3.55 \(\int x^4 \sqrt{d-c^2 d x^2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=262 \[ \frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}+\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt{1-c^2 x^2}}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{1-c^2 x^2}}+\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{1-c^2 x^2}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{32 c^3 \sqrt{1-c^2 x^2}} \]

[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])

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Rubi [A]  time = 0.28189, antiderivative size = 262, normalized size of antiderivative = 1., 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 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]

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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*}

Mathematica [A]  time = 0.124203, 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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Maple [B]  time = 0.448, size = 482, normalized size = 1.8 \begin{align*} -{\frac{a{x}^{3}}{6\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{8\,{c}^{4}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ax}{16\,{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{ad}{16\,{c}^{4}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{bc{x}^{6}}{36\,{c}^{2}{x}^{2}-36}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{x}^{4}}{96\,c \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{x}^{2}}{32\,{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b{c}^{2}\arcsin \left ( cx \right ){x}^{7}}{6\,{c}^{2}{x}^{2}-6}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{5\,b\arcsin \left ( cx \right ){x}^{5}}{24\,{c}^{2}{x}^{2}-24}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b\arcsin \left ( cx \right ){x}^{3}}{48\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b\arcsin \left ( cx \right ) x}{16\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{25\,b}{2304\,{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{32\,{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}

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]

-1/6*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*x*(-c^2*d*x^2+d)^(1/2)+1/1
6*a/c^4*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/36*b*(-d*(c^2*x^2-1))^(1/2)*c/(c^2*x^2-
1)*(-c^2*x^2+1)^(1/2)*x^6-1/96*b*(-d*(c^2*x^2-1))^(1/2)/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-1/32*b*(-d*(c^2*x
^2-1))^(1/2)/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)*c^2/(c^2*x^2-1)*arcsin(c*x)*x
^7-5/24*b*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*arcsin(c*x)*x^5-1/48*b*(-d*(c^2*x^2-1))^(1/2)/c^2/(c^2*x^2-1)*arc
sin(c*x)*x^3+1/16*b*(-d*(c^2*x^2-1))^(1/2)/c^4/(c^2*x^2-1)*arcsin(c*x)*x+25/2304*b*(-d*(c^2*x^2-1))^(1/2)/c^5/
(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-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

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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^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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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**4*(-c**2*d*x**2+d)**(1/2)*(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 \sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}

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)