Integrand size = 17, antiderivative size = 196 \[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=a x \sqrt {a+\frac {b}{c+d x^2}}+\frac {(b-a c) \sqrt {a+\frac {b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {a \sqrt {c} \sqrt {a+\frac {b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \] Output:
a*x*(a+b/(d*x^2+c))^(1/2)+(-a*c+b)*(a+b/(d*x^2+c))^(1/2)*EllipticE(d^(1/2) *x/c^(1/2)/(1+d*x^2/c)^(1/2),(b/(a*c+b))^(1/2))/c^(1/2)/d^(1/2)/(c*(a*d*x^ 2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+a*c^(1/2)*(a+b/(d*x^2+c))^(1/2)*InverseJ acobiAM(arctan(d^(1/2)*x/c^(1/2)),(b/(a*c+b))^(1/2))/d^(1/2)/(c*(a*d*x^2+a *c+b)/(a*c+b)/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 10.65 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.17 \[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {d}{c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (b \sqrt {\frac {d}{c}} x \left (b+a \left (c+d x^2\right )\right )+i \left (b^2-a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{d \left (b+a \left (c+d x^2\right )\right )} \] Input:
Integrate[(a + b/(c + d*x^2))^(3/2),x]
Output:
(Sqrt[d/c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b*Sqrt[d/c]*x*(b + a*(c + d*x^2)) + I*(b^2 - a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] - I*b*(b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*A rcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(d*(b + a*(c + d*x^2)))
Time = 0.79 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.66, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2057, 2058, 315, 27, 406, 320, 388, 313}
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.
\(\displaystyle \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \int \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}dx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \int \frac {\left (a d x^2+b+a c\right )^{3/2}}{\left (d x^2+c\right )^{3/2}}dx}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\int \frac {a d \left (c (b+a c)-(b-a c) d x^2\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c d}+\frac {b x \sqrt {a c+a d x^2+b}}{c \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {a \int \frac {c (b+a c)-(b-a c) d x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c}+\frac {b x \sqrt {a c+a d x^2+b}}{c \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 406 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {a \left (c (a c+b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx-d (b-a c) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )}{c}+\frac {b x \sqrt {a c+a d x^2+b}}{c \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {a \left (\frac {c^{3/2} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-d (b-a c) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )}{c}+\frac {b x \sqrt {a c+a d x^2+b}}{c \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {a \left (\frac {c^{3/2} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-d (b-a c) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )\right )}{c}+\frac {b x \sqrt {a c+a d x^2+b}}{c \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {a \left (\frac {c^{3/2} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-d (b-a c) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )\right )}{c}+\frac {b x \sqrt {a c+a d x^2+b}}{c \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
Input:
Int[(a + b/(c + d*x^2))^(3/2),x]
Output:
(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*((b*x*Sqrt[b + a*c + a*d*x^2])/(c*Sqrt[c + d*x^2]) + (a*(-((b - a*c)*d*((x*Sqrt[b + a*c + a*d *x^2])/(a*d*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticE[ ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d^(3/2)*Sqrt[c + d*x^2]*Sqrt [(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]))) + (c^(3/2)*Sqrt[b + a *c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(Sqrt[d ]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]))) /c))/Sqrt[b + a*c + a*d*x^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(514\) vs. \(2(185)=370\).
Time = 5.19 (sec) , antiderivative size = 515, normalized size of antiderivative = 2.63
method | result | size |
default | \(\frac {\left (\sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{2}+\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, a b d \,x^{3}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c -\sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c +\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, a b c x +\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, b^{2} x \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, c \left (a d \,x^{2}+a c +b \right )}\) | \(515\) |
Input:
int((a+b/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
(((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2 +c)/c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a^2*c^2 +(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b* d*x^3+2*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)* ((d*x^2+c)/c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))* a*b*c-((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*(( d*x^2+c)/c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a* b*c+(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(-a*d/(a*c+b))^(1/2)*a *b*c*x+(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(-a*d/(a*c+b))^(1/2 )*b^2*x)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+ a*c^2+b*c)^(1/2)/(-a*d/(a*c+b))^(1/2)/c/(a*d*x^2+a*c+b)
Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82 \[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {{\left (a c^{2} - b c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c^{2} - b c + {\left (a c + b\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c d x^{2} + a c^{2} - b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c d x} \] Input:
integrate((a+b/(d*x^2+c))^(3/2),x, algorithm="fricas")
Output:
-((a*c^2 - b*c)*sqrt(a)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (a*c^2 - b*c + (a*c + b)*d)*sqrt(a)*x*sqrt(-c/d)*elliptic_f (arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (a*c*d*x^2 + a*c^2 - b*c)*sqrt(( a*d*x^2 + a*c + b)/(d*x^2 + c)))/(c*d*x)
\[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int \left (a + \frac {b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a+b/(d*x**2+c))**(3/2),x)
Output:
Integral((a + b/(c + d*x**2))**(3/2), x)
\[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^(3/2),x, algorithm="maxima")
Output:
integrate((a + b/(d*x^2 + c))^(3/2), x)
\[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^(3/2),x, algorithm="giac")
Output:
integrate((a + b/(d*x^2 + c))^(3/2), x)
Timed out. \[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int {\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \] Input:
int((a + b/(c + d*x^2))^(3/2),x)
Output:
int((a + b/(c + d*x^2))^(3/2), x)
\[ \int \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a c x +\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b x -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, x^{4}}{a \,d^{3} x^{6}+3 a c \,d^{2} x^{4}+b \,d^{2} x^{4}+3 a \,c^{2} d \,x^{2}+2 b c d \,x^{2}+a \,c^{3}+b \,c^{2}}d x \right ) a b c \,d^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, x^{4}}{a \,d^{3} x^{6}+3 a c \,d^{2} x^{4}+b \,d^{2} x^{4}+3 a \,c^{2} d \,x^{2}+2 b c d \,x^{2}+a \,c^{3}+b \,c^{2}}d x \right ) a b \,d^{3} x^{2}}{c \left (d \,x^{2}+c \right )} \] Input:
int((a+b/(d*x^2+c))^(3/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a*c*x + sqrt(c + d*x**2)*sqrt(a *c + a*d*x**2 + b)*b*x - int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*x* *4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**2 + 2 *b*c*d*x**2 + b*d**2*x**4),x)*a*b*c*d**2 - int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3 *x**6 + b*c**2 + 2*b*c*d*x**2 + b*d**2*x**4),x)*a*b*d**3*x**2)/(c*(c + d*x **2))