\(\int \frac {(c+d x^2)^{3/2} (e+f x^2)^{3/2}}{a+b x^2} \, dx\) [71]

Optimal result

Integrand size = 32, antiderivative size = 659 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {(b c-a d)^2 f^2 x \sqrt {c+d x^2}}{b^3 d \sqrt {e+f x^2}}+\frac {2 (b c-a d) f (2 d e-c f) x \sqrt {c+d x^2}}{3 b^2 d \sqrt {e+f x^2}}+\frac {\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b d \sqrt {e+f x^2}}+\frac {(b c-a d) f x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}+\frac {2 (3 d e-c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b}+\frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}-\frac {\sqrt {e} \left (15 a^2 d^2 f^2-20 a b d f (d e+c f)+3 b^2 \left (d^2 e^2+9 c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} \left (15 a^2 d^2 f+3 b^2 c (8 d e+3 c f)-5 a b d (3 d e+5 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^2 e^{3/2} (b e-a f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

(-a*d+b*c)^2*f^2*x*(d*x^2+c)^(1/2)/b^3/d/(f*x^2+e)^(1/2)+2/3*(-a*d+b*c)*f*(-c*f+2*d*e)*x*(d*x^2+c)^(1/2)/b^2/d
/(f*x^2+e)^(1/2)+1/15*(-2*c^2*f^2+7*c*d*e*f+3*d^2*e^2)*x*(d*x^2+c)^(1/2)/b/d/(f*x^2+e)^(1/2)+1/15*e^(3/2)*(15*
a^2*d^2*f+3*b^2*c*(3*c*f+8*d*e)-5*a*b*d*(5*c*f+3*d*e))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticF(x*f^(
1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/b^3/c/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2
)/(f*x^2+e)^(1/2)+(-a*d+b*c)^2*e^(3/2)*(-a*f+b*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticPi(x*f^(1/2)
/e^(1/2)/(1+f*x^2/e)^(1/2),1-b*e/a/f,(1-d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/a/b^3/c/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+
e))^(1/2)/(f*x^2+e)^(1/2)-1/15*(15*a^2*d^2*f^2-20*a*b*d*f*(c*f+d*e)+3*b^2*(c^2*f^2+9*c*d*e*f+d^2*e^2))*(1/(1+f
*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*(d*x
^2+c)^(1/2)/b^3/d/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/5*f*x*(d*x^2+c)^(3/2)*(f*x^2+e)^(1
/2)/b+1/3*(-a*d+b*c)*f*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^2+2/15*(-c*f+3*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/
2)/b

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {559, 427, 542, 545, 429, 506, 422, 557, 553} \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {c^{3/2} \sqrt {e+f x^2} (b c-a d) (b e-a f)^2 \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{a b^3 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (b c-a d) (5 b e-3 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 d \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {f x \sqrt {c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e)}{3 b^3 d \sqrt {e+f x^2}}+\frac {f x \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)}{3 b^2}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right ) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {e^{3/2} \sqrt {c+d x^2} (9 d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 b \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )}{15 b d \sqrt {e+f x^2}}+\frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}+\frac {2 x \sqrt {c+d x^2} \sqrt {e+f x^2} (3 d e-c f)}{15 b} \]

[In]

Int[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2),x]

[Out]

((b*c - a*d)*f*(4*b*d*e + b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*d*Sqrt[e + f*x^2]) + ((3*d^2*e^2 + 7*c*d*
e*f - 2*c^2*f^2)*x*Sqrt[c + d*x^2])/(15*b*d*Sqrt[e + f*x^2]) + ((b*c - a*d)*f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2
])/(3*b^2) + (2*(3*d*e - c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b) + (f*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^
2])/(5*b) - ((b*c - a*d)*Sqrt[e]*Sqrt[f]*(4*b*d*e + b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]
*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*d*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(3*
d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*
b*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*Sqrt[e]*Sqrt[f]*(5*b*e - 3*a
*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(9*d*e - c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1
- (d*e)/(c*f)])/(15*b*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)*(
b*e - a*f)^2*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^3
*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[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]

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 506

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] - Dist[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]

Rule 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 553

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)), Ar
cTan[Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 557

Int[(((c_) + (d_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[(b*c - a*
d)^2/b^2, Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] + Dist[d/b^2, Int[(2*b*c - a*d + b*d*x^2)*
(Sqrt[e + f*x^2]/Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]

Rule 559

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/b, Int[
(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {d \int \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2} \, dx}{b}+\frac {(b c-a d) \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx}{b} \\ & = \frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}+\frac {\int \frac {\sqrt {c+d x^2} \left (e (5 d e-c f)+2 f (3 d e-c f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{5 b}+\frac {((b c-a d) f) \int \frac {\sqrt {c+d x^2} \left (2 b e-a f+b f x^2\right )}{\sqrt {e+f x^2}} \, dx}{b^3}+\frac {\left ((b c-a d) (b e-a f)^2\right ) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx}{b^3} \\ & = \frac {(b c-a d) f x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}+\frac {2 (3 d e-c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b}+\frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}+\frac {c^{3/2} (b c-a d) (b e-a f)^2 \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a b^3 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {(b c-a d) \int \frac {c f (5 b e-3 a f)+f (4 b d e+b c f-3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}+\frac {\int \frac {c e f (9 d e-c f)+f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f} \\ & = \frac {(b c-a d) f x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}+\frac {2 (3 d e-c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b}+\frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}+\frac {c^{3/2} (b c-a d) (b e-a f)^2 \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a b^3 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {(c (b c-a d) f (5 b e-3 a f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}+\frac {(c e (9 d e-c f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b}+\frac {((b c-a d) f (4 b d e+b c f-3 a d f)) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}+\frac {\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b} \\ & = \frac {(b c-a d) f (4 b d e+b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 d \sqrt {e+f x^2}}+\frac {\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b d \sqrt {e+f x^2}}+\frac {(b c-a d) f x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}+\frac {2 (3 d e-c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b}+\frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}+\frac {(b c-a d) \sqrt {e} \sqrt {f} (5 b e-3 a f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} (9 d e-c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {c^{3/2} (b c-a d) (b e-a f)^2 \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a b^3 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {((b c-a d) e f (4 b d e+b c f-3 a d f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b^3 d}-\frac {\left (e \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 b d} \\ & = \frac {(b c-a d) f (4 b d e+b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 d \sqrt {e+f x^2}}+\frac {\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b d \sqrt {e+f x^2}}+\frac {(b c-a d) f x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}+\frac {2 (3 d e-c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b}+\frac {f x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 b}-\frac {(b c-a d) \sqrt {e} \sqrt {f} (4 b d e+b c f-3 a d f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 d \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d) \sqrt {e} \sqrt {f} (5 b e-3 a f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} (9 d e-c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {c^{3/2} (b c-a d) (b e-a f)^2 \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a b^3 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.65 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {-i a b e \left (15 a^2 d^2 f^2-20 a b d f (d e+c f)+3 b^2 \left (d^2 e^2+9 c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a \left (-15 a^3 d^2 f^3+15 a^2 b d f^2 (d e+2 c f)-3 b^3 e \left (d^2 e^2+c d e f-7 c^2 f^2\right )+5 a b^2 f \left (d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+f \left (a b^2 \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-5 a d f+3 b \left (2 d e+2 c f+d f x^2\right )\right )-15 i (b c-a d)^2 (b e-a f)^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{15 a b^4 \sqrt {\frac {d}{c}} f \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

Integrate[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2),x]

[Out]

((-I)*a*b*e*(15*a^2*d^2*f^2 - 20*a*b*d*f*(d*e + c*f) + 3*b^2*(d^2*e^2 + 9*c*d*e*f + c^2*f^2))*Sqrt[1 + (d*x^2)
/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-15*a^3*d^2*f^3 + 15*a^2*b*d*f^2
*(d*e + 2*c*f) - 3*b^3*e*(d^2*e^2 + c*d*e*f - 7*c^2*f^2) + 5*a*b^2*f*(d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*Sqrt[1
+ (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*Sqrt[d/c]*x*(c + d
*x^2)*(e + f*x^2)*(-5*a*d*f + 3*b*(2*d*e + 2*c*f + d*f*x^2)) - (15*I)*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[1 + (d*
x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*a*b^4*Sqrt[d/c]
*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

Maple [A] (verified)

Time = 7.95 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {x \left (-3 b d f \,x^{2}+5 a d f -6 b c f -6 b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 b^{2}}+\frac {\left (-\frac {\left (15 a^{2} d^{2} f^{2}-20 a b c d \,f^{2}-20 a b \,d^{2} e f +3 b^{2} c^{2} f^{2}+27 b^{2} c d e f +3 b^{2} d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}-\frac {\left (15 a^{3} d^{2} f^{2}-30 a^{2} b c d \,f^{2}-30 a^{2} b \,d^{2} e f +15 a \,b^{2} c^{2} f^{2}+55 a \,b^{2} c d e f +15 a \,b^{2} d^{2} e^{2}-24 b^{3} c^{2} e f -24 b^{3} c d \,e^{2}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (15 a^{4} d^{2} f^{2}-30 a^{3} b c d \,f^{2}-30 a^{3} b \,d^{2} e f +15 a^{2} b^{2} c^{2} f^{2}+60 a^{2} b^{2} c d e f +15 b^{2} e^{2} d^{2} a^{2}-30 a \,b^{3} c^{2} e f -30 a \,b^{3} c d \,e^{2}+15 c^{2} e^{2} b^{4}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{b^{2} a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 b^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(663\)
default	\(\text {Expression too large to display}\)	\(1939\)
elliptic	\(\text {Expression too large to display}\)	\(2556\)

[In]

int((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/15*x*(-3*b*d*f*x^2+5*a*d*f-6*b*c*f-6*b*d*e)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^2+1/15/b^2*(-1/b*(15*a^2*d^2*
f^2-20*a*b*c*d*f^2-20*a*b*d^2*e*f+3*b^2*c^2*f^2+27*b^2*c*d*e*f+3*b^2*d^2*e^2)*e/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)
*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/f*(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-E
llipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2)))-(15*a^3*d^2*f^2-30*a^2*b*c*d*f^2-30*a^2*b*d^2*e*f+15*a*b^2*
c^2*f^2+55*a*b^2*c*d*e*f+15*a*b^2*d^2*e^2-24*b^3*c^2*e*f-24*b^3*c*d*e^2)/b^2/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1
+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+(15*a^4
*d^2*f^2-30*a^3*b*c*d*f^2-30*a^3*b*d^2*e*f+15*a^2*b^2*c^2*f^2+60*a^2*b^2*c*d*e*f+15*a^2*b^2*d^2*e^2-30*a*b^3*c
^2*e*f-30*a*b^3*c*d*e^2+15*b^4*c^2*e^2)/b^2/a/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^
2+d*e*x^2+c*e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2)))*((d*x^2+c)*(f*x^2+e))^(1/2)
/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\text {Timed out} \]

[In]

integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}{a + b x^{2}}\, dx \]

[In]

integrate((d*x**2+c)**(3/2)*(f*x**2+e)**(3/2)/(b*x**2+a),x)

[Out]

Integral((c + d*x**2)**(3/2)*(e + f*x**2)**(3/2)/(a + b*x**2), x)

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{b x^{2} + a} \,d x } \]

[In]

integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)/(b*x^2 + a), x)

Giac [F]

\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{b x^{2} + a} \,d x } \]

[In]

integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)/(b*x^2 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}}{b\,x^2+a} \,d x \]

[In]

int(((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2),x)

[Out]

int(((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2), x)