\(\int \frac {(a+b x^2) (e+f x^2)^{3/2}}{\sqrt {c+d x^2}} \, dx\) [30]

Optimal result

Integrand size = 30, antiderivative size = 400 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {e+f x^2}}+\frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}-\frac {\sqrt {e} \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c d^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

1/5*b*x*(f*x^2+e)^(3/2)*(d*x^2+c)^(1/2)/d+1/15*(10*a*d*f*(-c*f+2*d*e)+b*(8*c^2*f^2-13*c*d*e*f+3*d^2*e^2))*x*(d
*x^2+c)^(1/2)/d^3/(f*x^2+e)^(1/2)+1/15*e^(3/2)*(5*a*d*(-c*f+3*d*e)-b*(-4*c^2*f+6*c*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)/c/d^2/f^(1
/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/15*(10*a*d*f*(-c*f+2*d*e)+b*(8*c^2*f^2-13*c*d*e*f+3*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)/d^3/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/15*(5*a*d*f-4*b*c*f+3*b
*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d^2

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {542, 545, 429, 506, 422} \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {e^{3/2} \sqrt {c+d x^2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c d^2 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^3 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right )}{15 d^3 \sqrt {e+f x^2}}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d} \]

[In]

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

[Out]

((10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*x^2])/(15*d^3*Sqrt[e + f*x^2])
+ ((3*b*d*e - 4*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*d^2) + (b*x*Sqrt[c + d*x^2]*(e + f*x^
2)^(3/2))/(5*d) - (Sqrt[e]*(10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*Sqrt[c + d*x^2]*E
llipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*
Sqrt[e + f*x^2]) + (e^(3/2)*(5*a*d*(3*d*e - c*f) - b*(6*c*d*e - 4*c^2*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sq
rt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c*d^2*Sqrt[f]*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 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 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}+\frac {\int \frac {\sqrt {e+f x^2} \left (-((b c-5 a d) e)+(3 b d e-4 b c f+5 a d f) x^2\right )}{\sqrt {c+d x^2}} \, dx}{5 d} \\ & = \frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}+\frac {\int \frac {-e (2 b c (3 d e-2 c f)-5 a d (3 d e-c f))+\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d^2} \\ & = \frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}-\frac {(e (2 b c (3 d e-2 c f)-5 a d (3 d e-c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d^2}+\frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d^2} \\ & = \frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {e+f x^2}}+\frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}+\frac {e^{3/2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \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 c d^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (e \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 d^3} \\ & = \frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {e+f x^2}}+\frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}-\frac {\sqrt {e} \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\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 d^3 \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 (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \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 c d^2 \sqrt {f} \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 = 3.10 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (4 b c f-5 a d f-3 b d \left (2 e+f x^2\right )\right )-i e \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i e (-d e+c f) (-3 b d e+4 b c f-5 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{15 c^2 \left (\frac {d}{c}\right )^{5/2} f \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

(-(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(4*b*c*f - 5*a*d*f - 3*b*d*(2*e + f*x^2))) - I*e*(10*a*d*f*(2*d*e - c
*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*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*e*(-(d*e) + c*f)*(-3*b*d*e + 4*b*c*f - 5*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2
)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*c^2*(d/c)^(5/2)*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

Maple [A] (verified)

Time = 6.17 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.12

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b f \,x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d}+\frac {\left (a \,f^{2}+2 b f e -\frac {b f \left (4 c f +4 d e \right )}{5 d}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (e^{2} a -\frac {\left (a \,f^{2}+2 b f e -\frac {b f \left (4 c f +4 d e \right )}{5 d}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (2 a e f +e^{2} b -\frac {3 b f c e}{5 d}-\frac {\left (a \,f^{2}+2 b f e -\frac {b f \left (4 c f +4 d e \right )}{5 d}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(448\)
risch	\(\frac {x \left (3 b d f \,x^{2}+5 a d f -4 b c f +6 b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 d^{2}}-\frac {\left (-\frac {15 a \,d^{2} e^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {4 b \,c^{2} e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {6 b c d \,e^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {5 a c d e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (10 a c d \,f^{2}-20 a \,d^{2} e f -8 b \,c^{2} f^{2}+13 b c d e f -3 b \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 d^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(628\)
default	\(\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \left (3 \sqrt {-\frac {d}{c}}\, b \,d^{2} f^{3} x^{7}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} f^{3} x^{5}-\sqrt {-\frac {d}{c}}\, b c d \,f^{3} x^{5}+9 \sqrt {-\frac {d}{c}}\, b \,d^{2} e \,f^{2} x^{5}+5 \sqrt {-\frac {d}{c}}\, a c d \,f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} e \,f^{2} x^{3}-4 \sqrt {-\frac {d}{c}}\, b \,c^{2} f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, b c d e \,f^{2} x^{3}+6 \sqrt {-\frac {d}{c}}\, b \,d^{2} e^{2} f \,x^{3}+5 \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}-5 \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 -4 \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}+7 \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 -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 \,d^{2} e^{3}-10 \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}+20 \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 +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 \,c^{2} e \,f^{2}-13 \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 +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 ) b \,d^{2} e^{3}+5 \sqrt {-\frac {d}{c}}\, a c d e \,f^{2} x -4 \sqrt {-\frac {d}{c}}\, b \,c^{2} e \,f^{2} x +6 \sqrt {-\frac {d}{c}}\, b c d \,e^{2} f x \right )}{15 d^{2} f \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) \sqrt {-\frac {d}{c}}}\)	\(870\)

[In]

int((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),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)*(1/5*b*f/d*x^3*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)
+1/3*(a*f^2+2*b*f*e-1/5*b*f/d*(4*c*f+4*d*e))/d/f*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+(e^2*a-1/3*(a*f^2+2*b*f
*e-1/5*b*f/d*(4*c*f+4*d*e))/d/f*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))-(2*a*e*f+e^2*b-3/5*b*f/d*c*e-1/3*(a*f^2+2*b*f*e
-1/5*b*f/d*(4*c*f+4*d*e))/d/f*(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 = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (3 \, b d^{2} e^{3} - {\left (13 \, b c d - 20 \, a d^{2}\right )} e^{2} f + 2 \, {\left (4 \, b c^{2} - 5 \, a c d\right )} e f^{2}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (3 \, b d^{2} e^{3} - {\left (13 \, b c d - 20 \, a d^{2}\right )} e^{2} f + {\left (8 \, b c^{2} - 2 \, {\left (5 \, a + 3 \, b\right )} c d + 15 \, a d^{2}\right )} e f^{2} + {\left (4 \, b c^{2} - 5 \, a c d\right )} f^{3}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (3 \, b d^{2} f^{3} x^{4} + 3 \, b d^{2} e^{2} f - {\left (13 \, b c d - 20 \, a d^{2}\right )} e f^{2} + 2 \, {\left (4 \, b c^{2} - 5 \, a c d\right )} f^{3} + {\left (6 \, b d^{2} e f^{2} - {\left (4 \, b c d - 5 \, a d^{2}\right )} f^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{15 \, d^{3} f^{2} x} \]

[In]

integrate((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

-1/15*((3*b*d^2*e^3 - (13*b*c*d - 20*a*d^2)*e^2*f + 2*(4*b*c^2 - 5*a*c*d)*e*f^2)*sqrt(d*f)*x*sqrt(-e/f)*ellipt
ic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (3*b*d^2*e^3 - (13*b*c*d - 20*a*d^2)*e^2*f + (8*b*c^2 - 2*(5*a + 3*b)*
c*d + 15*a*d^2)*e*f^2 + (4*b*c^2 - 5*a*c*d)*f^3)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_f(arcsin(sqrt(-e/f)/x), c*f/(
d*e)) - (3*b*d^2*f^3*x^4 + 3*b*d^2*e^2*f - (13*b*c*d - 20*a*d^2)*e*f^2 + 2*(4*b*c^2 - 5*a*c*d)*f^3 + (6*b*d^2*
e*f^2 - (4*b*c*d - 5*a*d^2)*f^3)*x^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(d^3*f^2*x)

Sympy [F]

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

[In]

integrate((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(1/2),x)

[Out]

int(((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(1/2), x)