Integrand size = 32, antiderivative size = 342 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=-\frac {d^{3/2} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \sqrt {d} (b d e-2 b c f+a d f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a (b c-a d) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-d^(3/2)*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1- b*c/a/d)^(1/2))/c^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(c*(b*x^2+a)/a/(d*x^2+c))^(1 /2)/(d*x^2+c)^(1/2)+c^(1/2)*d^(1/2)*(a*d*f-2*b*c*f+b*d*e)*(b*x^2+a)^(1/2)* InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/(-a*d+b*c)/ (-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+c^(3/2)*f^2*( b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1 -b*c/a/d)^(1/2))/a/d^(1/2)/e/(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/ (d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 4.58 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {\frac {b}{a}} d^2 e x \left (a+b x^2\right )-i b c d e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c (-b c+a d) e (-d e+c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
(-(Sqrt[b/a]*d^2*e*x*(a + b*x^2)) - I*b*c*d*e*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I* ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*c*(-(b*c) + a*d)*e*(-(d*e) + c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.45 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {421, 25, 400, 313, 320, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\) |
\(\Big \downarrow \) 421 |
\(\displaystyle \frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\) |
\(\Big \downarrow \) 400 |
\(\displaystyle \frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d (d e-c f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}\right )}{(d e-c f)^2}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\) |
\(\Big \downarrow \) 414 |
\(\displaystyle \frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\) |
Input:
Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
(d*(-((Sqrt[d]*(d*e - c*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq rt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(b*d*e - 2*b*c*f + a*d*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d ]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(d* e - c*f)^2 + (c^(3/2)*f^2*Sqrt[a + b*x^2]*EllipticPi[1 - (c*f)/(d*e), ArcT an[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*e*(d*e - c*f)^2*Sqrt [(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
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[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[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
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[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Time = 8.66 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\left (-\sqrt {-\frac {b}{a}}\, b \,d^{2} e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a c d f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b \,c^{2} f -\sqrt {-\frac {b}{a}}\, a \,d^{2} e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{c \left (a d -b c \right ) e \sqrt {-\frac {b}{a}}\, \left (c f -d e \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(304\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) d x}{c \left (a d -b c \right ) \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) d}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, c \left (c f -d e \right )}+\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) a \,d^{2}}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, c \left (a d -b c \right ) \left (c f -d e \right )}-\frac {d b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\left (a d -b c \right ) \left (c f -d e \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {d b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\left (a d -b c \right ) \left (c f -d e \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{\left (c f -d e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(636\) |
Input:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
(-(-b/a)^(1/2)*b*d^2*e*x^3+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti cE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*d*e+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/ c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a* c*d*f-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a* f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*c^2*f-(-b/a)^(1/2)*a*d^2*e*x)*(d*x^2+ c)^(1/2)*(b*x^2+a)^(1/2)/c/(a*d-b*c)/e/(-b/a)^(1/2)/(c*f-d*e)/(b*d*x^4+a*d *x^2+b*c*x^2+a*c)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)**(3/2)/(f*x**2+e),x)
Output:
Integral(1/(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(e + f*x**2)), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="maxima ")
Output:
integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)),x)
Output:
int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{2} f \,x^{8}+a \,d^{2} f \,x^{6}+2 b c d f \,x^{6}+b \,d^{2} e \,x^{6}+2 a c d f \,x^{4}+a \,d^{2} e \,x^{4}+b \,c^{2} f \,x^{4}+2 b c d e \,x^{4}+a \,c^{2} f \,x^{2}+2 a c d e \,x^{2}+b \,c^{2} e \,x^{2}+a \,c^{2} e}d x \] Input:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2*e + a*c**2*f*x**2 + 2*a*c* d*e*x**2 + 2*a*c*d*f*x**4 + a*d**2*e*x**4 + a*d**2*f*x**6 + b*c**2*e*x**2 + b*c**2*f*x**4 + 2*b*c*d*e*x**4 + 2*b*c*d*f*x**6 + b*d**2*e*x**6 + b*d**2 *f*x**8),x)