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

Optimal. Leaf size=484 \[ -\frac{b (f x)^{m+2} \left (\frac{e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{(m+3) (m+5) (m+7)}+\frac{c^6 d^3 (m+3) (m+5) (m+7)}{m+1}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{c^5 f^2 (m+2) (m+3) (m+5) (m+7)}+\frac{3 d^2 e (f x)^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{f (m+1)}+\frac{3 d e^2 (f x)^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (m+7)}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2}+\frac{b e^2 \sqrt{1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^3 f^4 (m+5)^2 (m+7)^2}+\frac{b e^3 \sqrt{1-c^2 x^2} (f x)^{m+6}}{c f^6 (m+7)^2} \]

[Out]

(b*e*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m
^3 + m^4))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^5*f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2) + (b*e^2*(3*c^2*d*(7 + m)^
2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c^3*f^4*(5 + m)^2*(7 + m)^2) + (b*e^3*(f*x)^(6 + m)
*Sqrt[1 - c^2*x^2])/(c*f^6*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(
3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x]))/(f^5*(5 + m)) + (e^3*(
f*x)^(7 + m)*(a + b*ArcSin[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(3 + m)*(5 + m)*(7 + m))/(1 + m) + (e*(2 + m)*(
3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m
^4)))/((3 + m)*(5 + m)*(7 + m)))*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^5*f^2
*(2 + m)*(3 + m)*(5 + m)*(7 + m))

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Rubi [A]  time = 2.37629, antiderivative size = 455, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {270, 4731, 12, 1809, 1267, 459, 364} \[ \frac{3 d^2 e (f x)^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{f (m+1)}+\frac{3 d e^2 (f x)^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (m+7)}-\frac{b c (f x)^{m+2} \left (\frac{e \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^6 (m+3)^2 (m+5)^2 (m+7)^2}+\frac{d^3}{m^2+3 m+2}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{f^2}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2}+\frac{b e^2 \sqrt{1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^3 f^4 (m+5)^2 (m+7)^2}+\frac{b e^3 \sqrt{1-c^2 x^2} (f x)^{m+6}}{c f^6 (m+7)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m
^3 + m^4))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^5*f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2) + (b*e^2*(3*c^2*d*(7 + m)^
2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c^3*f^4*(5 + m)^2*(7 + m)^2) + (b*e^3*(f*x)^(6 + m)
*Sqrt[1 - c^2*x^2])/(c*f^6*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(
3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x]))/(f^5*(5 + m)) + (e^3*(
f*x)^(7 + m)*(a + b*ArcSin[c*x]))/(f^7*(7 + m)) - (b*c*(d^3/(2 + 3*m + m^2) + (e*(3*c^2*d*e*(7 + m)^2*(12 + 7*
m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)))/(c^6*(3 + m)^2*(5 + m)
^2*(7 + m)^2))*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/f^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 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 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 (f x)^m \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-(b c) \int \frac{(f x)^{1+m} \left (\frac{d^3}{1+m}+\frac{3 d^2 e x^2}{3+m}+\frac{3 d e^2 x^4}{5+m}+\frac{e^3 x^6}{7+m}\right )}{f \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac{(b c) \int \frac{(f x)^{1+m} \left (\frac{d^3}{1+m}+\frac{3 d^2 e x^2}{3+m}+\frac{3 d e^2 x^4}{5+m}+\frac{e^3 x^6}{7+m}\right )}{\sqrt{1-c^2 x^2}} \, dx}{f}\\ &=\frac{b e^3 (f x)^{6+m} \sqrt{1-c^2 x^2}}{c f^6 (7+m)^2}+\frac{d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{b \int \frac{(f x)^{1+m} \left (-\frac{c^2 d^3 (7+m)}{1+m}-\frac{3 c^2 d^2 e (7+m) x^2}{3+m}-\frac{e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) x^4}{(5+m) (7+m)}\right )}{\sqrt{1-c^2 x^2}} \, dx}{c f (7+m)}\\ &=\frac{b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt{1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac{b e^3 (f x)^{6+m} \sqrt{1-c^2 x^2}}{c f^6 (7+m)^2}+\frac{d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac{b \int \frac{(f x)^{1+m} \left (\frac{c^4 d^3 (5+m) (7+m)}{1+m}+\frac{e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt{1-c^2 x^2}} \, dx}{c^3 f (5+m) (7+m)}\\ &=\frac{b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \sqrt{1-c^2 x^2}}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2}+\frac{b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt{1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac{b e^3 (f x)^{6+m} \sqrt{1-c^2 x^2}}{c f^6 (7+m)^2}+\frac{d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac{\left (b \left (\frac{c^6 d^3}{1+m}+\frac{e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m)^2 (5+m)^2 (7+m)^2}\right )\right ) \int \frac{(f x)^{1+m}}{\sqrt{1-c^2 x^2}} \, dx}{c^5 f}\\ &=\frac{b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \sqrt{1-c^2 x^2}}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2}+\frac{b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt{1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac{b e^3 (f x)^{6+m} \sqrt{1-c^2 x^2}}{c f^6 (7+m)^2}+\frac{d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac{b \left (\frac{c^6 d^3}{1+m}+\frac{e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m)^2 (5+m)^2 (7+m)^2}\right ) (f x)^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{c^5 f^2 (2+m)}\\ \end{align*}

Mathematica [F]  time = 5.31832, size = 0, normalized size = 0. \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 22.783, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( e{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((f*x)^m*(e*x^2+d)^3*(a+b*arcsin(c*x)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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