3.55 \(\int \frac{(f+g x)^2 (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\)
Optimal. Leaf size=271 \[ \frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{2 f \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x \left (x \left (c^2 f^2+g^2\right )+2 f g\right )}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (2 c f-g) (c f+g) \log (1-c x)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g) (2 c f+g) \log (c x+1)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}} \]
[Out]
-(b*x*(2*f*g + (c^2*f^2 + g^2)*x))/(6*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*(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]) +
(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])
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Rubi [A] time = 0.390261, antiderivative size = 354, normalized size of antiderivative =
1.31, number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules
used = {4777, 729, 637, 4761, 819, 633, 31} \[ \frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{2 f \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (f+g x)^2}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b f \sqrt{1-c^2 x^2} (c f+g) \log (1-c x)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g \sqrt{1-c^2 x^2} (c f+g) \log (1-c x)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b f \sqrt{1-c^2 x^2} (c f-g) \log (c x+1)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b g \sqrt{1-c^2 x^2} (c f-g) \log (c x+1)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
Int[((f + g*x)^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
[Out]
-(b*(f + g*x)^2)/(6*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 4777
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ
[p - 1/2] && !GtQ[d, 0]
Rule 729
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 637
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 4761
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 819
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 633
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 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\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 ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{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 ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{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 ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{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 ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{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 ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{x (f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{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] time = 1.02236, size = 285, normalized size = 1.05 \[ \frac{c \sqrt{d-c^2 d x^2} \left (-\sqrt{-c^2} \left (4 a c^5 f^2 x^3-6 a c^3 f^2 x-2 a c^3 g^2 x^3-4 a c f g-b \left (1-c^2 x^2\right )^{3/2} \left (2 c^2 f^2-g^2\right ) \log \left (c^2 x^2-1\right )+2 b c \sin ^{-1}(c x) \left (c^2 f^2 x \left (2 c^2 x^2-3\right )-c^2 g^2 x^3-2 f g\right )+b c^2 f^2 \sqrt{1-c^2 x^2}+2 b c^2 f g x \sqrt{1-c^2 x^2}+b g^2 \sqrt{1-c^2 x^2}\right )+2 i b c^2 f g \left (1-c^2 x^2\right )^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),1\right )\right )}{6 \left (-c^2\right )^{5/2} d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
[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)
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Maple [C] time = 0.437, size = 3765, normalized size = 13.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)
[Out]
1/3*a*f^2/d*x/(-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^4*ar
csin(c*x)*x^6*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)*arcsin(c*x)*(-c^2*x^2+
1)*x^2*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*arcsin(c*x)*(-c^2*x^2+1)*x*g
^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+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+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)/c^2/d^3/(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)/c^2/d^3/(c^2*x^2-1)*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*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*f^2+4/3*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*ar
csin(c*x)/c/d^3/(c^2*x^2-1)*f^2-2/3*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)/c^3/d^3/(c^2*x^2
-1)*g^2+1/3*a*g^2/c^2/d*x/(-c^2*d*x^2+d)^(3/2)-1/3*a*g^2/c^2/d^2*x/(-c^2*d*x^2+d)^(1/2)+2/3*a*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*arcsin(c*x)*(-c^2*x^2+
1)*x^4*f*g+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*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4*f^2-1
0/3*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*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2*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*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2*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*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4*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^2-5/3*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+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-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*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*(-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*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*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^2*(-c^2*x^2+1)*x^5*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*arcsin(c*x)*(-c^2*x^2+1)*f*g+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-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)*(-c^2*x^2+1)*x^5*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*(-c
^2*x^2+1)^(1/2)*x^3*f*g-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^2-2*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+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^2-1/2*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)/c/d^3/(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)/c^3/d^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)/c/d^3/(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)/c^3/d^3/(c^2*x^2-1)*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*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-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+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^2-22/3*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+1
1*c^2*x^2-4)*arcsin(c*x)*x*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
+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^2-7/3*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+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+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^2-7
/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+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+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)*(-c^2*x^2+1)*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+1
1*c^2*x^2-4)/c^2*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*x^2+1)*x^3
*g^2+2/3*a*f^2/d^2*x/(-c^2*d*x^2+d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, b c f^{2}{\left (\frac{1}{c^{4} d^{\frac{5}{2}} x^{2} - c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac{5}{2}}}\right )} + \frac{1}{3} \, b f^{2}{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac{1}{3} \, a f^{2}{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} - \frac{1}{3} \, a g^{2}{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} - \sqrt{d} \int \frac{{\left (b g^{2} x^{2} + 2 \, b f g x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}\,{d x} + \frac{2 \, a f g}{3 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \arcsin \left (c x\right )\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
[Out]
integrate((g*x + f)^2*(b*arcsin(c*x) + a)/(-c^2*d*x^2 + d)^(5/2), x)