\(\int \frac {a+b \arcsin (c x)}{(d+e x^2)^{7/2}} \, dx\) [169]

Optimal result

Integrand size = 20, antiderivative size = 226 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x^2}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {8 b \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{15 d^3 \sqrt {e}} \] Output:

1/15*b*c*(-c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)^(3/2)+2/15*b*c*(3*c^2*d+ 
2*e)*(-c^2*x^2+1)^(1/2)/d^2/(c^2*d+e)^2/(e*x^2+d)^(1/2)+1/5*x*(a+b*arcsin( 
c*x))/d/(e*x^2+d)^(5/2)+4/15*x*(a+b*arcsin(c*x))/d^2/(e*x^2+d)^(3/2)+8/15* 
x*(a+b*arcsin(c*x))/d^3/(e*x^2+d)^(1/2)+8/15*b*arctan(e^(1/2)*(-c^2*x^2+1) 
^(1/2)/c/(e*x^2+d)^(1/2))/d^3/e^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )+\frac {b c d \sqrt {1-c^2 x^2} \left (d+e x^2\right ) \left (e \left (5 d+4 e x^2\right )+c^2 d \left (7 d+6 e x^2\right )\right )}{\left (c^2 d+e\right )^2}-4 b c x^2 \left (d+e x^2\right )^2 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )+b x \left (15 d^2+20 d e x^2+8 e^2 x^4\right ) \arcsin (c x)}{15 d^3 \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[(a + b*ArcSin[c*x])/(d + e*x^2)^(7/2),x]
 

Output:

(a*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4) + (b*c*d*Sqrt[1 - c^2*x^2]*(d + e*x 
^2)*(e*(5*d + 4*e*x^2) + c^2*d*(7*d + 6*e*x^2)))/(c^2*d + e)^2 - 4*b*c*x^2 
*(d + e*x^2)^2*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e* 
x^2)/d)] + b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcSin[c*x])/(15*d^3*(d + 
 e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5170, 27, 7266, 1193, 27, 87, 66, 218}

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.

\(\displaystyle \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5170

\(\displaystyle -b c \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{15 d^3 \sqrt {1-c^2 x^2} \left (e x^2+d\right )^{5/2}}dx+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{5/2}}dx}{15 d^3}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 7266

\(\displaystyle -\frac {b c \int \frac {8 e^2 x^4+20 d e x^2+15 d^2}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{5/2}}dx^2}{30 d^3}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1193

\(\displaystyle -\frac {b c \left (-\frac {2 \int -\frac {3 \left (4 e \left (d c^2+e\right ) x^2+d \left (7 d c^2+6 e\right )\right )}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx^2}{3 \left (c^2 d+e\right )}-\frac {2 d^2 \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \left (\frac {2 \int \frac {4 e \left (d c^2+e\right ) x^2+d \left (7 d c^2+6 e\right )}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx^2}{c^2 d+e}-\frac {2 d^2 \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {b c \left (\frac {2 \left (4 \left (c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2-\frac {2 d \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{c^2 d+e}-\frac {2 d^2 \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {b c \left (\frac {2 \left (8 \left (c^2 d+e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}-\frac {2 d \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{c^2 d+e}-\frac {2 d^2 \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {8 x (a+b \arcsin (c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \arcsin (c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \arcsin (c x))}{5 d \left (d+e x^2\right )^{5/2}}-\frac {b c \left (\frac {2 \left (-\frac {8 \left (c^2 d+e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}-\frac {2 d \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{c^2 d+e}-\frac {2 d^2 \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}\)

Input:

Int[(a + b*ArcSin[c*x])/(d + e*x^2)^(7/2),x]
 

Output:

(x*(a + b*ArcSin[c*x]))/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcSin[c*x]) 
)/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcSin[c*x]))/(15*d^3*Sqrt[d + 
e*x^2]) - (b*c*((-2*d^2*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(d + e*x^2)^(3/2)) 
 + (2*((-2*d*(3*c^2*d + 2*e)*Sqrt[1 - c^2*x^2])/((c^2*d + e)*Sqrt[d + e*x^ 
2]) - (8*(c^2*d + e)*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2] 
)])/(c*Sqrt[e])))/(c^2*d + e)))/(30*d^3)
 

Defintions of rubi rules used

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
 

rule 5170

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(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]] /; Fr 
eeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2 
, 0])
 

rule 7266

Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1)   Subst[Int[SubstFor[x^(m 
+ 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function 
OfQ[x^(m + 1), u, x]
 

Maple [F]

\[\int \frac {a +b \arcsin \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]

Input:

int((a+b*arcsin(c*x))/(e*x^2+d)^(7/2),x)
 

Output:

int((a+b*arcsin(c*x))/(e*x^2+d)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (192) = 384\).

Time = 0.24 (sec) , antiderivative size = 1321, normalized size of antiderivative = 5.85 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arcsin(c*x))/(e*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

[-1/15*(2*(b*c^4*d^5 + 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b* 
e^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 
 3*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*sqrt(-e)*log(8*c^4*e^2 
*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + 
c^3*d - c*e)*sqrt(-c^2*x^2 + 1)*sqrt(e*x^2 + d)*sqrt(-e) + e^2) - (8*(a*c^ 
4*d^2*e^3 + 2*a*c^2*d*e^4 + a*e^5)*x^5 + 20*(a*c^4*d^3*e^2 + 2*a*c^2*d^2*e 
^3 + a*d*e^4)*x^3 + 15*(a*c^4*d^4*e + 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x + (8* 
(b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b*e^5)*x^5 + 20*(b*c^4*d^3*e^2 + 2*b*c^2* 
d^2*e^3 + b*d*e^4)*x^3 + 15*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x) 
*arcsin(c*x) + (7*b*c^3*d^4*e + 5*b*c*d^3*e^2 + 2*(3*b*c^3*d^2*e^3 + 2*b*c 
*d*e^4)*x^4 + (13*b*c^3*d^3*e^2 + 9*b*c*d^2*e^3)*x^2)*sqrt(-c^2*x^2 + 1))* 
sqrt(e*x^2 + d))/(c^4*d^8*e + 2*c^2*d^7*e^2 + d^6*e^3 + (c^4*d^5*e^4 + 2*c 
^2*d^4*e^5 + d^3*e^6)*x^6 + 3*(c^4*d^6*e^3 + 2*c^2*d^5*e^4 + d^4*e^5)*x^4 
+ 3*(c^4*d^7*e^2 + 2*c^2*d^6*e^3 + d^5*e^4)*x^2), 1/15*(4*(b*c^4*d^5 + 2*b 
*c^2*d^4*e + (b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2 + 3*( 
b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 3*(b*c^4*d^4*e + 2*b*c^2* 
d^3*e^2 + b*d^2*e^3)*x^2)*sqrt(e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqr 
t(-c^2*x^2 + 1)*sqrt(e*x^2 + d)*sqrt(e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - 
c*e^2)*x^2)) + (8*(a*c^4*d^2*e^3 + 2*a*c^2*d*e^4 + a*e^5)*x^5 + 20*(a*c^4* 
d^3*e^2 + 2*a*c^2*d^2*e^3 + a*d*e^4)*x^3 + 15*(a*c^4*d^4*e + 2*a*c^2*d^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*asin(c*x))/(e*x**2+d)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))/(e*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

1/15*a*(8*x/(sqrt(e*x^2 + d)*d^3) + 4*x/((e*x^2 + d)^(3/2)*d^2) + 3*x/((e* 
x^2 + d)^(5/2)*d)) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1) 
)/((e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3)*sqrt(e*x^2 + d)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))/(e*x^2+d)^(7/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int((a + b*asin(c*x))/(d + e*x^2)^(7/2),x)
 

Output:

int((a + b*asin(c*x))/(d + e*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {15 \sqrt {e \,x^{2}+d}\, a \,d^{2} e x +20 \sqrt {e \,x^{2}+d}\, a d \,e^{2} x^{3}+8 \sqrt {e \,x^{2}+d}\, a \,e^{3} x^{5}-8 \sqrt {e}\, a \,d^{3}-24 \sqrt {e}\, a \,d^{2} e \,x^{2}-24 \sqrt {e}\, a d \,e^{2} x^{4}-8 \sqrt {e}\, a \,e^{3} x^{6}+15 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{3}+3 \sqrt {e \,x^{2}+d}\, d^{2} e \,x^{2}+3 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{4}+\sqrt {e \,x^{2}+d}\, e^{3} x^{6}}d x \right ) b \,d^{6} e +45 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{3}+3 \sqrt {e \,x^{2}+d}\, d^{2} e \,x^{2}+3 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{4}+\sqrt {e \,x^{2}+d}\, e^{3} x^{6}}d x \right ) b \,d^{5} e^{2} x^{2}+45 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{3}+3 \sqrt {e \,x^{2}+d}\, d^{2} e \,x^{2}+3 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{4}+\sqrt {e \,x^{2}+d}\, e^{3} x^{6}}d x \right ) b \,d^{4} e^{3} x^{4}+15 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{3}+3 \sqrt {e \,x^{2}+d}\, d^{2} e \,x^{2}+3 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{4}+\sqrt {e \,x^{2}+d}\, e^{3} x^{6}}d x \right ) b \,d^{3} e^{4} x^{6}}{15 d^{3} e \left (e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}\right )} \] Input:

int((a+b*asin(c*x))/(e*x^2+d)^(7/2),x)
 

Output:

(15*sqrt(d + e*x**2)*a*d**2*e*x + 20*sqrt(d + e*x**2)*a*d*e**2*x**3 + 8*sq 
rt(d + e*x**2)*a*e**3*x**5 - 8*sqrt(e)*a*d**3 - 24*sqrt(e)*a*d**2*e*x**2 - 
 24*sqrt(e)*a*d*e**2*x**4 - 8*sqrt(e)*a*e**3*x**6 + 15*int(asin(c*x)/(sqrt 
(d + e*x**2)*d**3 + 3*sqrt(d + e*x**2)*d**2*e*x**2 + 3*sqrt(d + e*x**2)*d* 
e**2*x**4 + sqrt(d + e*x**2)*e**3*x**6),x)*b*d**6*e + 45*int(asin(c*x)/(sq 
rt(d + e*x**2)*d**3 + 3*sqrt(d + e*x**2)*d**2*e*x**2 + 3*sqrt(d + e*x**2)* 
d*e**2*x**4 + sqrt(d + e*x**2)*e**3*x**6),x)*b*d**5*e**2*x**2 + 45*int(asi 
n(c*x)/(sqrt(d + e*x**2)*d**3 + 3*sqrt(d + e*x**2)*d**2*e*x**2 + 3*sqrt(d 
+ e*x**2)*d*e**2*x**4 + sqrt(d + e*x**2)*e**3*x**6),x)*b*d**4*e**3*x**4 + 
15*int(asin(c*x)/(sqrt(d + e*x**2)*d**3 + 3*sqrt(d + e*x**2)*d**2*e*x**2 + 
 3*sqrt(d + e*x**2)*d*e**2*x**4 + sqrt(d + e*x**2)*e**3*x**6),x)*b*d**3*e* 
*4*x**6)/(15*d**3*e*(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6))