3.93 \(\int x^5 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x)) \, dx\)
Optimal. Leaf size=354 \[ -\frac{\left (d-c^2 d x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{11 c^6 d^3}+\frac{2 \left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d}-\frac{b c^5 d^2 x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 x^9 \sqrt{d-c^2 d x^2}}{891 \sqrt{1-c^2 x^2}}-\frac{113 b c d^2 x^7 \sqrt{d-c^2 d x^2}}{4851 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^5 \sqrt{d-c^2 d x^2}}{1155 c \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x^3 \sqrt{d-c^2 d x^2}}{2079 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b d^2 x \sqrt{d-c^2 d x^2}}{693 c^5 \sqrt{1-c^2 x^2}} \]
[Out]
(8*b*d^2*x*Sqrt[d - c^2*d*x^2])/(693*c^5*Sqrt[1 - c^2*x^2]) + (4*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(2079*c^3*Sqrt
[1 - c^2*x^2]) + (b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(1155*c*Sqrt[1 - c^2*x^2]) - (113*b*c*d^2*x^7*Sqrt[d - c^2*d*
x^2])/(4851*Sqrt[1 - c^2*x^2]) + (23*b*c^3*d^2*x^9*Sqrt[d - c^2*d*x^2])/(891*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x
^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[1 - c^2*x^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^2) - ((d - c^2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^6*
d^3)
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Rubi [A] time = 0.246211, antiderivative size = 354, normalized size of antiderivative = 1.,
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} \[ -\frac{\left (d-c^2 d x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{11 c^6 d^3}+\frac{2 \left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d}-\frac{b c^5 d^2 x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 x^9 \sqrt{d-c^2 d x^2}}{891 \sqrt{1-c^2 x^2}}-\frac{113 b c d^2 x^7 \sqrt{d-c^2 d x^2}}{4851 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^5 \sqrt{d-c^2 d x^2}}{1155 c \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x^3 \sqrt{d-c^2 d x^2}}{2079 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b d^2 x \sqrt{d-c^2 d x^2}}{693 c^5 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
[Out]
(8*b*d^2*x*Sqrt[d - c^2*d*x^2])/(693*c^5*Sqrt[1 - c^2*x^2]) + (4*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(2079*c^3*Sqrt
[1 - c^2*x^2]) + (b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(1155*c*Sqrt[1 - c^2*x^2]) - (113*b*c*d^2*x^7*Sqrt[d - c^2*d*
x^2])/(4851*Sqrt[1 - c^2*x^2]) + (23*b*c^3*d^2*x^9*Sqrt[d - c^2*d*x^2])/(891*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x
^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[1 - c^2*x^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^2) - ((d - c^2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^6*
d^3)
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 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 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])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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]
Rubi steps
\begin{align*} \int x^5 \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 (1-c^2 x^2\right )^3 \left (-8-28 c^2 x^2-63 c^4 x^4\right )}{693 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 )^{5/2} \, dx\\ &=-\frac{\left (b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (-8-28 c^2 x^2-63 c^4 x^4\right ) \, dx}{693 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 )^{5/2} \, dx,x,x^2\right )\\ &=-\frac{\left (b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+113 c^6 x^6-161 c^8 x^8+63 c^{10} x^{10}\right ) \, dx}{693 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 )^{5/2}}{c^4}-\frac{2 \left (d-c^2 d x\right )^{7/2}}{c^4 d}+\frac{\left (d-c^2 d x\right )^{9/2}}{c^4 d^2}\right ) \, dx,x,x^2\right )\\ &=\frac{8 b d^2 x \sqrt{d-c^2 d x^2}}{693 c^5 \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x^3 \sqrt{d-c^2 d x^2}}{2079 c^3 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^5 \sqrt{d-c^2 d x^2}}{1155 c \sqrt{1-c^2 x^2}}-\frac{113 b c d^2 x^7 \sqrt{d-c^2 d x^2}}{4851 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 x^9 \sqrt{d-c^2 d x^2}}{891 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^{11} \sqrt{d-c^2 d x^2}}{121 \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^6 d}+\frac{2 \left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{11 c^6 d^3}\\ \end{align*}
Mathematica [A] time = 0.200128, size = 160, normalized size = 0.45 \[ -\frac{d^2 \sqrt{d-c^2 d x^2} \left (3465 a \left (63 c^4 x^4+28 c^2 x^2+8\right ) \left (1-c^2 x^2\right )^{7/2}+b c x \left (19845 c^{10} x^{10}-61985 c^8 x^8+55935 c^6 x^6-2079 c^4 x^4-4620 c^2 x^2-27720\right )+3465 b \left (63 c^4 x^4+28 c^2 x^2+8\right ) \left (1-c^2 x^2\right )^{7/2} \sin ^{-1}(c x)\right )}{2401245 c^6 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
[Out]
-(d^2*Sqrt[d - c^2*d*x^2]*(3465*a*(1 - c^2*x^2)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4) + b*c*x*(-27720 - 4620*c^2
*x^2 - 2079*c^4*x^4 + 55935*c^6*x^6 - 61985*c^8*x^8 + 19845*c^10*x^10) + 3465*b*(1 - c^2*x^2)^(7/2)*(8 + 28*c^
2*x^2 + 63*c^4*x^4)*ArcSin[c*x]))/(2401245*c^6*Sqrt[1 - c^2*x^2])
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Maple [C] time = 0.47, size = 1775, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x)
[Out]
a*(-1/11*x^4*(-c^2*d*x^2+d)^(7/2)/c^2/d+4/11/c^2*(-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*(1/247808*(-d*(c^2*x^2-1))^(1/2)*(1+11*I*(-c^2*x^2+1)^(1/2)*x*c+4096*c^8*x^8-2352*c^6*x^6+620*c^4*
x^4-61*c^2*x^2-220*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+2816*I*(-c^2*x^2+1)^(1/2)*x^9*c^9+1024*x^12*c^12-3328*c^10*x^1
0-1024*I*(-c^2*x^2+1)^(1/2)*x^11*c^11+1232*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-2816*I*(-c^2*x^2+1)^(1/2)*x^7*c^7)*(I+
11*arcsin(c*x))*d^2/c^6/(c^2*x^2-1)-1/165888*(-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^6/(c^2*x^2-1)-
5/100352*(-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^6/(c^2*x^2-1)+1/10240*(-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^2/c^6/(c^2*x^2-1)
+5/9216*(-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^6/(c^2*x^2-1)-5/1024*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(a
rcsin(c*x)+I)*d^2/c^6/(c^2*x^2-1)-5/1024*(-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^6/(c^2*x^2-1)+5/9216*(-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^6/(c^2*x^2-1)+1/10240*(-d*(c^2*x^2-1))^(1/2)*(16*I*(-c^2
*x^2+1)^(1/2)*x^5*c^5+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*
x^2-1)*(-I+5*arcsin(c*x))*d^2/c^6/(c^2*x^2-1)-5/100352*(-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^6/(c^2*x^2-1)-1/165888*(-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^2/c^6/(c^2*x^2-1)+1/247808*(-d*(c^2*x^2-1))^(1/2)*(1024*I*(-c^2*x^2+1)^(1/2)*x^11*c^11+1024*
x^12*c^12-2816*I*(-c^2*x^2+1)^(1/2)*x^9*c^9-3328*c^10*x^10+2816*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+4096*c^8*x^8-1232
*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-2352*c^6*x^6+220*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+620*c^4*x^4-11*I*(-c^2*x^2+1)^(1/2
)*x*c-61*c^2*x^2+1)*(-I+11*arcsin(c*x))*d^2/c^6/(c^2*x^2-1))
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.31399, size = 666, normalized size = 1.88 \begin{align*} \frac{{\left (19845 \, b c^{11} d^{2} x^{11} - 61985 \, b c^{9} d^{2} x^{9} + 55935 \, b c^{7} d^{2} x^{7} - 2079 \, b c^{5} d^{2} x^{5} - 4620 \, b c^{3} d^{2} x^{3} - 27720 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 3465 \,{\left (63 \, a c^{12} d^{2} x^{12} - 224 \, a c^{10} d^{2} x^{10} + 274 \, a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} - a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + 8 \, a d^{2} +{\left (63 \, b c^{12} d^{2} x^{12} - 224 \, b c^{10} d^{2} x^{10} + 274 \, b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} - b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + 8 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{2401245 \,{\left (c^{8} x^{2} - c^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")
[Out]
1/2401245*((19845*b*c^11*d^2*x^11 - 61985*b*c^9*d^2*x^9 + 55935*b*c^7*d^2*x^7 - 2079*b*c^5*d^2*x^5 - 4620*b*c^
3*d^2*x^3 - 27720*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 3465*(63*a*c^12*d^2*x^12 - 224*a*c^10*d
^2*x^10 + 274*a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 - a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + 8*a*d^2 + (63*b*c^12*d^2*x
^12 - 224*b*c^10*d^2*x^10 + 274*b*c^8*d^2*x^8 - 116*b*c^6*d^2*x^6 - b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + 8*b*d^2)
*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**5*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
[Out]
integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)*x^5, x)