*d^(5/2)*x^9 - 171*c^6*d^(5
/2)*x^7 + 189*c^4*d^(5/2)*x^5 - 21*c^2*d^(5/2)*x^3 - 126*d^(5/2)*x)*b/c^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

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)^(5/2),x)

[Out]

int(x^3*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)

[Out]

Timed out

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