\(\int \frac {(d+e x^2)^2 (a+b \arcsin (c x))}{x^4} \, dx\) [440]

Optimal result

Integrand size = 21, antiderivative size = 126 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=\frac {b e^2 \sqrt {1-c^2 x^2}}{c}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))-\frac {1}{6} b c d \left (c^2 d+12 e\right ) \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \] Output:

b*e^2*(-c^2*x^2+1)^(1/2)/c-1/6*b*c*d^2*(-c^2*x^2+1)^(1/2)/x^2-1/3*d^2*(a+b 
*arcsin(c*x))/x^3-2*d*e*(a+b*arcsin(c*x))/x+e^2*x*(a+b*arcsin(c*x))-1/6*b* 
c*d*(c^2*d+12*e)*arctanh((-c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a d^2}{x^3}-\frac {12 a d e}{x}+6 a e^2 x+6 b \left (\frac {e^2}{c}-\frac {c d^2}{6 x^2}\right ) \sqrt {1-c^2 x^2}-\frac {2 b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \arcsin (c x)}{x^3}+b c d \left (c^2 d+12 e\right ) \log (x)-b c d \left (c^2 d+12 e\right ) \log \left (1+\sqrt {1-c^2 x^2}\right )\right ) \] Input:

Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^4,x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5230, 27, 1578, 1192, 1471, 25, 299, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx\)

\(\Big \downarrow \) 5230

\(\displaystyle -b c \int -\frac {-3 e^2 x^4+6 d e x^2+d^2}{3 x^3 \sqrt {1-c^2 x^2}}dx-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} b c \int \frac {-3 e^2 x^4+6 d e x^2+d^2}{x^3 \sqrt {1-c^2 x^2}}dx-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {1}{6} b c \int \frac {-3 e^2 x^4+6 d e x^2+d^2}{x^4 \sqrt {1-c^2 x^2}}dx^2-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 1192

\(\displaystyle -\frac {b \int \frac {-3 e^2 x^8-6 \left (c^2 d-e\right ) e x^4+c^4 d^2-3 e^2+6 c^2 d e}{\left (1-x^4\right )^2}d\sqrt {1-c^2 x^2}}{3 c}-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 1471

\(\displaystyle -\frac {b \left (\frac {c^4 d^2 \sqrt {1-c^2 x^2}}{2 \left (1-x^4\right )}-\frac {1}{2} \int -\frac {d^2 c^4+12 d e c^2+6 e^2 x^4-6 e^2}{1-x^4}d\sqrt {1-c^2 x^2}\right )}{3 c}-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \left (\frac {1}{2} \int \frac {d^2 c^4+12 d e c^2+6 e^2 x^4-6 e^2}{1-x^4}d\sqrt {1-c^2 x^2}+\frac {c^4 d^2 \sqrt {1-c^2 x^2}}{2 \left (1-x^4\right )}\right )}{3 c}-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 299

\(\displaystyle -\frac {b \left (\frac {1}{2} \left (c^2 d \left (c^2 d+12 e\right ) \int \frac {1}{1-x^4}d\sqrt {1-c^2 x^2}-6 e^2 \sqrt {1-c^2 x^2}\right )+\frac {c^4 d^2 \sqrt {1-c^2 x^2}}{2 \left (1-x^4\right )}\right )}{3 c}-\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {d^2 (a+b \arcsin (c x))}{3 x^3}-\frac {2 d e (a+b \arcsin (c x))}{x}+e^2 x (a+b \arcsin (c x))-\frac {b \left (\frac {1}{2} \left (c^2 d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \left (c^2 d+12 e\right )-6 e^2 \sqrt {1-c^2 x^2}\right )+\frac {c^4 d^2 \sqrt {1-c^2 x^2}}{2 \left (1-x^4\right )}\right )}{3 c}\)

Input:

Int[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^4,x]
 

Output:

-1/3*(d^2*(a + b*ArcSin[c*x]))/x^3 - (2*d*e*(a + b*ArcSin[c*x]))/x + e^2*x 
*(a + b*ArcSin[c*x]) - (b*((c^4*d^2*Sqrt[1 - c^2*x^2])/(2*(1 - x^4)) + (-6 
*e^2*Sqrt[1 - c^2*x^2] + c^2*d*(c^2*d + 12*e)*ArcTanh[Sqrt[1 - c^2*x^2]])/ 
2))/(3*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1192

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

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[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] && Int 
egerQ[(m - 1)/2]
 

rule 5230

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]}, Simp 
[(a + b*ArcSin[c*x])   u, x] - Simp[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]))
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(c^{3} \left (\frac {a \left (e^{2} c x -\frac {2 c d e}{x}-\frac {c \,d^{2}}{3 x^{3}}\right )}{c^{4}}+\frac {b \left (\arcsin \left (c x \right ) e^{2} c x -\frac {2 \arcsin \left (c x \right ) c d e}{x}-\frac {\arcsin \left (c x \right ) c \,d^{2}}{3 x^{3}}+\frac {c^{4} d^{2} \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2}\right )}{3}+e^{2} \sqrt {-c^{2} x^{2}+1}-2 c^{2} d e \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{4}}\right )\)	\(156\)
default	\(c^{3} \left (\frac {a \left (e^{2} c x -\frac {2 c d e}{x}-\frac {c \,d^{2}}{3 x^{3}}\right )}{c^{4}}+\frac {b \left (\arcsin \left (c x \right ) e^{2} c x -\frac {2 \arcsin \left (c x \right ) c d e}{x}-\frac {\arcsin \left (c x \right ) c \,d^{2}}{3 x^{3}}+\frac {c^{4} d^{2} \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2}\right )}{3}+e^{2} \sqrt {-c^{2} x^{2}+1}-2 c^{2} d e \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{4}}\right )\)	\(156\)
parts	\(a \left (e^{2} x -\frac {d^{2}}{3 x^{3}}-\frac {2 d e}{x}\right )+b \,c^{3} \left (\frac {\arcsin \left (c x \right ) x \,e^{2}}{c^{3}}-\frac {\arcsin \left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {2 \arcsin \left (c x \right ) d e}{c^{3} x}-\frac {-3 e^{2} \sqrt {-c^{2} x^{2}+1}-c^{4} d^{2} \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2}\right )+6 c^{2} d e \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{3 c^{4}}\right )\)	\(159\)

Input:

int((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x,method=_RETURNVERBOSE)
 

Output:

c^3*(a/c^4*(e^2*c*x-2*c*d*e/x-1/3*c*d^2/x^3)+b/c^4*(arcsin(c*x)*e^2*c*x-2* 
arcsin(c*x)*c*d*e/x-1/3*arcsin(c*x)*c*d^2/x^3+1/3*c^4*d^2*(-1/2/c^2/x^2*(- 
c^2*x^2+1)^(1/2)-1/2*arctanh(1/(-c^2*x^2+1)^(1/2)))+e^2*(-c^2*x^2+1)^(1/2) 
-2*c^2*d*e*arctanh(1/(-c^2*x^2+1)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.38 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=\frac {12 \, a c e^{2} x^{4} - 24 \, a c d e x^{2} - {\left (b c^{4} d^{2} + 12 \, b c^{2} d e\right )} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + {\left (b c^{4} d^{2} + 12 \, b c^{2} d e\right )} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) - 4 \, a c d^{2} + 4 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2}\right )} \arcsin \left (c x\right ) - 2 \, {\left (b c^{2} d^{2} x - 6 \, b e^{2} x^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, c x^{3}} \] Input:

integrate((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x, algorithm="fricas")
 

Output:

1/12*(12*a*c*e^2*x^4 - 24*a*c*d*e*x^2 - (b*c^4*d^2 + 12*b*c^2*d*e)*x^3*log 
(sqrt(-c^2*x^2 + 1) + 1) + (b*c^4*d^2 + 12*b*c^2*d*e)*x^3*log(sqrt(-c^2*x^ 
2 + 1) - 1) - 4*a*c*d^2 + 4*(3*b*c*e^2*x^4 - 6*b*c*d*e*x^2 - b*c*d^2)*arcs 
in(c*x) - 2*(b*c^2*d^2*x - 6*b*e^2*x^3)*sqrt(-c^2*x^2 + 1))/(c*x^3)
 

Sympy [A] (verification not implemented)

Time = 2.99 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.73 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=- \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x + \frac {b c d^{2} \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {c}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 c x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} + 2 b c d e \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{2} \operatorname {asin}{\left (c x \right )}}{3 x^{3}} - \frac {2 b d e \operatorname {asin}{\left (c x \right )}}{x} + b e^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((e*x**2+d)**2*(a+b*asin(c*x))/x**4,x)
 

Output:

-a*d**2/(3*x**3) - 2*a*d*e/x + a*e**2*x + b*c*d**2*Piecewise((-c**2*acosh( 
1/(c*x))/2 + c/(2*x*sqrt(-1 + 1/(c**2*x**2))) - 1/(2*c*x**3*sqrt(-1 + 1/(c 
**2*x**2))), 1/Abs(c**2*x**2) > 1), (I*c**2*asin(1/(c*x))/2 - I*c*sqrt(1 - 
 1/(c**2*x**2))/(2*x), True))/3 + 2*b*c*d*e*Piecewise((-acosh(1/(c*x)), 1/ 
Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b*d**2*asin(c*x)/(3*x**3) 
- 2*b*d*e*asin(c*x)/x + b*e**2*Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqr 
t(-c**2*x**2 + 1)/c, True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=-\frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d^{2} - 2 \, {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b e^{2}}{c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \] Input:

integrate((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x, algorithm="maxima")
 

Output:

-1/6*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1 
)/x^2)*c + 2*arcsin(c*x)/x^3)*b*d^2 - 2*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) 
 + 2/abs(x)) + arcsin(c*x)/x)*b*d*e + a*e^2*x + (c*x*arcsin(c*x) + sqrt(-c 
^2*x^2 + 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2534 vs. \(2 (114) = 228\).

Time = 1.34 (sec) , antiderivative size = 2534, normalized size of antiderivative = 20.11 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x, algorithm="giac")
 

Output:

-1/24*b*c^12*d^2*x^8*arcsin(c*x)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^ 
4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) - 1/24*a*c^1 
2*d^2*x^8/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 
1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) + 1/24*b*c^11*d^2*x^7/((c^6*x^5/(sq 
rt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x 
^2 + 1) + 1)^7) - 1/6*b*c^10*d^2*x^6*arcsin(c*x)/((c^6*x^5/(sqrt(-c^2*x^2 
+ 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1) 
^6) - 1/6*a*c^10*d^2*x^6/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(s 
qrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) + 1/6*b*c^9*d^2*x^5* 
log(abs(c)*abs(x))/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c 
^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 1/6*b*c^9*d^2*x^5*log(sq 
rt(-c^2*x^2 + 1) + 1)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt 
(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) + 1/24*b*c^9*d^2*x^5/(( 
c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*( 
sqrt(-c^2*x^2 + 1) + 1)^5) - b*c^8*d*e*x^6*arcsin(c*x)/((c^6*x^5/(sqrt(-c^ 
2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1 
) + 1)^6) - 1/4*b*c^8*d^2*x^4*arcsin(c*x)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 
1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) - a 
*c^8*d*e*x^6/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 
 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) - 1/4*a*c^8*d^2*x^4/((c^6*x^5...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x^4} \,d x \] Input:

int(((a + b*asin(c*x))*(d + e*x^2)^2)/x^4,x)
 

Output:

int(((a + b*asin(c*x))*(d + e*x^2)^2)/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arcsin (c x))}{x^4} \, dx=\frac {-2 \mathit {asin} \left (c x \right ) b c \,d^{2}-12 \mathit {asin} \left (c x \right ) b c d e \,x^{2}+6 \mathit {asin} \left (c x \right ) b c \,e^{2} x^{4}-\sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2} x +6 \sqrt {-c^{2} x^{2}+1}\, b \,e^{2} x^{3}+\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) b \,c^{4} d^{2} x^{3}+12 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) b \,c^{2} d e \,x^{3}-2 a c \,d^{2}-12 a c d e \,x^{2}+6 a c \,e^{2} x^{4}}{6 c \,x^{3}} \] Input:

int((e*x^2+d)^2*(a+b*asin(c*x))/x^4,x)
 

Output:

( - 2*asin(c*x)*b*c*d**2 - 12*asin(c*x)*b*c*d*e*x**2 + 6*asin(c*x)*b*c*e** 
2*x**4 - sqrt( - c**2*x**2 + 1)*b*c**2*d**2*x + 6*sqrt( - c**2*x**2 + 1)*b 
*e**2*x**3 + log(tan(asin(c*x)/2))*b*c**4*d**2*x**3 + 12*log(tan(asin(c*x) 
/2))*b*c**2*d*e*x**3 - 2*a*c*d**2 - 12*a*c*d*e*x**2 + 6*a*c*e**2*x**4)/(6* 
c*x**3)