\(\int \frac {(a+b x^2) (c+d x^2)^{3/2}}{(e+f x^2)^{3/2}} \, dx\) [43]

Optimal result

Integrand size = 30, antiderivative size = 358 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{3/2}} \, dx=-\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) x \sqrt {c+d x^2}}{3 e f^2 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}+\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (4 b d e-3 b c f-3 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-(-a*f+b*e)*x*(d*x^2+c)^(3/2)/e/f/(f*x^2+e)^(1/2)-1/3*(b*e*(-7*c*f+8*d*e)-3*a*f*(-c*f+2*d*e))*x*(d*x^2+c)^(1/2
)/e/f^2/(f*x^2+e)^(1/2)+1/3*(b*e*(-7*c*f+8*d*e)-3*a*f*(-c*f+2*d*e))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*El
lipticE(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)/f^(5/2)/e^(1/2)/(e*(d*x^2+c)/c/
(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/3*(-3*a*d*f-3*b*c*f+4*b*d*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*Ellipt
icF(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)/f^(5/2)/(e*(d*x^2+c)/c/(f*x
^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d*(-3*a*f+4*b*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/e/f^2

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {540, 542, 545, 429, 506, 422} \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{3/2}} \, dx=-\frac {\sqrt {e} \sqrt {c+d x^2} (-3 a d f-3 b c f+4 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (4 b e-3 a f)}{3 e f^2}-\frac {x \sqrt {c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f))}{3 e f^2 \sqrt {e+f x^2}}-\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{e f \sqrt {e+f x^2}} \]

[In]

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

[Out]

-1/3*((b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*x*Sqrt[c + d*x^2])/(e*f^2*Sqrt[e + f*x^2]) - ((b*e - a*f)*x*
(c + d*x^2)^(3/2))/(e*f*Sqrt[e + f*x^2]) + (d*(4*b*e - 3*a*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f^2) + (
(b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(
c*f)])/(3*Sqrt[e]*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(4*b*d*e - 3*b*c*f
- 3*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(5/2)*Sqrt[(e*(c + d
*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*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 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 540

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

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}-\frac {\int \frac {\sqrt {c+d x^2} \left (-b c e-d (4 b e-3 a f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{e f} \\ & = -\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}-\frac {\int \frac {c e (4 b d e-3 b c f-3 a d f)+d (b e (8 d e-7 c f)-3 a f (2 d e-c f)) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f^2} \\ & = -\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}-\frac {(c (4 b d e-3 b c f-3 a d f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 f^2}-\frac {(d (b e (8 d e-7 c f)-3 a f (2 d e-c f))) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f^2} \\ & = -\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) x \sqrt {c+d x^2}}{3 e f^2 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}-\frac {\sqrt {e} (4 b d e-3 b c f-3 a d 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 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 f^2} \\ & = -\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c f)) x \sqrt {c+d x^2}}{3 e f^2 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{e f \sqrt {e+f x^2}}+\frac {d (4 b e-3 a f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f^2}+\frac {(b e (8 d e-7 c f)-3 a f (2 d e-c 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 \sqrt {e} f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (4 b d e-3 b c f-3 a d 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 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.87 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (3 a f (-d e+c f)+b e \left (4 d e-3 c f+d f x^2\right )\right )-i d e (-3 a f (-2 d e+c f)+b e (-8 d e+7 c f)) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i e (-d e+c f) (-8 b d e+3 b c f+6 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{3 \sqrt {\frac {d}{c}} e f^3 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(3*a*f*(-(d*e) + c*f) + b*e*(4*d*e - 3*c*f + d*f*x^2)) - I*d*e*(-3*a*f*(-2*d*e + c*
f) + b*e*(-8*d*e + 7*c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*
e)] - I*e*(-(d*e) + c*f)*(-8*b*d*e + 3*b*c*f + 6*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*Ar
cSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*Sqrt[d/c]*e*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

Maple [A] (verified)

Time = 7.41 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.52

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+c f \right ) \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{f^{3} e \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {b d x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 f^{2}}+\frac {\left (\frac {2 a c d \,f^{2}-a \,d^{2} e f +b \,c^{2} f^{2}-2 b c d e f +b \,d^{2} e^{2}}{f^{3}}+\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \left (c f -d e \right )}{f^{3} e}-\frac {c \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right )}{f^{2} e}-\frac {b d c e}{3 f^{2}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (\frac {d \left (a d f +2 b c f -b d e \right )}{f^{2}}-\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) d}{f^{2} e}-\frac {b d \left (2 c f +2 d e \right )}{3 f^{2}}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(543\)
risch	\(\frac {b x \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, d}{3 f^{2}}+\frac {\left (-\frac {d \left (3 a d f +4 b c f -5 b d e \right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}+\frac {\left (6 a c d \,f^{2}-3 a \,d^{2} e f +3 b \,c^{2} f^{2}-7 b c d e f +3 b \,d^{2} e^{2}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{f \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (3 c^{2} a \,f^{3}-6 a c d e \,f^{2}+3 a \,d^{2} e^{2} f -3 b \,c^{2} e \,f^{2}+6 b c d \,e^{2} f -3 b \,d^{2} e^{3}\right ) \left (\frac {\left (d f \,x^{2}+c f \right ) x}{e \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {\left (\frac {1}{e}-\frac {c f}{e \left (c f -d e \right )}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\left (c f -d e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{f}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{3 f^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(685\)
default	\(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, \left (\sqrt {-\frac {d}{c}}\, b \,d^{2} e \,f^{2} x^{5}+3 \sqrt {-\frac {d}{c}}\, a c d \,f^{3} x^{3}-3 \sqrt {-\frac {d}{c}}\, a \,d^{2} e \,f^{2} x^{3}-2 \sqrt {-\frac {d}{c}}\, b c d e \,f^{2} x^{3}+4 \sqrt {-\frac {d}{c}}\, b \,d^{2} e^{2} f \,x^{3}+6 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e \,f^{2}-6 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2} f +3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e \,f^{2}-11 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2} f +8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,d^{2} e^{3}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e \,f^{2}+6 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2} f +7 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2} f -8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,d^{2} e^{3}+3 \sqrt {-\frac {d}{c}}\, a \,c^{2} f^{3} x -3 \sqrt {-\frac {d}{c}}\, a c d e \,f^{2} x -3 \sqrt {-\frac {d}{c}}\, b \,c^{2} e \,f^{2} x +4 \sqrt {-\frac {d}{c}}\, b c d \,e^{2} f x \right )}{3 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) f^{3} e \sqrt {-\frac {d}{c}}}\)	\(750\)

[In]

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

[Out]

((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*((d*f*x^2+c*f)*(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/f
^3/e*x/((x^2+e/f)*(d*f*x^2+c*f))^(1/2)+1/3*b*d/f^2*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+((2*a*c*d*f^2-a*d^2*e
*f+b*c^2*f^2-2*b*c*d*e*f+b*d^2*e^2)/f^3+(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/f^3*(c*f-d*e)/e-c/f^2*(a*c*f^2-a*d*e
*f-b*c*e*f+b*d*e^2)/e-1/3*b*d/f^2*c*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)*EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-(d/f^2*(a*d*f+2*b*c*f-b*d*e)-(a*c*f^2-a*d*e*f
-b*c*e*f+b*d*e^2)/f^2*d/e-1/3*b*d/f^2*(2*c*f+2*d*e))*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))-EllipticE(x*(-d/c)^(1/2),(
-1+(c*f+d*e)/e/d)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (8 \, b d^{2} e^{3} f + 3 \, a c d e f^{3} - {\left (7 \, b c d + 6 \, a d^{2}\right )} e^{2} f^{2}\right )} x^{3} + {\left (8 \, b d^{2} e^{4} + 3 \, a c d e^{2} f^{2} - {\left (7 \, b c d + 6 \, a d^{2}\right )} e^{3} f\right )} x\right )} \sqrt {d f} \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left ({\left (8 \, b d^{2} e^{3} f + {\left (3 \, a + 4 \, b\right )} c d e f^{3} - {\left (7 \, b c d + 6 \, a d^{2}\right )} e^{2} f^{2} - 3 \, {\left (b c^{2} + a c d\right )} f^{4}\right )} x^{3} + {\left (8 \, b d^{2} e^{4} + {\left (3 \, a + 4 \, b\right )} c d e^{2} f^{2} - {\left (7 \, b c d + 6 \, a d^{2}\right )} e^{3} f - 3 \, {\left (b c^{2} + a c d\right )} e f^{3}\right )} x\right )} \sqrt {d f} \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) + {\left (b d^{2} e f^{3} x^{4} - 8 \, b d^{2} e^{3} f - 3 \, a c d e f^{3} + {\left (7 \, b c d + 6 \, a d^{2}\right )} e^{2} f^{2} - {\left (4 \, b d^{2} e^{2} f^{2} - {\left (4 \, b c d + 3 \, a d^{2}\right )} e f^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{3 \, {\left (d e f^{5} x^{3} + d e^{2} f^{4} x\right )}} \]

[In]

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

[Out]

1/3*(((8*b*d^2*e^3*f + 3*a*c*d*e*f^3 - (7*b*c*d + 6*a*d^2)*e^2*f^2)*x^3 + (8*b*d^2*e^4 + 3*a*c*d*e^2*f^2 - (7*
b*c*d + 6*a*d^2)*e^3*f)*x)*sqrt(d*f)*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - ((8*b*d^2*e^3*f
+ (3*a + 4*b)*c*d*e*f^3 - (7*b*c*d + 6*a*d^2)*e^2*f^2 - 3*(b*c^2 + a*c*d)*f^4)*x^3 + (8*b*d^2*e^4 + (3*a + 4*b
)*c*d*e^2*f^2 - (7*b*c*d + 6*a*d^2)*e^3*f - 3*(b*c^2 + a*c*d)*e*f^3)*x)*sqrt(d*f)*sqrt(-e/f)*elliptic_f(arcsin
(sqrt(-e/f)/x), c*f/(d*e)) + (b*d^2*e*f^3*x^4 - 8*b*d^2*e^3*f - 3*a*c*d*e*f^3 + (7*b*c*d + 6*a*d^2)*e^2*f^2 -
(4*b*d^2*e^2*f^2 - (4*b*c*d + 3*a*d^2)*e*f^3)*x^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(d*e*f^5*x^3 + d*e^2*f^4*x
)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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