3.2.37 \(\int \frac {a+b \text {ArcSin}(c x)}{x^2 (d-c^2 d x^2)^{5/2}} \, dx\) [137]

Optimal. Leaf size=224 \[ -\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {a+b \text {ArcSin}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {ArcSin}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^3 \sqrt {1-c^2 x^2}}+\frac {5 b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt {1-c^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {277, 198, 197, 4779, 12, 1265, 907} \begin {gather*} \frac {8 c^2 x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {ArcSin}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {ArcSin}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{d^3 \sqrt {1-c^2 x^2}}+\frac {5 b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*c*Sqrt[d - c^2*d*x^2])/(d^3*(1 - c^2*x^2)^(3/2)) - (a + b*ArcSin[c*x])/(d*x*(d - c^2*d*x^2)^(3/2)) + (
4*c^2*x*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c^2*d
*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^3*Sqrt[1 - c^2*x^2]) + (5*b*c*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2
])/(6*d^3*Sqrt[1 - c^2*x^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 197

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 198

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

Rule 277

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

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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 4779

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[Si
mplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p
- 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps

\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\left (4 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \sin ^{-1}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (8 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b c}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {a+b \sin ^{-1}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{x}+\frac {c^2}{\left (-1+c^2 x\right )^2}-\frac {c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b c^3 \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 c}{6 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {a+b \sin ^{-1}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log (x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b c \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 188, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {d-c^2 d x^2} \left (6 a-24 a c^2 x^2+16 a c^4 x^4+b c x \sqrt {1-c^2 x^2}+2 b \left (3-12 c^2 x^2+8 c^4 x^4\right ) \text {ArcSin}(c x)-3 b c x \left (1-c^2 x^2\right )^{3/2} \log \left (x^2\right )-5 b c x \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )+5 b c^3 x^3 \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )\right )}{6 d^3 x \left (-1+c^2 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(Sqrt[d - c^2*d*x^2]*(6*a - 24*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(3 - 12*c^2*x^2 +
8*c^4*x^4)*ArcSin[c*x] - 3*b*c*x*(1 - c^2*x^2)^(3/2)*Log[x^2] - 5*b*c*x*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] +
5*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2]))/(d^3*x*(-1 + c^2*x^2)^2)

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Maple [C] Result contains complex when optimal does not.
time = 0.25, size = 1346, normalized size = 6.01

method	result	size
default	\(a \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )+\frac {20 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{4}}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {136 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {32 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{7} \left (-c^{2} x^{2}+1\right ) c^{8}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{2}}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{2}}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {140 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{6}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {64 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{4} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {24 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{2 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {9 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3} x}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {56 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arcsin \left (c x \right ) c^{4}}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {44 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arcsin \left (c x \right ) c^{2}}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c}{3 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d^{3} \left (c^{2} x^{2}-1\right )}-\frac {64 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} \arcsin \left (c x \right ) c^{6}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {24 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{4}}{\left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}-\frac {112 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{7} c^{8}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {16 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{3 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {80 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}+\frac {32 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{9} c^{10}}{3 \left (8 c^{6} x^{6}-25 c^{4} x^{4}+26 c^{2} x^{2}-9\right ) d^{3}}\)	\(1346\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

a*(-1/d/x/(-c^2*d*x^2+d)^(3/2)+4*c^2*(1/3*x/d/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2)))+20*I*b*(-d
*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*(-c^2*x^2+1)*c^4+136/3*I*b*(-d*(c^2*x^2-1))^(1
/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+32/3*I*b*(-d*(c^2*x^2-1))^(
1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^7*(-c^2*x^2+1)*c^8+4*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c
^4*x^4+26*c^2*x^2-9)/d^3*x*c^2-4*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*(-c^2*x^
2+1)*c^2+140/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*c^6-64/3*I*b*(-d*(c^2*x^
2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^4*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5-24*I*b*(-d*(c^2*x^2
-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c+3/2*b*(-d*(c^2*x^2-1))^(1/
2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*(-c^2*x^2+1)^(1/2)*c+9*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x
^4+26*c^2*x^2-9)/d^3/x*arcsin(c*x)-4/3*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^2*(-
c^2*x^2+1)^(1/2)*c^3+56*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*arcsin(c*x)*c^4-4
4*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arcsin(c*x)*c^2-5/3*b*(-d*(c^2*x^2-1))^(1
/2)*(-c^2*x^2+1)^(1/2)/d^3/(c^2*x^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*c-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2
+1)^(1/2)/d^3/(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c-64/3*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4
*x^4+26*c^2*x^2-9)/d^3*x^5*arcsin(c*x)*c^6-24*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d
^3*x^3*c^4-112/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^7*c^8+16/3*I*b*(-d*(c^2*
x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/(c^2*x^2-1)*arcsin(c*x)*c-80/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c
^4*x^4+26*c^2*x^2-9)/d^3*x^5*(-c^2*x^2+1)*c^6+32/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2
-9)/d^3*x^9*c^10

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

[Out]

1/3*a*(8*c^2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 4*c^2*x/((-c^2*d*x^2 + d)^(3/2)*d) - 3/((-c^2*d*x^2 + d)^(3/2)*d*x
)) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(c*x
+ 1)*sqrt(-c*x + 1)), x)/sqrt(d)

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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((a+b*arcsin(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asin(c*x))/(x**2*(-d*(c*x - 1)*(c*x + 1))**(5/2)), 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((a+b*arcsin(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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