3.7.55 \(\int (f x)^m (d+e x^2)^2 (a+b \text {ArcSin}(c x)) \, dx\) [655]
Optimal. Leaf size=293 \[ \frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^3 f^2 (3+m)^2 (5+m)^2}+\frac {b e^2 (f x)^{4+m} \sqrt {1-c^2 x^2}}{c f^4 (5+m)^2}+\frac {d^2 (f x)^{1+m} (a+b \text {ArcSin}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {ArcSin}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {ArcSin}(c x))}{f^5 (5+m)}-\frac {b \left (\frac {c^4 d^2 (3+m) (5+m)}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m) (5+m)}\right ) (f x)^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{c^3 f^2 (2+m) (3+m) (5+m)} \]
[Out]
d^2*(f*x)^(1+m)*(a+b*arcsin(c*x))/f/(1+m)+2*d*e*(f*x)^(3+m)*(a+b*arcsin(c*x))/f^3/(3+m)+e^2*(f*x)^(5+m)*(a+b*a
rcsin(c*x))/f^5/(5+m)-b*(c^4*d^2*(3+m)*(5+m)/(1+m)+e*(2+m)*(2*c^2*d*(5+m)^2+e*(m^2+7*m+12))/(3+m)/(5+m))*(f*x)
^(2+m)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/c^3/f^2/(2+m)/(3+m)/(5+m)+b*e*(2*c^2*d*(5+m)^2+e*(m^2+7*m+1
2))*(f*x)^(2+m)*(-c^2*x^2+1)^(1/2)/c^3/f^2/(3+m)^2/(5+m)^2+b*e^2*(f*x)^(4+m)*(-c^2*x^2+1)^(1/2)/c/f^4/(5+m)^2
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Rubi [A]
time = 0.31, antiderivative size = 272, normalized size of antiderivative = 0.93, number of steps
used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {276, 4815, 12,
1281, 470, 371} \begin {gather*} \frac {d^2 (f x)^{m+1} (a+b \text {ArcSin}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {ArcSin}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {ArcSin}(c x))}{f^5 (m+5)}+\frac {b e^2 \sqrt {1-c^2 x^2} (f x)^{m+4}}{c f^4 (m+5)^2}-\frac {b c (f x)^{m+2} \left (\frac {e \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^4 (m+3)^2 (m+5)^2}+\frac {d^2}{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 (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^3 f^2 (m+3)^2 (m+5)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]
[Out]
(b*e*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^3*f^2*(3 + m)^2*(5 + m)^2) +
(b*e^2*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c*f^4*(5 + m)^2) + (d^2*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m
)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x]))/(f^5*(5
+ m)) - (b*c*(d^2/(2 + 3*m + m^2) + (e*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2)))/(c^4*(3 + m)^2*(5 + m)^2))*(
f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/f^2
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 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 470
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 1281
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 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 (f x)^m \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-(b c) \int \frac {(f x)^{1+m} \left (\frac {d^2}{1+m}+\frac {2 d e x^2}{3+m}+\frac {e^2 x^4}{5+m}\right )}{f \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {(b c) \int \frac {(f x)^{1+m} \left (\frac {d^2}{1+m}+\frac {2 d e x^2}{3+m}+\frac {e^2 x^4}{5+m}\right )}{\sqrt {1-c^2 x^2}} \, dx}{f}\\ &=\frac {b e^2 (f x)^{4+m} \sqrt {1-c^2 x^2}}{c f^4 (5+m)^2}+\frac {d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {b \int \frac {(f x)^{1+m} \left (-\frac {c^2 d^2 (5+m)}{1+m}-\frac {e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) x^2}{(3+m) (5+m)}\right )}{\sqrt {1-c^2 x^2}} \, dx}{c f (5+m)}\\ &=\frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^3 f^2 (3+m)^2 (5+m)^2}+\frac {b e^2 (f x)^{4+m} \sqrt {1-c^2 x^2}}{c f^4 (5+m)^2}+\frac {d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b \left (\frac {c^4 d^2}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m)^2 (5+m)^2}\right )\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{c^3 f}\\ &=\frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^3 f^2 (3+m)^2 (5+m)^2}+\frac {b e^2 (f x)^{4+m} \sqrt {1-c^2 x^2}}{c f^4 (5+m)^2}+\frac {d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {b \left (\frac {c^4 d^2}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m)^2 (5+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^3 f^2 (2+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 5.56, size = 2792, normalized size = 9.53 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]
[Out]
(x*(f*x)^m*(-2*(d + e*x^2)^2*(-((2 + m)*(a + b*ArcSin[c*x])) + b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2,
c^2*x^2]) + 8*e*x^2*(d + e*x^2)*(-a - b*ArcSin[c*x] + (b*c*x*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*
x^2])/(3 + m) + b*c*x*Gamma[2 + m/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {2 + m/2}, c^2*x^2] - (b*c*x
*Gamma[2 + m/2]*Gamma[(3 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(5 + m)/2}, c^2*x^2])/Gamma[1
+ m/2]) - (4*e*x^2*(d + 3*e*x^2)*((-2*Gamma[1 + m/2]*((4 + m)*(a + b*ArcSin[c*x]) - b*c*x*Hypergeometric2F1[1
/2, 2 + m/2, 3 + m/2, c^2*x^2]))/(4 + m) + b*c*(3 + m)*x*Gamma[1 + m/2]*Gamma[2 + m/2]*HypergeometricPFQRegula
rized[{1/2, 1 + m/2}, {3 + m/2}, c^2*x^2] + (2*b*c*x*Gamma[2 + m/2]*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] -
Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {3 + m/2}, c^2*x^2])/Gamma[(3 +
m)/2] - 2*b*c*x*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] - Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQ
Regularized[{1/2, (3 + m)/2}, {(5 + m)/2}, c^2*x^2]))/((3 + m)*Gamma[1 + m/2]) - (8*e^2*x^4*(30*a*Gamma[1 + m/
2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] + 6*a*m*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] + 30*b*ArcSin[c*x]*G
amma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] + 6*b*m*ArcSin[c*x]*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/
2] - 6*b*c*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, c^2*x
^2] - b*c*(60 + 47*m + 12*m^2 + m^3)*x*Gamma[1 + m/2]*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*HypergeometricPFQRegul
arized[{1/2, 1 + m/2}, {3 + m/2}, c^2*x^2] + b*c*(5 + m)*x*Gamma[2 + m/2]*(-6*(12 + 7*m + m^2)*Gamma[2 + m/2]^
2*Gamma[(3 + m)/2] + Gamma[1 + m/2]*(-6*Gamma[3 + m/2]*Gamma[(3 + m)/2] + (4 + m)*Gamma[2 + m/2]*((6 + 5*m + m
^2)*Gamma[(3 + m)/2] + 6*Gamma[(5 + m)/2])))*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {3 + m/2}, c^2*x^2]
+ 180*b*c*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^
2*x^2] + 141*b*c*m*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 +
m)/2}, c^2*x^2] + 36*b*c*m^2*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/
2}, {(7 + m)/2}, c^2*x^2] + 3*b*c*m^3*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2,
(3 + m)/2}, {(7 + m)/2}, c^2*x^2] - 60*b*c*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*
HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] - 27*b*c*m*x*Gamma[1 + m/2]*Gamma[2 + m/2
]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] - 3*b
*c*m^2*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (3
+ m)/2}, {(7 + m)/2}, c^2*x^2] + 240*b*c*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricP
FQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 188*b*c*m*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5
+ m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 48*b*c*m^2*x*Gamma[2 + m/2]^2*
Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 4*b*c
*m^3*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 +
m)/2}, c^2*x^2] + 30*b*c*x*Gamma[1 + m/2]*Gamma[3 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQR
egularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 6*b*c*m*x*Gamma[1 + m/2]*Gamma[3 + m/2]*Gamma[(3 + m)/2]*
Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] - 120*b*c*x*Gamma[1 + m/
2]*Gamma[2 + m/2]*Gamma[(5 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] - 54
*b*c*m*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(5 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 +
m)/2}, c^2*x^2] - 6*b*c*m^2*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(5 + m)/2]^2*HypergeometricPFQRegularized[{
1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2]))/((3 + m)*(4 + m)*(5 + m)*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/
2]) + (4*e^2*x^4*(-(b*c*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)*x*Gamma[1 + m/2]*Gamma[2 + m/2]^2*Gamma[(3 + m)
/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {4 + m/2}, c^2*x^2]) + b*c*(30 + 11*m + m^2)
*x*Gamma[2 + m/2]*Gamma[(5 + m)/2]*(-6*(12 + 7*m + m^2)*Gamma[2 + m/2]^2*Gamma[(3 + m)/2] + Gamma[1 + m/2]*(-6
*Gamma[3 + m/2]*Gamma[(3 + m)/2] + (4 + m)*Gamma[2 + m/2]*((6 + 5*m + m^2)*Gamma[(3 + m)/2] + 6*Gamma[(5 + m)/
2])))*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {4 + m/2}, c^2*x^2] - 4*b*c*(6 + m)*x*Gamma[3 + m/2]*((60 +
47*m + 12*m^2 + m^3)*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2] + 3*(5 + m)*Gamma[1 + m/2]*Gamma[3 +
m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2] - Gamma[1 + m/2]*Gamma[2 + m/2]*(2*(20 + 9*m + m^2)*Gamma[(5 + m)/2]^2
+ 3*Gamma[(3 + m)/2]*Gamma[(7 + m)/2]))*HypergeometricPFQRegularized[{1/2, 3 + m/2}, {4 + m/2}, c^2*x^2] + 2*G
amma[(5 + m)/2]*(6*Gamma[1 + m/2]*Gamma[2 + m/2...
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Maple [F]
time = 8.01, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x)^m*(e*x^2+d)^2*(a+b*arcsin(c*x)),x)
[Out]
int((f*x)^m*(e*x^2+d)^2*(a+b*arcsin(c*x)),x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^m*(e*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="maxima")
[Out]
a*f^m*x^5*e^(m*log(x) + 2)/(m + 5) + 2*a*d*f^m*x^3*e^(m*log(x) + 1)/(m + 3) + (f*x)^(m + 1)*a*d^2/(f*(m + 1))
+ (((b*f^m*m^2*e^2 + 4*b*f^m*m*e^2 + 3*b*f^m*e^2)*x^5 + 2*(b*d*f^m*m^2*e + 6*b*d*f^m*m*e + 5*b*d*f^m*e)*x^3 +
(b*d^2*f^m*m^2 + 8*b*d^2*f^m*m + 15*b*d^2*f^m)*x)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (m^3 + 9*m^
2 + 23*m + 15)*integrate(-((b*c*f^m*m^2*e^2 + 4*b*c*f^m*m*e^2 + 3*b*c*f^m*e^2)*x^5 + 2*(b*c*d*f^m*m^2*e + 6*b*
c*d*f^m*m*e + 5*b*c*d*f^m*e)*x^3 + (b*c*d^2*f^m*m^2 + 8*b*c*d^2*f^m*m + 15*b*c*d^2*f^m)*x)*sqrt(c*x + 1)*sqrt(
-c*x + 1)*x^m/(m^3 - (c^2*m^3 + 9*c^2*m^2 + 23*c^2*m + 15*c^2)*x^2 + 9*m^2 + 23*m + 15), x))/(m^3 + 9*m^2 + 23
*m + 15)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^m*(e*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="fricas")
[Out]
integral((a*x^4*e^2 + 2*a*d*x^2*e + a*d^2 + (b*x^4*e^2 + 2*b*d*x^2*e + b*d^2)*arcsin(c*x))*(f*x)^m, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{m} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)**m*(e*x**2+d)**2*(a+b*asin(c*x)),x)
[Out]
Integral((f*x)**m*(a + b*asin(c*x))*(d + e*x**2)**2, x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^m*(e*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="giac")
[Out]
integrate((e*x^2 + d)^2*(b*arcsin(c*x) + a)*(f*x)^m, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*asin(c*x))*(f*x)^m*(d + e*x^2)^2,x)
[Out]
int((a + b*asin(c*x))*(f*x)^m*(d + e*x^2)^2, x)
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