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

Optimal. Leaf size=315 \[ -\frac{3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac{3 c^2 d^3 x^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{m+3}+\frac{3 c^4 d^3 x^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{m+5}-\frac{c^6 d^3 x^{m+7} \left (a+b \sin ^{-1}(c x)\right )}{m+7}+\frac{d^3 x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{m+1}-\frac{b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt{1-c^2 x^2} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}+\frac{b c^3 d^3 (m+9) (2 m+13) \sqrt{1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)^2}-\frac{b c^5 d^3 \sqrt{1-c^2 x^2} x^{m+6}}{(m+7)^2} \]

[Out]

-((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((3 + m)^2*(5 + m)^2*(7 + m)^
2)) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 - c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*
Sqrt[1 - c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (3*c^2*d^3*x^(3 + m)*(a + b*ArcSi
n[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSin[c*x]))/(5 + m) - (c^6*d^3*x^(7 + m)*(a + b*ArcSin[c*x]))
/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2
, c^2*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)^2)

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Rubi [A]  time = 2.16443, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {270, 4687, 12, 1809, 1267, 459, 364} \[ -\frac{3 c^2 d^3 x^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{m+3}+\frac{3 c^4 d^3 x^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{m+5}-\frac{c^6 d^3 x^{m+7} \left (a+b \sin ^{-1}(c x)\right )}{m+7}+\frac{d^3 x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{m+1}-\frac{3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt{1-c^2 x^2} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}+\frac{b c^3 d^3 (m+9) (2 m+13) \sqrt{1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)^2}-\frac{b c^5 d^3 \sqrt{1-c^2 x^2} x^{m+6}}{(m+7)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((3 + m)^2*(5 + m)^2*(7 + m)^
2)) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 - c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*
Sqrt[1 - c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (3*c^2*d^3*x^(3 + m)*(a + b*ArcSi
n[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSin[c*x]))/(5 + m) - (c^6*d^3*x^(7 + m)*(a + b*ArcSin[c*x]))
/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2
, c^2*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)^2)

Rule 270

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

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(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] && EqQ[c^2*d + e, 0] && IGtQ[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 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{d^3 x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac{3 c^2 d^3 x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{5+m}-\frac{c^6 d^3 x^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{7+m}-(b c) \int \frac{d^3 x^{1+m} \left (\frac{1}{1+m}-\frac{3 c^2 x^2}{3+m}+\frac{3 c^4 x^4}{5+m}-\frac{c^6 x^6}{7+m}\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{d^3 x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac{3 c^2 d^3 x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{5+m}-\frac{c^6 d^3 x^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{7+m}-\left (b c d^3\right ) \int \frac{x^{1+m} \left (\frac{1}{1+m}-\frac{3 c^2 x^2}{3+m}+\frac{3 c^4 x^4}{5+m}-\frac{c^6 x^6}{7+m}\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b c^5 d^3 x^{6+m} \sqrt{1-c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac{3 c^2 d^3 x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{5+m}-\frac{c^6 d^3 x^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{7+m}+\frac{\left (b d^3\right ) \int \frac{x^{1+m} \left (-\frac{c^2 (7+m)}{1+m}+\frac{3 c^4 (7+m) x^2}{3+m}-\frac{c^6 (9+m) (13+2 m) x^4}{(5+m) (7+m)}\right )}{\sqrt{1-c^2 x^2}} \, dx}{c (7+m)}\\ &=\frac{b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt{1-c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac{b c^5 d^3 x^{6+m} \sqrt{1-c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac{3 c^2 d^3 x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{5+m}-\frac{c^6 d^3 x^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{7+m}-\frac{\left (b d^3\right ) \int \frac{x^{1+m} \left (\frac{c^4 (5+m) (7+m)}{1+m}-\frac{c^6 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt{1-c^2 x^2}} \, dx}{c^3 (5+m) (7+m)}\\ &=-\frac{b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt{1-c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}+\frac{b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt{1-c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac{b c^5 d^3 x^{6+m} \sqrt{1-c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac{3 c^2 d^3 x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{5+m}-\frac{c^6 d^3 x^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{7+m}-\frac{\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right )\right ) \int \frac{x^{1+m}}{\sqrt{1-c^2 x^2}} \, dx}{(1+m) (3+m)^2 (5+m)^2 (7+m)^2}\\ &=-\frac{b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt{1-c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}+\frac{b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt{1-c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac{b c^5 d^3 x^{6+m} \sqrt{1-c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac{3 c^2 d^3 x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{5+m}-\frac{c^6 d^3 x^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{7+m}-\frac{3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{(1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2}\\ \end{align*}

Mathematica [A]  time = 0.550445, size = 256, normalized size = 0.81 \[ \frac{x^{m+1} \left (\frac{6 d \left (-\frac{4 d^2 \left (b c (m+1) x \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )+2 b c x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )+(m+2) \left (m \left (c^2 x^2-1\right )+c^2 x^2-3\right ) \left (a+b \sin ^{-1}(c x)\right )\right )}{(m+1) (m+2) (m+3)}-\frac{b c d^2 x \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )}{m+2}+\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )\right )}{m+5}-\frac{b c d^3 x \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )}{m+2}+\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )\right )}{m+7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]) - (b*c*d^3*x*Hypergeometric2F1[-5/2, 1 + m/2, 2 + m/2, c^2*x
^2])/(2 + m) + (6*d*((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]) - (b*c*d^2*x*Hypergeometric2F1[-3/2, 1 + m/2, 2 + m
/2, c^2*x^2])/(2 + m) - (4*d^2*((2 + m)*(-3 + c^2*x^2 + m*(-1 + c^2*x^2))*(a + b*ArcSin[c*x]) + b*c*(1 + m)*x*
Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, c^2*x^2] + 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c^2*x^2]
))/((1 + m)*(2 + m)*(3 + m))))/(5 + m)))/(7 + m)

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Maple [F]  time = 8.282, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{3} \left ( a+b\arcsin \left ( cx \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^m*(-c^2*d*x^2+d)^3*(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 (a c^{6} d^{3} x^{6} - 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} - a d^{3} +{\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \arcsin \left (c x\right )\right )} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^6*d^3*x^6 - 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 - a*d^3 + (b*c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*
c^2*d^3*x^2 - b*d^3)*arcsin(c*x))*x^m, 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**m*(-c**2*d*x**2+d)**3*(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 )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(c^2*d*x^2 - d)^3*(b*arcsin(c*x) + a)*x^m, x)