3.7.7 \(\int x^2 (d+e x^2)^2 (a+b \text {ArcSin}(c x)) \, dx\) [607]

Optimal. Leaf size=198 \[ \frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {1-c^2 x^2}}{105 c^7}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{315 c^7}+\frac {b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+\frac {1}{3} d^2 x^3 (a+b \text {ArcSin}(c x))+\frac {2}{5} d e x^5 (a+b \text {ArcSin}(c x))+\frac {1}{7} e^2 x^7 (a+b \text {ArcSin}(c x)) \]

[Out]

-1/315*b*(35*c^4*d^2+84*c^2*d*e+45*e^2)*(-c^2*x^2+1)^(3/2)/c^7+1/175*b*e*(14*c^2*d+15*e)*(-c^2*x^2+1)^(5/2)/c^
7-1/49*b*e^2*(-c^2*x^2+1)^(7/2)/c^7+1/3*d^2*x^3*(a+b*arcsin(c*x))+2/5*d*e*x^5*(a+b*arcsin(c*x))+1/7*e^2*x^7*(a
+b*arcsin(c*x))+1/105*b*(35*c^4*d^2+42*c^2*d*e+15*e^2)*(-c^2*x^2+1)^(1/2)/c^7

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Rubi [A]
time = 0.16, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {276, 4815, 12, 1265, 785} \begin {gather*} \frac {1}{3} d^2 x^3 (a+b \text {ArcSin}(c x))+\frac {2}{5} d e x^5 (a+b \text {ArcSin}(c x))+\frac {1}{7} e^2 x^7 (a+b \text {ArcSin}(c x))+\frac {b e \left (1-c^2 x^2\right )^{5/2} \left (14 c^2 d+15 e\right )}{175 c^7}-\frac {b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7}+\frac {b \sqrt {1-c^2 x^2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*Sqrt[1 - c^2*x^2])/(105*c^7) - (b*(35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(1
- c^2*x^2)^(3/2))/(315*c^7) + (b*e*(14*c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^2*(1 - c^2*x^2)^(7/
2))/(49*c^7) + (d^2*x^3*(a + b*ArcSin[c*x]))/3 + (2*d*e*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*x^7*(a + b*ArcSin[c*
x]))/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 276

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 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1265

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

Rule 4815

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]))

Rubi steps

\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{105} (b c) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{210} (b c) \text {Subst}\left (\int \frac {x \left (35 d^2+42 d e x+15 e^2 x^2\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{210} (b c) \text {Subst}\left (\int \left (\frac {35 c^4 d^2+42 c^2 d e+15 e^2}{c^6 \sqrt {1-c^2 x}}+\frac {\left (-35 c^4 d^2-84 c^2 d e-45 e^2\right ) \sqrt {1-c^2 x}}{c^6}+\frac {3 e \left (14 c^2 d+15 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac {15 e^2 \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=\frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {1-c^2 x^2}}{105 c^7}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{315 c^7}+\frac {b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+\frac {1}{3} d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 158, normalized size = 0.80 \begin {gather*} \frac {105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+\frac {b \sqrt {1-c^2 x^2} \left (720 e^2+24 c^2 e \left (98 d+15 e x^2\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \text {ArcSin}(c x)}{11025} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + (b*Sqrt[1 - c^2*x^2]*(720*e^2 + 24*c^2*e*(98*d + 15*e*x^2) + 2
*c^4*(1225*d^2 + 588*d*e*x^2 + 135*e^2*x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7 + 105*b*x^3
*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcSin[c*x])/11025

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Maple [A]
time = 0.14, size = 279, normalized size = 1.41

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arcsin \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {d^{2} c^{4} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {2 d \,c^{2} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}\right )}{c^{4}}}{c^{3}}\)	\(279\)
default	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arcsin \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {d^{2} c^{4} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {2 d \,c^{2} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}\right )}{c^{4}}}{c^{3}}\)	\(279\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x^2+d)^2*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^3*(a/c^4*(1/3*d^2*c^7*x^3+2/5*d*c^7*e*x^5+1/7*e^2*c^7*x^7)+b/c^4*(1/3*arcsin(c*x)*d^2*c^7*x^3+2/5*arcsin(c
*x)*d*c^7*e*x^5+1/7*arcsin(c*x)*e^2*c^7*x^7-1/3*d^2*c^4*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2
))-2/5*d*c^2*e*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-1/7*e
^2*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/35*(-c^
2*x^2+1)^(1/2))))

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Maxima [A]
time = 0.50, size = 253, normalized size = 1.28 \begin {gather*} \frac {1}{7} \, a x^{7} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e + \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*x^7*e^2 + 2/5*a*d*x^5*e + 1/3*a*d^2*x^3 + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqr
t(-c^2*x^2 + 1)/c^4))*b*d^2 + 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*
x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d*e + 1/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*s
qrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*e^2

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Fricas [A]
time = 1.64, size = 185, normalized size = 0.93 \begin {gather*} \frac {1575 \, a c^{7} x^{7} e^{2} + 4410 \, a c^{7} d x^{5} e + 3675 \, a c^{7} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} x^{7} e^{2} + 42 \, b c^{7} d x^{5} e + 35 \, b c^{7} d^{2} x^{3}\right )} \arcsin \left (c x\right ) + {\left (1225 \, b c^{6} d^{2} x^{2} + 2450 \, b c^{4} d^{2} + 45 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} e^{2} + 294 \, {\left (3 \, b c^{6} d x^{4} + 4 \, b c^{4} d x^{2} + 8 \, b c^{2} d\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{11025 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11025*(1575*a*c^7*x^7*e^2 + 4410*a*c^7*d*x^5*e + 3675*a*c^7*d^2*x^3 + 105*(15*b*c^7*x^7*e^2 + 42*b*c^7*d*x^5
*e + 35*b*c^7*d^2*x^3)*arcsin(c*x) + (1225*b*c^6*d^2*x^2 + 2450*b*c^4*d^2 + 45*(5*b*c^6*x^6 + 6*b*c^4*x^4 + 8*
b*c^2*x^2 + 16*b)*e^2 + 294*(3*b*c^6*d*x^4 + 4*b*c^4*d*x^2 + 8*b*c^2*d)*e)*sqrt(-c^2*x^2 + 1))/c^7

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Sympy [A]
time = 0.87, size = 333, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {2 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {6 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {16 b d e \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {8 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{3}}{3} + \frac {2 d e x^{5}}{5} + \frac {e^{2} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**3/3 + 2*a*d*e*x**5/5 + a*e**2*x**7/7 + b*d**2*x**3*asin(c*x)/3 + 2*b*d*e*x**5*asin(c*x)/5
+ b*e**2*x**7*asin(c*x)/7 + b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 2*b*d*e*x**4*sqrt(-c**2*x**2 + 1)/(25*c)
+ b*e**2*x**6*sqrt(-c**2*x**2 + 1)/(49*c) + 2*b*d**2*sqrt(-c**2*x**2 + 1)/(9*c**3) + 8*b*d*e*x**2*sqrt(-c**2*
x**2 + 1)/(75*c**3) + 6*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) + 16*b*d*e*sqrt(-c**2*x**2 + 1)/(75*c**5)
+ 8*b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) + 16*b*e**2*sqrt(-c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d*
*2*x**3/3 + 2*d*e*x**5/5 + e**2*x**7/7), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (176) = 352\).
time = 0.41, size = 429, normalized size = 2.17 \begin {gather*} \frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b d^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d e x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b d e x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{9 \, c^{3}} + \frac {2 \, b d e x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2}}{3 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e}{25 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e}{15 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{49 \, c^{7}} + \frac {b e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e}{5 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{35 \, c^{7}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{7 \, c^{7}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2}}{7 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/3*(c^2*x^2 - 1)*b*d^2*x*arcsin(c*x)/c^2 + 1/3*b*d^2*x*arcsin
(c*x)/c^2 + 2/5*(c^2*x^2 - 1)^2*b*d*e*x*arcsin(c*x)/c^4 + 4/5*(c^2*x^2 - 1)*b*d*e*x*arcsin(c*x)/c^4 + 1/7*(c^2
*x^2 - 1)^3*b*e^2*x*arcsin(c*x)/c^6 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^2/c^3 + 2/5*b*d*e*x*arcsin(c*x)/c^4 + 3/7*(
c^2*x^2 - 1)^2*b*e^2*x*arcsin(c*x)/c^6 + 1/3*sqrt(-c^2*x^2 + 1)*b*d^2/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2
+ 1)*b*d*e/c^5 + 3/7*(c^2*x^2 - 1)*b*e^2*x*arcsin(c*x)/c^6 - 4/15*(-c^2*x^2 + 1)^(3/2)*b*d*e/c^5 + 1/49*(c^2*
x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^2/c^7 + 1/7*b*e^2*x*arcsin(c*x)/c^6 + 2/5*sqrt(-c^2*x^2 + 1)*b*d*e/c^5 + 3/3
5*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^2/c^7 - 1/7*(-c^2*x^2 + 1)^(3/2)*b*e^2/c^7 + 1/7*sqrt(-c^2*x^2 + 1)*b
*e^2/c^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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