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

Optimal. Leaf size=149 \[ \frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^3 \sqrt{1-c^2 x^2} \left (9 c^2 d+5 e\right )}{144 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac{b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}+\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c} \]

[Out]

(b*(9*c^2*d + 5*e)*x*Sqrt[1 - c^2*x^2])/(96*c^5) + (b*(9*c^2*d + 5*e)*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e*
x^5*Sqrt[1 - c^2*x^2])/(36*c) - (b*(9*c^2*d + 5*e)*ArcSin[c*x])/(96*c^6) + (d*x^4*(a + b*ArcSin[c*x]))/4 + (e*
x^6*(a + b*ArcSin[c*x]))/6

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Rubi [A]  time = 0.117026, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 4731, 12, 459, 321, 216} \[ \frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^3 \sqrt{1-c^2 x^2} \left (9 c^2 d+5 e\right )}{144 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac{b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}+\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(9*c^2*d + 5*e)*x*Sqrt[1 - c^2*x^2])/(96*c^5) + (b*(9*c^2*d + 5*e)*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e*
x^5*Sqrt[1 - c^2*x^2])/(36*c) - (b*(9*c^2*d + 5*e)*ArcSin[c*x])/(96*c^6) + (d*x^4*(a + b*ArcSin[c*x]))/4 + (e*
x^6*(a + b*ArcSin[c*x]))/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 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 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x^4 \left (3 d+2 e x^2\right )}{12 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{12} (b c) \int \frac{x^4 \left (3 d+2 e x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{36} \left (b c \left (9 d+\frac{5 e}{c^2}\right )\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{48 c^3}\\ &=\frac{b \left (9 c^2 d+5 e\right ) x \sqrt{1-c^2 x^2}}{96 c^5}+\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac{b \left (9 c^2 d+5 e\right ) x \sqrt{1-c^2 x^2}}{96 c^5}+\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}-\frac{b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0816949, size = 116, normalized size = 0.78 \[ \frac{24 a c^6 x^4 \left (3 d+2 e x^2\right )+b c x \sqrt{1-c^2 x^2} \left (2 c^4 \left (9 d x^2+4 e x^4\right )+c^2 \left (27 d+10 e x^2\right )+15 e\right )+3 b \sin ^{-1}(c x) \left (8 c^6 \left (3 d x^4+2 e x^6\right )-9 c^2 d-5 e\right )}{288 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a*c^6*x^4*(3*d + 2*e*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(15*e + c^2*(27*d + 10*e*x^2) + 2*c^4*(9*d*x^2 + 4*e*x
^4)) + 3*b*(-9*c^2*d - 5*e + 8*c^6*(3*d*x^4 + 2*e*x^6))*ArcSin[c*x])/(288*c^6)

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Maple [A]  time = 0.006, size = 177, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{6}{x}^{6}}{6}}+{\frac{{x}^{4}{c}^{6}d}{4}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{\arcsin \left ( cx \right ) e{c}^{6}{x}^{6}}{6}}+{\frac{\arcsin \left ( cx \right ){c}^{6}{x}^{4}d}{4}}-{\frac{e}{6} \left ( -{\frac{{c}^{5}{x}^{5}}{6}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,{c}^{3}{x}^{3}}{24}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,cx}{16}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{5\,\arcsin \left ( cx \right ) }{16}} \right ) }-{\frac{{c}^{2}d}{4} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)*(a+b*arcsin(c*x)),x)

[Out]

1/c^4*(a/c^2*(1/6*e*c^6*x^6+1/4*x^4*c^6*d)+b/c^2*(1/6*arcsin(c*x)*e*c^6*x^6+1/4*arcsin(c*x)*c^6*x^4*d-1/6*e*(-
1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^2+1)^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))-1
/4*c^2*d*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8*arcsin(c*x))))

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Maxima [A]  time = 1.47481, size = 252, normalized size = 1.69 \begin{align*} \frac{1}{6} \, a e x^{6} + \frac{1}{4} \, a d x^{4} + \frac{1}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d + \frac{1}{288} \,{\left (48 \, x^{6} \arcsin \left (c x\right ) +{\left (\frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{6}} - \frac{15 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*e*x^6 + 1/4*a*d*x^4 + 1/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c
^4 - 3*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d + 1/288*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5
/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^6
))*c)*b*e

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Fricas [A]  time = 2.34499, size = 286, normalized size = 1.92 \begin{align*} \frac{48 \, a c^{6} e x^{6} + 72 \, a c^{6} d x^{4} + 3 \,{\left (16 \, b c^{6} e x^{6} + 24 \, b c^{6} d x^{4} - 9 \, b c^{2} d - 5 \, b e\right )} \arcsin \left (c x\right ) +{\left (8 \, b c^{5} e x^{5} + 2 \,{\left (9 \, b c^{5} d + 5 \, b c^{3} e\right )} x^{3} + 3 \,{\left (9 \, b c^{3} d + 5 \, b c e\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{288 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(48*a*c^6*e*x^6 + 72*a*c^6*d*x^4 + 3*(16*b*c^6*e*x^6 + 24*b*c^6*d*x^4 - 9*b*c^2*d - 5*b*e)*arcsin(c*x) +
(8*b*c^5*e*x^5 + 2*(9*b*c^5*d + 5*b*c^3*e)*x^3 + 3*(9*b*c^3*d + 5*b*c*e)*x)*sqrt(-c^2*x^2 + 1))/c^6

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Sympy [A]  time = 5.26755, size = 206, normalized size = 1.38 \begin{align*} \begin{cases} \frac{a d x^{4}}{4} + \frac{a e x^{6}}{6} + \frac{b d x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b e x^{6} \operatorname{asin}{\left (c x \right )}}{6} + \frac{b d x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} + \frac{b e x^{5} \sqrt{- c^{2} x^{2} + 1}}{36 c} + \frac{3 b d x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} + \frac{5 b e x^{3} \sqrt{- c^{2} x^{2} + 1}}{144 c^{3}} - \frac{3 b d \operatorname{asin}{\left (c x \right )}}{32 c^{4}} + \frac{5 b e x \sqrt{- c^{2} x^{2} + 1}}{96 c^{5}} - \frac{5 b e \operatorname{asin}{\left (c x \right )}}{96 c^{6}} & \text{for}\: c \neq 0 \\a \left (\frac{d x^{4}}{4} + \frac{e x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d*x**4/4 + a*e*x**6/6 + b*d*x**4*asin(c*x)/4 + b*e*x**6*asin(c*x)/6 + b*d*x**3*sqrt(-c**2*x**2 +
1)/(16*c) + b*e*x**5*sqrt(-c**2*x**2 + 1)/(36*c) + 3*b*d*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 5*b*e*x**3*sqrt(-c
**2*x**2 + 1)/(144*c**3) - 3*b*d*asin(c*x)/(32*c**4) + 5*b*e*x*sqrt(-c**2*x**2 + 1)/(96*c**5) - 5*b*e*asin(c*x
)/(96*c**6), Ne(c, 0)), (a*(d*x**4/4 + e*x**6/6), True))

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Giac [B]  time = 1.26794, size = 454, normalized size = 3.05 \begin{align*} -\frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d x}{16 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right )}{4 \, c^{4}} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} b d x}{32 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b x e}{36 \, c^{5}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a d}{4 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )} b d \arcsin \left (c x\right )}{2 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b \arcsin \left (c x\right ) e}{6 \, c^{6}} - \frac{13 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b x e}{144 \, c^{5}} + \frac{{\left (c^{2} x^{2} - 1\right )} a d}{2 \, c^{4}} + \frac{5 \, b d \arcsin \left (c x\right )}{32 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a e}{6 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b \arcsin \left (c x\right ) e}{2 \, c^{6}} + \frac{11 \, \sqrt{-c^{2} x^{2} + 1} b x e}{96 \, c^{5}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a e}{2 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right ) e}{2 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )} a e}{2 \, c^{6}} + \frac{11 \, b \arcsin \left (c x\right ) e}{96 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

-1/16*(-c^2*x^2 + 1)^(3/2)*b*d*x/c^3 + 1/4*(c^2*x^2 - 1)^2*b*d*arcsin(c*x)/c^4 + 5/32*sqrt(-c^2*x^2 + 1)*b*d*x
/c^3 + 1/36*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*x*e/c^5 + 1/4*(c^2*x^2 - 1)^2*a*d/c^4 + 1/2*(c^2*x^2 - 1)*b*d
*arcsin(c*x)/c^4 + 1/6*(c^2*x^2 - 1)^3*b*arcsin(c*x)*e/c^6 - 13/144*(-c^2*x^2 + 1)^(3/2)*b*x*e/c^5 + 1/2*(c^2*
x^2 - 1)*a*d/c^4 + 5/32*b*d*arcsin(c*x)/c^4 + 1/6*(c^2*x^2 - 1)^3*a*e/c^6 + 1/2*(c^2*x^2 - 1)^2*b*arcsin(c*x)*
e/c^6 + 11/96*sqrt(-c^2*x^2 + 1)*b*x*e/c^5 + 1/2*(c^2*x^2 - 1)^2*a*e/c^6 + 1/2*(c^2*x^2 - 1)*b*arcsin(c*x)*e/c
^6 + 1/2*(c^2*x^2 - 1)*a*e/c^6 + 11/96*b*arcsin(c*x)*e/c^6