\(\int \frac {x (a+b \arcsin (c x))}{(d+e x^2)^3} \, dx\) [469]

Optimal result

Integrand size = 19, antiderivative size = 133 \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.06 \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {-\frac {2 a}{e}+\frac {b c x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \arcsin (c x)}{e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{3/2} e \left (c^2 d+e\right )^{3/2}}\right ) \] Input:

Integrate[(x*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5228, 296, 291, 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.

\(\displaystyle \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 5228

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

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[(x*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]
 

Output:

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

Defintions of rubi rules used

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 291

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

rule 296

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d))   Int[ 
(a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N 
eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1 
]) && NeQ[p, -1]
 

rule 5228

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(997\) vs. \(2(115)=230\).

Time = 19.00 (sec) , antiderivative size = 998, normalized size of antiderivative = 7.50

method	result	size
parts	\(-\frac {a}{4 e \left (e \,x^{2}+d \right )^{2}}+\frac {b \left (-\frac {c^{6} \arcsin \left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {c^{6} \left (-\frac {-\frac {e \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{\left (c^{2} d +e \right ) \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}-\frac {\sqrt {-c^{2} d e}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} d e}}{e}}\right )}{\left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}}{4 d \,c^{2} e}-\frac {-\frac {e \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{\left (c^{2} d +e \right ) \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}+\frac {\sqrt {-c^{2} d e}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} d e}}{e}}\right )}{\left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}}{4 d \,c^{2} e}-\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} d e}}{e}}\right )}{4 d \,c^{2} \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}+\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} d e}}{e}}\right )}{4 d \,c^{2} \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}\right )}{4 e}\right )}{c^{2}}\)	\(998\)
derivativedivides	\(\text {Expression too large to display}\)	\(1006\)
default	\(\text {Expression too large to display}\)	\(1006\)

Input:

int(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (115) = 230\).

Time = 0.27 (sec) , antiderivative size = 783, normalized size of antiderivative = 5.89 \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

[-1/32*(8*a*c^4*d^4 + 16*a*c^2*d^3*e + 8*a*d^2*e^2 + (2*b*c^3*d^3 + b*c*d^ 
2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + b*c*d*e^2)*x^2)*s 
qrt(-c^2*d^2 - d*e)*log(((8*c^4*d^2 + 8*c^2*d*e + e^2)*x^4 - 2*(4*c^2*d^2 
+ 3*d*e)*x^2 - 4*sqrt(-c^2*d^2 - d*e)*sqrt(-c^2*x^2 + 1)*((2*c^2*d + e)*x^ 
3 - d*x) + d^2)/(e^2*x^4 + 2*d*e*x^2 + d^2)) + 8*(b*c^4*d^4 + 2*b*c^2*d^3* 
e + b*d^2*e^2)*arcsin(c*x) - 4*sqrt(-c^2*x^2 + 1)*((b*c^3*d^2*e^2 + b*c*d* 
e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x))/(c^4*d^6*e + 2*c^2*d^5*e^2 + d^ 
4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c 
^2*d^4*e^3 + d^3*e^4)*x^2), -1/16*(4*a*c^4*d^4 + 8*a*c^2*d^3*e + 4*a*d^2*e 
^2 + (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3 
*d^2*e + b*c*d*e^2)*x^2)*sqrt(c^2*d^2 + d*e)*arctan(1/2*sqrt(c^2*d^2 + d*e 
)*sqrt(-c^2*x^2 + 1)*((2*c^2*d + e)*x^2 - d)/((c^4*d^2 + c^2*d*e)*x^3 - (c 
^2*d^2 + d*e)*x)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arcsin(c*x) 
- 2*sqrt(-c^2*x^2 + 1)*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b 
*c*d^2*e^2)*x))/(c^4*d^6*e + 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^ 
2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)*x^2)]
 

Sympy [F]

\[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \] Input:

integrate(x*(a+b*asin(c*x))/(e*x**2+d)**3,x)
 

Output:

Integral(x*(a + b*asin(c*x))/(d + e*x**2)**3, x)
 

Maxima [F]

\[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,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. \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((x*(a + b*asin(c*x)))/(d + e*x^2)^3,x)
 

Output:

int((x*(a + b*asin(c*x)))/(d + e*x^2)^3, x)
 

Reduce [F]

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

int(x*(a+b*asin(c*x))/(e*x^2+d)^3,x)
 

Output:

(4*int((asin(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x) 
*b*d**2*e + 8*int((asin(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e* 
*3*x**6),x)*b*d*e**2*x**2 + 4*int((asin(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3* 
d*e**2*x**4 + e**3*x**6),x)*b*e**3*x**4 - a)/(4*e*(d**2 + 2*d*e*x**2 + e** 
2*x**4))