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

Optimal. Leaf size=301 \[ -\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}+\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/c^6/d+2/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/c^6/d^2-1/9*(-c^2
*d*x^2+d)^(9/2)*(a+b*arcsin(c*x))/c^6/d^3+8/315*b*d*x*(-c^2*d*x^2+d)^(1/2)/c^5/(-c^2*x^2+1)^(1/2)+4/945*b*d*x^
3*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/525*b*d*x^5*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-10/441*b
*c*d*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/81*b*c^3*d*x^9*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 301, 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, 1153} \[ -\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}}+\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b*d*x*Sqrt[d - c^2*d*x^2])/(315*c^5*Sqrt[1 - c^2*x^2]) + (4*b*d*x^3*Sqrt[d - c^2*d*x^2])/(945*c^3*Sqrt[1 -
c^2*x^2]) + (b*d*x^5*Sqrt[d - c^2*d*x^2])/(525*c*Sqrt[1 - c^2*x^2]) - (10*b*c*d*x^7*Sqrt[d - c^2*d*x^2])/(441*
Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*
ArcSin[c*x]))/(5*c^6*d) + (2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6*d^2) - ((d - c^2*d*x^2)^(9/2)*(
a + b*ArcSin[c*x]))/(9*c^6*d^3)

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 1153

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

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^5 \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 (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right )}{315 c^6} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^5 \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 (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right ) \, dx}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname {Subst}\left (\int x^2 \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 (-8-4 c^2 x^2-3 c^4 x^4+50 c^6 x^6-35 c^8 x^8\right ) \, dx}{315 c^5 \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 )^{3/2}}{c^4}-\frac {2 \left (d-c^2 d x\right )^{5/2}}{c^4 d}+\frac {\left (d-c^2 d x\right )^{7/2}}{c^4 d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b c^3 d 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 )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 150, normalized size = 0.50 \[ \frac {d \sqrt {d-c^2 d x^2} \left (-315 a \left (35 c^4 x^4+20 c^2 x^2+8\right ) \left (1-c^2 x^2\right )^{5/2}-315 b \left (35 c^4 x^4+20 c^2 x^2+8\right ) \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)+b c x \left (1225 c^8 x^8-2250 c^6 x^6+189 c^4 x^4+420 c^2 x^2+2520\right )\right )}{99225 c^6 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*(-315*a*(1 - c^2*x^2)^(5/2)*(8 + 20*c^2*x^2 + 35*c^4*x^4) + b*c*x*(2520 + 420*c^2*x^2 +
189*c^4*x^4 - 2250*c^6*x^6 + 1225*c^8*x^8) - 315*b*(1 - c^2*x^2)^(5/2)*(8 + 20*c^2*x^2 + 35*c^4*x^4)*ArcSin[c
*x]))/(99225*c^6*Sqrt[1 - c^2*x^2])

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fricas [A]  time = 2.06, size = 219, normalized size = 0.73 \[ -\frac {{\left (1225 \, b c^{9} d x^{9} - 2250 \, b c^{7} d x^{7} + 189 \, b c^{5} d x^{5} + 420 \, b c^{3} d x^{3} + 2520 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 315 \, {\left (35 \, a c^{10} d x^{10} - 85 \, a c^{8} d x^{8} + 53 \, a c^{6} d x^{6} + a c^{4} d x^{4} + 4 \, a c^{2} d x^{2} - 8 \, a d + {\left (35 \, b c^{10} d x^{10} - 85 \, b c^{8} d x^{8} + 53 \, b c^{6} d x^{6} + b c^{4} d x^{4} + 4 \, b c^{2} d x^{2} - 8 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{99225 \, {\left (c^{8} x^{2} - c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/99225*((1225*b*c^9*d*x^9 - 2250*b*c^7*d*x^7 + 189*b*c^5*d*x^5 + 420*b*c^3*d*x^3 + 2520*b*c*d*x)*sqrt(-c^2*d
*x^2 + d)*sqrt(-c^2*x^2 + 1) + 315*(35*a*c^10*d*x^10 - 85*a*c^8*d*x^8 + 53*a*c^6*d*x^6 + a*c^4*d*x^4 + 4*a*c^2
*d*x^2 - 8*a*d + (35*b*c^10*d*x^10 - 85*b*c^8*d*x^8 + 53*b*c^6*d*x^6 + b*c^4*d*x^4 + 4*b*c^2*d*x^2 - 8*b*d)*ar
csin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)

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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^5*(-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,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 = 0.54, size = 1254, normalized size = 4.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-1/9*x^4*(-c^2*d*x^2+d)^(5/2)/c^2/d+4/9/c^2*(-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*(-1/41472*(-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/c^6/(c^2*x^2-1)-1/25088*(-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/c^6/(c^2*x^2-1)+1/3200
*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1
/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+5*arcsin(c*x))*d/c^6/(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/c^6/(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/c^6/(c^2*x^2-1)+1/1152*(-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^6/(c^2*x^2-1)-1/25088*(-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/c^6/(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^10*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/c^6/(c^2*x^2-1)-1/14400*(-d*(c^2*x^2-1))^(1/2
)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(17*I+15*arcsin(c*x))*cos(4*arcsin(c*x))*d/c^6/(c^2*x^2-1)-1/3600*(-d*(
c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(2*I+15*arcsin(c*x))*sin(4*arcsin(c*x))*d/c^6/(c^2*x^2-
1))

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maxima [A]  time = 0.56, size = 208, normalized size = 0.69 \[ -\frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} a + \frac {{\left (1225 \, c^{8} d^{\frac {3}{2}} x^{9} - 2250 \, c^{6} d^{\frac {3}{2}} x^{7} + 189 \, c^{4} d^{\frac {3}{2}} x^{5} + 420 \, c^{2} d^{\frac {3}{2}} x^{3} + 2520 \, d^{\frac {3}{2}} x\right )} b}{99225 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(35*(-c^2*d*x^2 + d)^(5/2)*x^4/(c^2*d) + 20*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(5/
2)/(c^6*d))*b*arcsin(c*x) - 1/315*(35*(-c^2*d*x^2 + d)^(5/2)*x^4/(c^2*d) + 20*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^4*
d) + 8*(-c^2*d*x^2 + d)^(5/2)/(c^6*d))*a + 1/99225*(1225*c^8*d^(3/2)*x^9 - 2250*c^6*d^(3/2)*x^7 + 189*c^4*d^(3
/2)*x^5 + 420*c^2*d^(3/2)*x^3 + 2520*d^(3/2)*x)*b/c^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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