3.55 \(\int x^4 \sqrt {d-c^2 d x^2} (a+b \sin ^{-1}(c x)) \, dx\)
Optimal. Leaf size=262 \[ \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {1-c^2 x^2}} \]
[Out]
-1/16*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/24*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/6*x^5
*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))+1/32*b*x^2*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/96*b*x^4*(-c^
2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/36*b*c*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/32*(a+b*arcsin(c*
x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^5/(-c^2*x^2+1)^(1/2)
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Rubi [A] time = 0.28, antiderivative size = 262, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used =
{4697, 4707, 4641, 30} \[ \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]
[Out]
(b*x^2*Sqrt[d - c^2*d*x^2])/(32*c^3*Sqrt[1 - c^2*x^2]) + (b*x^4*Sqrt[d - c^2*d*x^2])/(96*c*Sqrt[1 - c^2*x^2])
- (b*c*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c^4)
- (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(24*c^2) + (x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/6 + (
Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c^5*Sqrt[1 - c^2*x^2])
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 4641
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]
Rule 4697
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 -
c^2*x^2]), Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
Rule 4707
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]
Rubi steps
\begin {align*} \int x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{24 c \sqrt {1-c^2 x^2}}\\ &=\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^4 \sqrt {1-c^2 x^2}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 169, normalized size = 0.65 \[ \frac {\sqrt {d-c^2 d x^2} \left (9 a^2+6 a b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-2 c^2 x^2-3\right )+6 b \sin ^{-1}(c x) \left (3 a+b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-2 c^2 x^2-3\right )\right )+b^2 c^2 x^2 \left (-8 c^4 x^4+3 c^2 x^2+9\right )+9 b^2 \sin ^{-1}(c x)^2\right )}{288 b c^5 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]
[Out]
(Sqrt[d - c^2*d*x^2]*(9*a^2 + b^2*c^2*x^2*(9 + 3*c^2*x^2 - 8*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^
2*x^2 + 8*c^4*x^4) + 6*b*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + 9*b^2*ArcS
in[c*x]^2))/(288*b*c^5*Sqrt[1 - c^2*x^2])
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{4} \arcsin \left (c x\right ) + a x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")
[Out]
integral((b*x^4*arcsin(c*x) + a*x^4)*sqrt(-c^2*d*x^2 + d), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
[Out]
integrate(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)*x^4, x)
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maple [C] time = 0.71, size = 1690, normalized size = 6.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x)),x)
[Out]
-1/192*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))/c^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-1/64*I*b
*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))/c^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-1/6*b*(-d*(c^2*x^2
-1))^(1/2)/(c^2*x^2-1)*arcsin(c*x)*x^5+3/512*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))/c^5/(c^2*x^2-1)+1/288
*b*(-d*(c^2*x^2-1))^(1/2)/c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+7/4608*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x)
)/c^5/(c^2*x^2-1)-1/36*I*b*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*x^5-1/6*a*x^3*(-c^2*d*x^2+d)^(3/2)/c^2/d-1/8*a/c
^4*x*(-c^2*d*x^2+d)^(3/2)/d+1/16*a/c^4*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/16*a/c^4
*x*(-c^2*d*x^2+d)^(1/2)+1/12*b*(-d*(c^2*x^2-1))^(1/2)/c^2/(c^2*x^2-1)*arcsin(c*x)*x^3+1/192*b*(-d*(c^2*x^2-1))
^(1/2)*sin(5*arcsin(c*x))/c^5/(c^2*x^2-1)*arcsin(c*x)-3/512*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))/c^3/(c
^2*x^2-1)*x^2+1/64*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))/c^5/(c^2*x^2-1)*arcsin(c*x)-1/32*b*(-d*(c^2*x^2
-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/(c^2*x^2-1)*arcsin(c*x)^2+1/12*b*(-d*(c^2*x^2-1))^(1/2)*c^2/(c^2*x^2-1)*arcs
in(c*x)*x^7+1/72*b*(-d*(c^2*x^2-1))^(1/2)*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6-1/48*b*(-d*(c^2*x^2-1))^(1/2)/c
/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-7/4608*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))/c^3/(c^2*x^2-1)*x^2+7/2
88*I*b*(-d*(c^2*x^2-1))^(1/2)/c^2/(c^2*x^2-1)*x^3-1/96*I*b*(-d*(c^2*x^2-1))^(1/2)/c^4/(c^2*x^2-1)*x+11/4608*I*
b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))/c^5/(c^2*x^2-1)+1/512*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x)
)/c^5/(c^2*x^2-1)+1/72*I*b*(-d*(c^2*x^2-1))^(1/2)*c^2/(c^2*x^2-1)*x^7-1/192*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arc
sin(c*x))/c^3/(c^2*x^2-1)*arcsin(c*x)*x^2-1/64*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))/c^3/(c^2*x^2-1)*arc
sin(c*x)*x^2+11/4608*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))/c^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+1/512*b*
(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))/c^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+1/96*I*b*(-d*(c^2*x^2-1))^(1/2)
/c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-1/96*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))/c^5/(c^2*x^
2-1)*arcsin(c*x)-11/4608*I*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))/c^3/(c^2*x^2-1)*x^2-1/512*I*b*(-d*(c^2*
x^2-1))^(1/2)*sin(3*arcsin(c*x))/c^3/(c^2*x^2-1)*x^2-1/96*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))/c^4/(c^2
*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-3/512*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))/c^4/(c^2*x^2-1)*(
-c^2*x^2+1)^(1/2)*x-1/12*I*b*(-d*(c^2*x^2-1))^(1/2)*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6+1/8*I*b*(
-d*(c^2*x^2-1))^(1/2)/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4-1/16*I*b*(-d*(c^2*x^2-1))^(1/2)/c^3/(c^
2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2+1/96*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))/c^3/(c^2*x^2-1)
*arcsin(c*x)*x^2-7/4608*I*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))/c^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \sqrt {d} \int \sqrt {c x + 1} \sqrt {-c x + 1} x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} - \frac {1}{48} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}}{c^{2} d} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} x}{c^{4}} + \frac {6 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{4} d} - \frac {3 \, \sqrt {d} \arcsin \left (c x\right )}{c^{5}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")
[Out]
b*sqrt(d)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x) - 1/48*(8*
(-c^2*d*x^2 + d)^(3/2)*x^3/(c^2*d) - 3*sqrt(-c^2*d*x^2 + d)*x/c^4 + 6*(-c^2*d*x^2 + d)^(3/2)*x/(c^4*d) - 3*sqr
t(d)*arcsin(c*x)/c^5)*a
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(1/2),x)
[Out]
int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4*(-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x)),x)
[Out]
Integral(x**4*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x)), x)
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