Integrand size = 19, antiderivative size = 133 \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2}} \]
1/4*(-a-b*arcsin(c*x))/e/(e*x^2+d)^2+1/8*b*c*(2*c^2*d+e)*arctan(x*(c^2*d+e )^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(3/2)/e/(c^2*d+e)^(3/2)+1/8*b*c*x*(- c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)
Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.06 \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {-\frac {2 a}{e}+\frac {b c x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \arcsin (c x)}{e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{3/2} e \left (c^2 d+e\right )^{3/2}}\right ) \]
(((-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
Time = 0.29 (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 |
| \(\displaystyle \frac {b c \left (\frac {\left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {e x \sqrt {1-c^2 x^2}}{2 d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 e}-\frac {a+b \arcsin (c x)}{4 e \left (d+e x^2\right )^2}\) |
-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)
3.7.43.3.1 Defintions of rubi rules used
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(997\) vs. \(2(118)=236\).
Time = 0.21 (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} e d}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{\left (c^{2} d +e \right ) \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}-\frac {\sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{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} e d}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{\left (c^{2} d +e \right ) \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}+\frac {\sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{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} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{4 d \,c^{2} \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}+\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{4 d \,c^{2} \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}\right )}{4 e}\right )}{c^{2}}\) | \(998\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1006\) |
| default | \(\text {Expression too large to display}\) | \(1006\) |
-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*e*d)^(1/2)/e)*(-(c*x-(-c^2 *e*d)^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e )^(1/2)-(-c^2*e*d)^(1/2)/(c^2*d+e)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e-2 *(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x- (-c^2*e*d)^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d +e)/e)^(1/2))/(c*x-(-c^2*e*d)^(1/2)/e)))-1/4/d/c^2/e*(-1/(c^2*d+e)*e/(c*x+ (-c^2*e*d)^(1/2)/e)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x +(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2)+(-c^2*e*d)^(1/2)/(c^2*d+e)/((c^2*d +e)/e)^(1/2)*ln((2*(c^2*d+e)/e+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/ e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e *(c*x+(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*e*d)^(1/2)/e)))-1 /4/d/c^2/(-c^2*e*d)^(1/2)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e-2*(-c^2*e* d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x-(-c^2*e*d )^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1 /2))/(c*x-(-c^2*e*d)^(1/2)/e))+1/4/d/c^2/(-c^2*e*d)^(1/2)/((c^2*d+e)/e)^(1 /2)*ln((2*(c^2*d+e)/e+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/e)+2*((c^ 2*d+e)/e)^(1/2)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c ^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*e*d)^(1/2)/e))))
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (115) = 230\).
Time = 0.37 (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=\left [-\frac {8 \, a c^{4} d^{4} + 16 \, a c^{2} d^{3} e + 8 \, a d^{2} e^{2} + {\left (2 \, b c^{3} d^{3} + b c d^{2} e + {\left (2 \, b c^{3} d e^{2} + b c e^{3}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} e + b c d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d^{2} - d e} \log \left (\frac {{\left (8 \, c^{4} d^{2} + 8 \, c^{2} d e + e^{2}\right )} x^{4} - 2 \, {\left (4 \, c^{2} d^{2} + 3 \, d e\right )} x^{2} - 4 \, \sqrt {-c^{2} d^{2} - d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{3} - d x\right )} + d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \arcsin \left (c x\right ) - 4 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{3} + {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x\right )}}{32 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + {\left (2 \, b c^{3} d^{3} + b c d^{2} e + {\left (2 \, b c^{3} d e^{2} + b c e^{3}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} e + b c d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d^{2} + d e} \arctan \left (\frac {\sqrt {c^{2} d^{2} + d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{2} - d\right )}}{2 \, {\left ({\left (c^{4} d^{2} + c^{2} d e\right )} x^{3} - {\left (c^{2} d^{2} + d e\right )} x\right )}}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \arcsin \left (c x\right ) - 2 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{3} + {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x\right )}}{16 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}\right ] \]
[-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)]
\[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \]
\[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
-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)
\[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]