3.77 \(\int x^7 (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x)) \, dx\)
Optimal. Leaf size=375 \[ \frac{\left (d-c^2 d x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{11 c^8 d^4}-\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^8 d^3}+\frac{3 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^8 d^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^8 d}+\frac{b c^3 d x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{1-c^2 x^2}}-\frac{4 b c d x^9 \sqrt{d-c^2 d x^2}}{297 \sqrt{1-c^2 x^2}}+\frac{b d x^7 \sqrt{d-c^2 d x^2}}{1617 c \sqrt{1-c^2 x^2}}+\frac{2 b d x^5 \sqrt{d-c^2 d x^2}}{1925 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b d x^3 \sqrt{d-c^2 d x^2}}{3465 c^5 \sqrt{1-c^2 x^2}}+\frac{16 b d x \sqrt{d-c^2 d x^2}}{1155 c^7 \sqrt{1-c^2 x^2}} \]
[Out]
(16*b*d*x*Sqrt[d - c^2*d*x^2])/(1155*c^7*Sqrt[1 - c^2*x^2]) + (8*b*d*x^3*Sqrt[d - c^2*d*x^2])/(3465*c^5*Sqrt[1
- c^2*x^2]) + (2*b*d*x^5*Sqrt[d - c^2*d*x^2])/(1925*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^7*Sqrt[d - c^2*d*x^2])/(1
617*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*x^9*Sqrt[d - c^2*d*x^2])/(297*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^11*Sqrt[d -
c^2*d*x^2])/(121*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^8*d) + (3*(d - c^2*d*x^
2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^8*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(3*c^8*d^3) + ((d - c^
2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^8*d^4)
________________________________________________________________________________________
Rubi [A] time = 0.293262, antiderivative size = 375, 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, 1810} \[ \frac{\left (d-c^2 d x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{11 c^8 d^4}-\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^8 d^3}+\frac{3 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^8 d^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^8 d}+\frac{b c^3 d x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{1-c^2 x^2}}-\frac{4 b c d x^9 \sqrt{d-c^2 d x^2}}{297 \sqrt{1-c^2 x^2}}+\frac{b d x^7 \sqrt{d-c^2 d x^2}}{1617 c \sqrt{1-c^2 x^2}}+\frac{2 b d x^5 \sqrt{d-c^2 d x^2}}{1925 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b d x^3 \sqrt{d-c^2 d x^2}}{3465 c^5 \sqrt{1-c^2 x^2}}+\frac{16 b d x \sqrt{d-c^2 d x^2}}{1155 c^7 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]
[Out]
(16*b*d*x*Sqrt[d - c^2*d*x^2])/(1155*c^7*Sqrt[1 - c^2*x^2]) + (8*b*d*x^3*Sqrt[d - c^2*d*x^2])/(3465*c^5*Sqrt[1
- c^2*x^2]) + (2*b*d*x^5*Sqrt[d - c^2*d*x^2])/(1925*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^7*Sqrt[d - c^2*d*x^2])/(1
617*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*x^9*Sqrt[d - c^2*d*x^2])/(297*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^11*Sqrt[d -
c^2*d*x^2])/(121*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^8*d) + (3*(d - c^2*d*x^
2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^8*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(3*c^8*d^3) + ((d - c^
2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^8*d^4)
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 1810
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rubi steps
\begin{align*} \int x^7 \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 (-16-40 c^2 x^2-70 c^4 x^4-105 c^6 x^6\right )}{1155 c^8} \, dx}{\sqrt{1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^7 \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 (-16-40 c^2 x^2-70 c^4 x^4-105 c^6 x^6\right ) \, dx}{1155 c^7 \sqrt{1-c^2 x^2}}+\frac{1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname{Subst}\left (\int x^3 \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 (-16-8 c^2 x^2-6 c^4 x^4-5 c^6 x^6+140 c^8 x^8-105 c^{10} x^{10}\right ) \, dx}{1155 c^7 \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^6}-\frac{3 \left (d-c^2 d x\right )^{5/2}}{c^6 d}+\frac{3 \left (d-c^2 d x\right )^{7/2}}{c^6 d^2}-\frac{\left (d-c^2 d x\right )^{9/2}}{c^6 d^3}\right ) \, dx,x,x^2\right )\\ &=\frac{16 b d x \sqrt{d-c^2 d x^2}}{1155 c^7 \sqrt{1-c^2 x^2}}+\frac{8 b d x^3 \sqrt{d-c^2 d x^2}}{3465 c^5 \sqrt{1-c^2 x^2}}+\frac{2 b d x^5 \sqrt{d-c^2 d x^2}}{1925 c^3 \sqrt{1-c^2 x^2}}+\frac{b d x^7 \sqrt{d-c^2 d x^2}}{1617 c \sqrt{1-c^2 x^2}}-\frac{4 b c d x^9 \sqrt{d-c^2 d x^2}}{297 \sqrt{1-c^2 x^2}}+\frac{b c^3 d 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 )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^8 d}+\frac{3 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^8 d^2}-\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^8 d^3}+\frac{\left (d-c^2 d x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{11 c^8 d^4}\\ \end{align*}
Mathematica [A] time = 0.18655, size = 174, normalized size = 0.46 \[ \frac{d \sqrt{d-c^2 d x^2} \left (-3465 a \left (105 c^6 x^6+70 c^4 x^4+40 c^2 x^2+16\right ) \left (1-c^2 x^2\right )^{5/2}+b c x \left (33075 c^{10} x^{10}-53900 c^8 x^8+2475 c^6 x^6+4158 c^4 x^4+9240 c^2 x^2+55440\right )-3465 b \left (105 c^6 x^6+70 c^4 x^4+40 c^2 x^2+16\right ) \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)\right )}{4002075 c^8 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]
[Out]
(d*Sqrt[d - c^2*d*x^2]*(-3465*a*(1 - c^2*x^2)^(5/2)*(16 + 40*c^2*x^2 + 70*c^4*x^4 + 105*c^6*x^6) + b*c*x*(5544
0 + 9240*c^2*x^2 + 4158*c^4*x^4 + 2475*c^6*x^6 - 53900*c^8*x^8 + 33075*c^10*x^10) - 3465*b*(1 - c^2*x^2)^(5/2)
*(16 + 40*c^2*x^2 + 70*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]))/(4002075*c^8*Sqrt[1 - c^2*x^2])
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Maple [C] time = 0.585, size = 1781, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x)
[Out]
a*(-1/11*x^6*(-c^2*d*x^2+d)^(5/2)/c^2/d+6/11/c^2*(-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*(-1/247808*(-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^10-1024*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/c^8/(c^2*x^2-1)-1/55296*(-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^8/(c^2*x^2-1)+1/100352*(-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/c^8/(c^2*x^2-1)+11/51200*(-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^8/(c^2*x^2-1)+1/3072*(-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/c^8/(c^2*x^2-1)-7/1024*(-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^8/(c^2*x^2-1)-7/1024*(-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/c^8/(c^2*x^2-1)+1/3072*(-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^8/(c^2*x^2-1)+11/51200*(-
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/c^8/(c^2*x^2-1)+1/100352*(-d*(c^2*x^2-1))^(1/2)*(6
4*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^8/(c^2*x^2-1)-1/55296*(-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^8/(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/c^8/(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^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.9603, size = 613, normalized size = 1.63 \begin{align*} -\frac{{\left (33075 \, b c^{11} d x^{11} - 53900 \, b c^{9} d x^{9} + 2475 \, b c^{7} d x^{7} + 4158 \, b c^{5} d x^{5} + 9240 \, b c^{3} d x^{3} + 55440 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 3465 \,{\left (105 \, a c^{12} d x^{12} - 245 \, a c^{10} d x^{10} + 145 \, a c^{8} d x^{8} + a c^{6} d x^{6} + 2 \, a c^{4} d x^{4} + 8 \, a c^{2} d x^{2} - 16 \, a d +{\left (105 \, b c^{12} d x^{12} - 245 \, b c^{10} d x^{10} + 145 \, b c^{8} d x^{8} + b c^{6} d x^{6} + 2 \, b c^{4} d x^{4} + 8 \, b c^{2} d x^{2} - 16 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{4002075 \,{\left (c^{10} x^{2} - c^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")
[Out]
-1/4002075*((33075*b*c^11*d*x^11 - 53900*b*c^9*d*x^9 + 2475*b*c^7*d*x^7 + 4158*b*c^5*d*x^5 + 9240*b*c^3*d*x^3
+ 55440*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 3465*(105*a*c^12*d*x^12 - 245*a*c^10*d*x^10 + 145*a
*c^8*d*x^8 + a*c^6*d*x^6 + 2*a*c^4*d*x^4 + 8*a*c^2*d*x^2 - 16*a*d + (105*b*c^12*d*x^12 - 245*b*c^10*d*x^10 + 1
45*b*c^8*d*x^8 + b*c^6*d*x^6 + 2*b*c^4*d*x^4 + 8*b*c^2*d*x^2 - 16*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^1
0*x^2 - c^8)
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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**7*(-c**2*d*x**2+d)**(3/2)*(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{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
[Out]
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)*x^7, x)