3.56 \(\int \frac {(f+g x) (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=228 \[ \frac {\left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (f+g x)}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

2/3*f*x*(a+b*arcsin(c*x))/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*(c^2*f*x+g)*(a+b*arcsin(c*x))/c^2/d^2/(-c^2*x^2+1)/(-c^
2*d*x^2+d)^(1/2)-1/6*b*(g*x+f)/c/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/6*b*g*arctanh(c*x)*(-c^2*x^2+1)
^(1/2)/c^2/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*b*f*ln(-c^2*x^2+1)*(-c^2*x^2+1)^(1/2)/c/d^2/(-c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4777, 639, 191, 4761, 206, 260} \[ \frac {\left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (f+g x)}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

-(b*(f + g*x))/(6*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (2*f*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c
^2*d*x^2]) + ((g + c^2*f*x)*(a + b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (b*g*Sqrt[1 -
c^2*x^2]*ArcTanh[c*x])/(6*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b*f*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(3*c*d^2*Sq
rt[d - c^2*d*x^2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

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 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]

Rubi steps

\begin {align*} \int \frac {(f+g x) \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) \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 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 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 {g+c^2 f x}{3 c^2 \left (1-c^2 x^2\right )^2}+\frac {2 f x}{3 \left (1-c^2 x^2\right )}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 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 {g+c^2 f x}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c f \sqrt {1-c^2 x^2}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x)}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{6 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x)}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.83, size = 208, normalized size = 0.91 \[ -\frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (-4 a c^4 f x^3+6 a c^2 f x+2 a g+2 b \sin ^{-1}(c x) \left (c^2 f x \left (3-2 c^2 x^2\right )+g\right )-b c f \sqrt {1-c^2 x^2}+2 b c f \left (1-c^2 x^2\right )^{3/2} \log \left (c^2 x^2-1\right )-b c g x \sqrt {1-c^2 x^2}\right )+i b c g \left (1-c^2 x^2\right )^{3/2} F\left (\left .i \sinh ^{-1}\left (\sqrt {-c^2} x\right )\right |1\right )\right )}{6 \left (-c^2\right )^{3/2} d^3 \left (c^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

-1/6*(Sqrt[d - c^2*d*x^2]*(I*b*c*g*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1] + Sqrt[-c^2]*(2*a
*g + 6*a*c^2*f*x - 4*a*c^4*f*x^3 - b*c*f*Sqrt[1 - c^2*x^2] - b*c*g*x*Sqrt[1 - c^2*x^2] + 2*b*(g + c^2*f*x*(3 -
2*c^2*x^2))*ArcSin[c*x] + 2*b*c*f*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2])))/((-c^2)^(3/2)*d^3*(-1 + c^2*x^2)^2
)

________________________________________________________________________________________

fricas [F]  time = 2.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a g x + a f + {\left (b g x + b f\right )} \arcsin \left (c x\right )\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(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*x + a*f + (b*g*x + b*f)*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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)/(-c^2*d*x^2 + d)^(5/2), x)

________________________________________________________________________________________

maple [C]  time = 0.76, size = 2236, normalized size = 9.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-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)*f*x*arcsin(c*x)+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-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*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)*f*x-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^2*g+1/3*a*f*x/d/(-c^2*d*x^2+d)^(3/2)+2/
3*a*f/d^2*x/(-c^2*d*x^2+d)^(1/2)+1/3*a*g/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/3*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1)
)^(1/2)/c/d^3/(c^2*x^2-1)*f*arcsin(c*x)-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-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+2/3*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-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^2*(-c^2*x^2+1)*arcsin(c*x)*x^4*g-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^4*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^6*g-1/6*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*
x^2-1))^(1/2)/c^2/d^3/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*g-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+1/6*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)*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*arcsin(c*x)*x^4*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+2/3*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-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^6*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^4*x^5*f+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^8*
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^4*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-4/3*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^4*arcsin(c*x)*x^6*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*arcs
in(c*x)*x^5*f-2/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c/d^3/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)
*f+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+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*x^2+1)*arcsin(c*x)*x^2*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^2*(-c^2*x^2+1)*arcsin(c*x)*g+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)*x*g-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^3*g-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+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*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+1
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*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*f

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, b c f {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} - c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b f {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac {1}{3} \, a f {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + \frac {\frac {b g \int \frac {x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c x + 1\right )}^{\frac {5}{2}} {\left (c x - 1\right )}^{2} \sqrt {-c x + 1}}\,{d x}}{d^{2}}}{\sqrt {d}} + \frac {a g}{3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/6*b*c*f*(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*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*x/(sqrt(-c^2*
d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + b*g*integrate(x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(
(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d) + 1/3*a*g/((-c^2*d*x^2 + d)^(3/2
)*c^2*d)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2),x)

[Out]

int(((f + g*x)*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x)/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)

________________________________________________________________________________________