\(\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [55]
Optimal result
Integrand size = 31, antiderivative size = 271 \[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b x \left (2 f g+\left (c^2 f^2+g^2\right ) x\right )}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f \left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b (2 c f-g) (c f+g) \sqrt {1-c^2 x^2} \log (1-c x)}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g) (2 c f+g) \sqrt {1-c^2 x^2} \log (1+c x)}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}
\]
[Out]
2/3*f*(c^2*f*x+g)*(a+b*arcsin(c*x))/c^2/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*x*(g*x+f)^2*(a+b*arcsin(c*x))/d^2/(-c^2*x
^2+1)/(-c^2*d*x^2+d)^(1/2)-1/6*b*x*(2*f*g+(c^2*f^2+g^2)*x)/c/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/6*b
*(2*c*f-g)*(c*f+g)*ln(-c*x+1)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(-c^2*d*x^2+d)^(1/2)+1/6*b*(c*f-g)*(2*c*f+g)*ln(c*x+1
)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(-c^2*d*x^2+d)^(1/2)
Rubi [A] (verified)
Time = 0.28 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.31, number of
steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4861, 743, 651, 4845, 833,
647, 31} \[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 f \left (c^2 f x+g\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (f+g x)^2}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} (c f+g) \log (1-c x)}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} (c f-g) \log (c x+1)}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} (c f+g) \log (1-c x)}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b g \sqrt {1-c^2 x^2} (c f-g) \log (c x+1)}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}
\]
[In]
Int[((f + g*x)^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
[Out]
-1/6*(b*(f + g*x)^2)/(c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (2*f*(g + c^2*f*x)*(a + b*ArcSin[c*x]))/(
3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (x*(f + g*x)^2*(a + b*ArcSin[c*x]))/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])
+ (b*f*(c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (b*g*(c*f + g)*Sqrt[1 - c^
2*x^2]*Log[1 - c*x])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b*f*(c*f - g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(3*c^2*d
^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)*g*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2])
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 647
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q)),
Int[1/(-q + c*x), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[(-a)*c]
Rule 651
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((-a)*e + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]
Rule 743
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^m)*(2*c*x)*((a + c*x^2)
^(p + 1)/(4*a*c*(p + 1))), x] - Dist[m*((2*c*d)/(4*a*c*(p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x
], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
Rule 833
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
- 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Rule 4845
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With
[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 - c^
2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2,
0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
Rule 4861
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; F
reeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && !GtQ[d, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {2 f \left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \left (\frac {x (f+g x)^2}{3 \left (1-c^2 x^2\right )^2}+\frac {2 f \left (g+c^2 f x\right )}{3 c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {2 f \left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {x (f+g x)^2}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f \sqrt {1-c^2 x^2}\right ) \int \frac {g+c^2 f x}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b (f+g x)^2}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f \left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {2 f g+2 g^2 x}{1-c^2 x^2} \, dx}{6 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b f (c f-g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b f (c f+g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b (f+g x)^2}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f \left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b f (c f+g) \sqrt {1-c^2 x^2} \log (1-c x)}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b f (c f-g) \sqrt {1-c^2 x^2} \log (1+c x)}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) g \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b g (c f+g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{6 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b (f+g x)^2}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f \left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (f+g x)^2 (a+b \arcsin (c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b f (c f+g) \sqrt {1-c^2 x^2} \log (1-c x)}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g (c f+g) \sqrt {1-c^2 x^2} \log (1-c x)}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b f (c f-g) \sqrt {1-c^2 x^2} \log (1+c x)}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g) g \sqrt {1-c^2 x^2} \log (1+c x)}{6 c^3 d^2 \sqrt {d-c^2 d x^2}} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.75 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.05
\[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {c \sqrt {d-c^2 d x^2} \left (2 i b c^2 f g \left (1-c^2 x^2\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )-\sqrt {-c^2} \left (-4 a c f g-6 a c^3 f^2 x+4 a c^5 f^2 x^3-2 a c^3 g^2 x^3+b c^2 f^2 \sqrt {1-c^2 x^2}+b g^2 \sqrt {1-c^2 x^2}+2 b c^2 f g x \sqrt {1-c^2 x^2}+2 b c \left (-2 f g-c^2 g^2 x^3+c^2 f^2 x \left (-3+2 c^2 x^2\right )\right ) \arcsin (c x)-b \left (2 c^2 f^2-g^2\right ) \left (1-c^2 x^2\right )^{3/2} \log \left (-1+c^2 x^2\right )\right )\right )}{6 \left (-c^2\right )^{5/2} d^3 \left (-1+c^2 x^2\right )^2}
\]
[In]
Integrate[((f + g*x)^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
[Out]
(c*Sqrt[d - c^2*d*x^2]*((2*I)*b*c^2*f*g*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1] - Sqrt[-c^2]
*(-4*a*c*f*g - 6*a*c^3*f^2*x + 4*a*c^5*f^2*x^3 - 2*a*c^3*g^2*x^3 + b*c^2*f^2*Sqrt[1 - c^2*x^2] + b*g^2*Sqrt[1
- c^2*x^2] + 2*b*c^2*f*g*x*Sqrt[1 - c^2*x^2] + 2*b*c*(-2*f*g - c^2*g^2*x^3 + c^2*f^2*x*(-3 + 2*c^2*x^2))*ArcSi
n[c*x] - b*(2*c^2*f^2 - g^2)*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2])))/(6*(-c^2)^(5/2)*d^3*(-1 + c^2*x^2)^2)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 3783, normalized size of antiderivative =
13.96
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\(\text {Expression too large to display}\) |
\(3783\) |
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\(\text {Expression too large to display}\) |
\(3783\) |
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[In]
int((g*x+f)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
[Out]
-14/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*x^6*f*g+2/3*I*b*(-d*(c^2*x^2-1))^
(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^2+1)*x^5*g^2+16/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*
c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*x^4*f*g-8/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x
^2-4)/c*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*f^2+4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4
)/c*(-c^2*x^2+1)^(1/2)*x*f*g-8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*arcsin(c
*x)*x^6*f*g+4*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*(-c^2*x^2+1)*arcsin(c*x)*x^3*g^
2+7*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*arcsin(c*x)*x^5*g^2-2*b*(-d*(c^2*x^2-
1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*arcsin(c*x)*x^7*g^2-2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^
6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*arcsin(c*x)*x^5*f^2+I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11
*c^2*x^2-4)*(-c^2*x^2+1)*x*f^2-6*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*arcsin(c*x)*
x^2*f*g+17/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*arcsin(c*x)*x^3*f^2-1/2*b*(-
d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c*(-c^2*x^2+1)^(1/2)*x^2*f^2-1/2*b*(-d*(c^2*x^2-1
))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c*(-c^2*x^2+1)^(1/2)*x^2*g^2+2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(
3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c^2*arcsin(c*x)*x*g^2-2/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ln(I*
c*x+(-c^2*x^2+1)^(1/2)+I)/d^3/c/(c^2*x^2-1)*f^2+1/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ln(I*c*x+(-c^2
*x^2+1)^(1/2)+I)/d^3/c^3/(c^2*x^2-1)*g^2-2/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ln(I*c*x+(-c^2*x^2+1)
^(1/2)-I)/d^3/c/(c^2*x^2-1)*f^2+1/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)
/d^3/c^3/(c^2*x^2-1)*g^2-7/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*x^5*f^2-7/
3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*x^5*g^2+2/3*I*b*(-d*(c^2*x^2-1))^(1/2
)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^6*x^7*f^2+2/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4
+11*c^2*x^2-4)*c^4*x^7*g^2-5/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*(-c^2*x^2+1)
*x^3*g^2-2*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*x^2*f*g+8/3*I*b*(-d*(c^2*x^2-1))
^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*x^3*f^2-I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^
4+11*c^2*x^2-4)/c^2*x*g^2-10/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^
2+1)*x^4*f*g+14/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c*(-c^2*x^2+1)^(1/2)*arcs
in(c*x)*x^2*f^2+4/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c^3*(-c^2*x^2+1)^(1/2)*
arcsin(c*x)*g^2+2*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*(-c^2*x^2+1)*x^2*f*g+4/3*
I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)/d^3/c/(c^2*x^2-1)*f^2+8*b*(-d*(c^2*x^2-1))^(1/2)/d^3
/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*arcsin(c*x)*x^4*f*g+16/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c
^4*x^4+11*c^2*x^2-4)*(-c^2*x^2+1)*arcsin(c*x)*x^2*f*g-2/3*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*arcsin
(c*x)/d^3/c^3/(c^2*x^2-1)*g^2+4/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^6*x^8*f
*g+2/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*(-c^2*x^2+1)*x^5*f^2+2/3*b*(-d*(
c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c*(-c^2*x^2+1)^(1/2)*f^2+2/3*b*(-d*(c^2*x^2-1))^(1/2
)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c^3*(-c^2*x^2+1)^(1/2)*g^2-22/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*
x^6-10*c^4*x^4+11*c^2*x^2-4)*arcsin(c*x)*x^3*g^2-4*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x
^2-4)*arcsin(c*x)*x*f^2+8/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*x^3*g^2-I*b*(-d
*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*x*f^2+I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10
*c^4*x^4+11*c^2*x^2-4)*c*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4*g^2-8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10
*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^2+1)*arcsin(c*x)*x^4*f*g-7/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c
^4*x^4+11*c^2*x^2-4)/c*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*g^2+4/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10
*c^4*x^4+11*c^2*x^2-4)*c^4*(-c^2*x^2+1)*x^6*f*g-2*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*
x^2-4)*c^3*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4*f^2+a*(f^2*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d
)^(1/2))+g^2*(1/2*x/c^2/d/(-c^2*d*x^2+d)^(3/2)-1/2/c^2*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^
(1/2)))+2/3*f*g/c^2/d/(-c^2*d*x^2+d)^(3/2))-8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-
4)/c^2*(-c^2*x^2+1)*arcsin(c*x)*f*g-b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c*(-c^2*x
^2+1)^(1/2)*x^3*f*g+1/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/d^3/c^2/(c^
2*x^2-1)*f*g-1/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)/d^3/c^2/(c^2*x^2-1
)*f*g-2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c^2*(-c^2*x^2+1)*arcsin(c*x)*x*g^2-2*
b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^2+1)*arcsin(c*x)*x^5*g^2+I*b*(-d*
(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c^2*(-c^2*x^2+1)*x*g^2-5/3*I*b*(-d*(c^2*x^2-1))^(1/
2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^2+1)*x^3*f^2
Fricas [F]
\[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((g*x+f)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
[Out]
integral(-sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 + 2*b*f*g*x + b*f^2)*arcsin(c*x))/(
c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
Sympy [F]
\[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx
\]
[In]
integrate((g*x+f)**2*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)
[Out]
Integral((a + b*asin(c*x))*(f + g*x)**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
Maxima [F]
\[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((g*x+f)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
[Out]
1/6*b*c*f^2*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^(5/2)) + 2*log(c*x - 1)/(c^2*d^(5/2)))
+ 1/3*b*f^2*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arcsin(c*x) + 1/3*a*f^2*(2*x/(sqrt
(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) - 1/3*a*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2
*d*x^2 + d)^(3/2)*c^2*d)) - sqrt(d)*integrate((b*g^2*x^2 + 2*b*f*g*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x
, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x) + 2/3*a*f*g/((-c^2*d*x
^2 + d)^(3/2)*c^2*d)
Giac [F(-2)]
Exception generated. \[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate((g*x+f)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value
Mupad [F(-1)]
Timed out. \[
\int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x
\]
[In]
int(((f + g*x)^2*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2),x)
[Out]
int(((f + g*x)^2*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2), x)