Integrand size = 20, antiderivative size = 146 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {2 b \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \]
1/3*x*(a+b*arcsin(c*x))/d/(e*x^2+d)^(3/2)+2/3*b*arctan(e^(1/2)*(-c^2*x^2+1 )^(1/2)/c/(e*x^2+d)^(1/2))/d^2/e^(1/2)+2/3*x*(a+b*arcsin(c*x))/d^2/(e*x^2+ d)^(1/2)+1/3*b*c*(-c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\sqrt {d+e x^2} \left (\frac {a x}{3 d \left (d+e x^2\right )^2}+\frac {2 a x}{3 d^2 \left (d+e x^2\right )}\right )-\frac {b c x^2 \sqrt {\frac {d+e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b x \left (3 d+2 e x^2\right ) \arcsin (c x)}{3 d^2 \left (d+e x^2\right )^{3/2}} \]
(b*c*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2]) + Sqrt[d + e*x^2 ]*((a*x)/(3*d*(d + e*x^2)^2) + (2*a*x)/(3*d^2*(d + e*x^2))) - (b*c*x^2*Sqr t[(d + e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/(3*d^2*S qrt[d + e*x^2]) + (b*x*(3*d + 2*e*x^2)*ArcSin[c*x])/(3*d^2*(d + e*x^2)^(3/ 2))
Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5170, 27, 435, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5170 |
| \(\displaystyle -b c \int \frac {x \left (2 e x^2+3 d\right )}{3 d^2 \sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx+\frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {x \left (2 e x^2+3 d\right )}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx}{3 d^2}+\frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {b c \int \frac {2 e x^2+3 d}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx^2}{6 d^2}+\frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {b c \left (2 \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2-\frac {2 d \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}+\frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {b c \left (4 \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}-\frac {2 d \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}+\frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 x (a+b \arcsin (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arcsin (c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \left (-\frac {4 \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}-\frac {2 d \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}\) |
(x*(a + b*ArcSin[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSin[c*x]) )/(3*d^2*Sqrt[d + e*x^2]) - (b*c*((-2*d*Sqrt[1 - c^2*x^2])/((c^2*d + e)*Sq rt[d + e*x^2]) - (4*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2]) ])/(c*Sqrt[e])))/(6*d^2)
3.7.52.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]))))
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[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
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])
\[\int \frac {a +b \arcsin \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (122) = 244\).
Time = 0.31 (sec) , antiderivative size = 683, normalized size of antiderivative = 4.68 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} + 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {e x^{2} + d} \sqrt {-e} + e^{2}\right ) - 2 \, {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x + {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arcsin \left (c x\right ) + {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {e x^{2} + d} \sqrt {e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x + {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arcsin \left (c x\right ) + {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \]
[-1/6*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d*e^2)*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) - 2*(2*(a*c^2*d*e^2 + a*e^3)*x^3 + 3*(a*c^2*d^2 *e + a*d*e^2)*x + (2*(b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^2*d^2*e + b*d*e^2) *x)*arcsin(c*x) + (b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(-c^2*x^2 + 1))*sqrt(e*x ^2 + d))/(c^2*d^5*e + d^4*e^2 + (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e ^2 + d^3*e^3)*x^2), 1/3*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d*e^2)*x^2)*sqrt(e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(-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)) + (2*(a*c^2*d*e^2 + a*e^3)*x^3 + 3*(a*c^2*d^2*e + a*d* e^2)*x + (2*(b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^2*d^2*e + b*d*e^2)*x)*arcsi n(c*x) + (b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(-c^2*x^2 + 1))*sqrt(e*x^2 + d))/ (c^2*d^5*e + d^4*e^2 + (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e^2 + d^3* e^3)*x^2)]
\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate( arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((e^2*x^4 + 2*d*e*x^2 + d^2)*sq rt(e*x^2 + d)), x)
\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]