Integrand size = 33, antiderivative size = 305 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\frac {b \sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} f \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}-\frac {b \sqrt {c} (b e-a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} f^2 \sqrt {a+b x^2} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b e-a f)^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {c f}{d e},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} e f^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:
b*c^(1/2)*(b*x^2+a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2),(- b*c/a/d)^(1/2))/d^(1/2)/f/(1+b*x^2/a)^(1/2)/(-d*x^2+c)^(1/2)-b*c^(1/2)*(-a *f+b*e)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(- b*c/a/d)^(1/2))/d^(1/2)/f^2/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)+c^(1/2)*(-a*f +b*e)^2*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2),- c*f/d/e,(-b*c/a/d)^(1/2))/d^(1/2)/e/f^2/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 4.03 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \left (-b^2 c e f E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-b e (b d e-b c f-2 a d f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+d (b e-a f)^2 \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} d e f^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Input:
Integrate[(a + b*x^2)^(3/2)/(Sqrt[c - d*x^2]*(e + f*x^2)),x]
Output:
((-I)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*(-(b^2*c*e*f*EllipticE[I*Arc Sinh[Sqrt[b/a]*x], -((a*d)/(b*c))]) - b*e*(b*d*e - b*c*f - 2*a*d*f)*Ellipt icF[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))] + d*(b*e - a*f)^2*EllipticPi[( a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))]))/(Sqrt[b/a]*d*e*f^2*S qrt[a + b*x^2]*Sqrt[c - d*x^2])
Time = 0.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.68, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {420, 331, 330, 327, 414}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2}}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {b \sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {d x^2}{c}}}dx}{f \sqrt {c-d x^2}}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{f \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {b \sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} f \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} f \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}-\frac {a^{3/2} \sqrt {c-d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),\frac {a d}{b c}+1\right )}{\sqrt {b} c e f \sqrt {a+b x^2} \sqrt {\frac {a \left (c-d x^2\right )}{c \left (a+b x^2\right )}}}\) |
Input:
Int[(a + b*x^2)^(3/2)/(Sqrt[c - d*x^2]*(e + f*x^2)),x]
Output:
(b*Sqrt[c]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x )/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*f*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2 ]) - (a^(3/2)*(b*e - a*f)*Sqrt[c - d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcT an[(Sqrt[b]*x)/Sqrt[a]], 1 + (a*d)/(b*c)])/(Sqrt[b]*c*e*f*Sqrt[a + b*x^2]* Sqrt[(a*(c - d*x^2))/(c*(a + b*x^2))])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Time = 4.42 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {\left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a b e f -\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b^{2} e^{2}+\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a b e f +\operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a^{2} f^{2}-2 \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a b e f +\operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) b^{2} e^{2}\right ) \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{e \sqrt {\frac {d}{c}}\, f^{2} \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\) | \(300\) |
| elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (\frac {b \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) a}{f \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {b^{2} \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) e}{f^{2} \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {b a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{f \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a^{2}}{e \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {2 \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a b}{f \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) b^{2}}{f^{2} \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\) | \(629\) |
Input:
int((b*x^2+a)^(3/2)/(-d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
(EllipticF(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*b*e*f-EllipticF(x*(1/c*d)^( 1/2),(-b*c/a/d)^(1/2))*b^2*e^2+EllipticE(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2)) *a*b*e*f+EllipticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b/a)^(1/2)/(1/c*d)^(1/2))*a ^2*f^2-2*EllipticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b/a)^(1/2)/(1/c*d)^(1/2))*a *b*e*f+EllipticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b/a)^(1/2)/(1/c*d)^(1/2))*b^2 *e^2)*((b*x^2+a)/a)^(1/2)*((-d*x^2+c)/c)^(1/2)*(-d*x^2+c)^(1/2)*(b*x^2+a)^ (1/2)/e/(1/c*d)^(1/2)/f^2/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)/(-d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fricas" )
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c - d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate((b*x**2+a)**(3/2)/(-d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral((a + b*x**2)**(3/2)/(sqrt(c - d*x**2)*(e + f*x**2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {-d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(-d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxima" )
Output:
integrate((b*x^2 + a)^(3/2)/(sqrt(-d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {-d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(-d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)/(sqrt(-d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {c-d\,x^2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((a + b*x^2)^(3/2)/((c - d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int((a + b*x^2)^(3/2)/((c - d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c-d x^2} \left (e+f x^2\right )} \, dx=\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}d x \right ) b +\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}d x \right ) a \] Input:
int((b*x^2+a)^(3/2)/(-d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
int((sqrt(c - d*x**2)*sqrt(a + b*x**2)*x**2)/(c*e + c*f*x**2 - d*e*x**2 - d*f*x**4),x)*b + int((sqrt(c - d*x**2)*sqrt(a + b*x**2))/(c*e + c*f*x**2 - d*e*x**2 - d*f*x**4),x)*a