\(\int (a+b x^2) (c+d x^2)^{3/2} \sqrt {e+f x^2} \, dx\) [23]

Optimal result

Integrand size = 30, antiderivative size = 544 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx=-\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\sqrt {e} \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{105 d f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-1/105*(7*a*d*f*(-3*c^2*f^2-7*c*d*e*f+2*d^2*e^2)-b*(-6*c^3*f^3+9*c^2*d*e*f^2-19*c*d^2*e^2*f+8*d^3*e^3))*x*(d*x
^2+c)^(1/2)/d^2/f^2/(f*x^2+e)^(1/2)-1/105*e^(3/2)*(7*a*d*f*(-9*c*f+d*e)-b*(-3*c^2*f^2-9*c*d*e*f+4*d^2*e^2))*(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)/d/f^(5/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/105*(7*a*d*f*(-3*c^2*f^2-7*c*d*e*f+2*d^2*
e^2)-b*(-6*c^3*f^3+9*c^2*d*e*f^2-19*c*d^2*e^2*f+8*d^3*e^3))*(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^2/f^(5/2)/(e*(d*x^2+c)/c/(f*x
^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/35*(7*a*d*f-2*b*c*f+b*d*e)*x*(d*x^2+c)^(3/2)*(f*x^2+e)^(1/2)/d/f+1/7*b*x*(d*x^2
+c)^(5/2)*(f*x^2+e)^(1/2)/d+1/105*(7*a*d*f*(3*c*f+d*e)-b*(6*c^2*f^2-6*c*d*e*f+4*d^2*e^2))*x*(d*x^2+c)^(1/2)*(f
*x^2+e)^(1/2)/d/f^2

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {542, 545, 429, 506, 422} \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx=-\frac {e^{3/2} \sqrt {c+d x^2} \left (7 a d f (d e-9 c f)-b \left (-3 c^2 f^2-9 c d e f+4 d^2 e^2\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{105 d f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} \left (7 a d f (3 c f+d e)-b \left (6 c^2 f^2-6 c d e f+4 d^2 e^2\right )\right )}{105 d f^2}-\frac {x \sqrt {c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right )\right )}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d} \]

[In]

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

[Out]

-1/105*((7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f
^3))*x*Sqrt[c + d*x^2])/(d^2*f^2*Sqrt[e + f*x^2]) + ((7*a*d*f*(d*e + 3*c*f) - b*(4*d^2*e^2 - 6*c*d*e*f + 6*c^2
*f^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d*f^2) + ((b*d*e - 2*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[
e + f*x^2])/(35*d*f) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*d) + (Sqrt[e]*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*
f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[
(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]
) - (e^(3/2)*(7*a*d*f*(d*e - 9*c*f) - b*(4*d^2*e^2 - 9*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[
(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*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 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 \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (-((b c-7 a d) e)+(b d e-2 b c f+7 a d f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{7 d} \\ & = \frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\int \frac {\sqrt {c+d x^2} \left (-c e (b d e+3 b c f-28 a d f)+\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x^2\right )}{\sqrt {e+f x^2}} \, dx}{35 d f} \\ & = \frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\int \frac {-c e \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right )+\left (-7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d f^2} \\ & = \frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}-\frac {\left (c e \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d f^2}-\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d f^2} \\ & = -\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}-\frac {e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\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 )}{105 d 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 {\left (e \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{105 d^2 f^2} \\ & = -\frac {\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^2 f^2 \sqrt {e+f x^2}}+\frac {\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d f^2}+\frac {(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d f}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 d}+\frac {\sqrt {e} \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\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 )}{105 d^2 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 {e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\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 )}{105 d 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 = 3.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.69 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx=\frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (6 c f+d \left (e+3 f x^2\right )\right )+b \left (3 c^2 f^2+3 c d f \left (3 e+8 f x^2\right )+d^2 \left (-4 e^2+3 e f x^2+15 f^2 x^4\right )\right )\right )+i e \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+b \left (-8 d^3 e^3+19 c d^2 e^2 f-9 c^2 d e f^2+6 c^3 f^3\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i e (-d e+c f) \left (-14 a d f (d e-3 c f)+b \left (8 d^2 e^2-15 c d e f+3 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{105 d \sqrt {\frac {d}{c}} f^3 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(6*c*f + d*(e + 3*f*x^2)) + b*(3*c^2*f^2 + 3*c*d*f*(3*e + 8*f*
x^2) + d^2*(-4*e^2 + 3*e*f*x^2 + 15*f^2*x^4))) + I*e*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + b*(-8*d^3*
e^3 + 19*c*d^2*e^2*f - 9*c^2*d*e*f^2 + 6*c^3*f^3))*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)*(-14*a*d*f*(d*e - 3*c*f) + b*(8*d^2*e^2 - 15*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)])/(105*d*Sqrt[d/c]
*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

Maple [A] (verified)

Time = 5.01 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.28

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b d \,x^{5} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{7}+\frac {\left (a \,d^{2} f +2 b c f d +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d f}+\frac {\left (2 a c d f +a e \,d^{2}+c^{2} b f +\frac {9 b c d e}{7}-\frac {\left (a \,d^{2} f +2 b c f d +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (c^{2} a e -\frac {\left (2 a c d f +a e \,d^{2}+c^{2} b f +\frac {9 b c d e}{7}-\frac {\left (a \,d^{2} f +2 b c f d +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (c^{2} a f +2 a c d e +b \,c^{2} e -\frac {3 \left (a \,d^{2} f +2 b c f d +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) c e}{5 d f}-\frac {\left (2 a c d f +a e \,d^{2}+c^{2} b f +\frac {9 b c d e}{7}-\frac {\left (a \,d^{2} f +2 b c f d +b \,d^{2} e -\frac {b d \left (6 c f +6 d e \right )}{7}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(695\)
risch	\(\frac {x \left (15 b \,x^{4} d^{2} f^{2}+21 a \,d^{2} f^{2} x^{2}+24 b c d \,f^{2} x^{2}+3 b \,d^{2} e f \,x^{2}+42 a c d \,f^{2}+7 a \,d^{2} e f +3 b \,c^{2} f^{2}+9 b c d e f -4 b \,d^{2} e^{2}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{105 d \,f^{2}}+\frac {\left (-\frac {3 b \,c^{3} e \,f^{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 \,d^{2} e^{3} \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 {7 a c \,d^{2} e^{2} 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 {9 b \,c^{2} d \,e^{2} 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 {63 a \,c^{2} d e \,f^{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 {\left (21 a \,c^{2} d \,f^{3}+49 a c \,d^{2} e \,f^{2}-14 a \,d^{3} e^{2} f -6 b \,c^{3} f^{3}+9 b \,c^{2} d e \,f^{2}-19 b c \,d^{2} e^{2} f +8 b \,d^{3} e^{3}\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 )}}{105 d \,f^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(834\)
default	\(\text {Expression too large to display}\)	\(1332\)

[In]

int((b*x^2+a)*(d*x^2+c)^(3/2)*(f*x^2+e)^(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)*(1/7*b*d*x^5*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+1
/5*(a*d^2*f+2*b*c*f*d+b*d^2*e-1/7*b*d*(6*c*f+6*d*e))/d/f*x^3*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+1/3*(2*a*c*d*
f+a*e*d^2+c^2*b*f+9/7*b*c*d*e-1/5*(a*d^2*f+2*b*c*f*d+b*d^2*e-1/7*b*d*(6*c*f+6*d*e))/d/f*(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)+(c^2*a*e-1/3*(2*a*c*d*f+a*e*d^2+c^2*b*f+9/7*b*c*d*e-1/5*(a*d^2*f+2*b*c*f*d+
b*d^2*e-1/7*b*d*(6*c*f+6*d*e))/d/f*(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))-(c^2*a*f+2*a*c*d*e+b*c^2*
e-3/5*(a*d^2*f+2*b*c*f*d+b*d^2*e-1/7*b*d*(6*c*f+6*d*e))/d/f*c*e-1/3*(2*a*c*d*f+a*e*d^2+c^2*b*f+9/7*b*c*d*e-1/5
*(a*d^2*f+2*b*c*f*d+b*d^2*e-1/7*b*d*(6*c*f+6*d*e))/d/f*(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 = 463, normalized size of antiderivative = 0.85 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx=-\frac {{\left (8 \, b d^{3} e^{4} - {\left (19 \, b c d^{2} + 14 \, a d^{3}\right )} e^{3} f + {\left (9 \, b c^{2} d + 49 \, a c d^{2}\right )} e^{2} f^{2} - 3 \, {\left (2 \, b c^{3} - 7 \, a c^{2} d\right )} e f^{3}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (8 \, b d^{3} e^{4} - {\left (19 \, b c d^{2} + 14 \, a d^{3}\right )} e^{3} f + {\left (9 \, b c^{2} d + {\left (49 \, a + 4 \, b\right )} c d^{2}\right )} e^{2} f^{2} - {\left (6 \, b c^{3} - 3 \, {\left (7 \, a - 3 \, b\right )} c^{2} d + 7 \, a c d^{2}\right )} e f^{3} - 3 \, {\left (b c^{3} - 21 \, a c^{2} d\right )} f^{4}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (15 \, b d^{3} f^{4} x^{6} + 8 \, b d^{3} e^{3} f - {\left (19 \, b c d^{2} + 14 \, a d^{3}\right )} e^{2} f^{2} + {\left (9 \, b c^{2} d + 49 \, a c d^{2}\right )} e f^{3} - 3 \, {\left (2 \, b c^{3} - 7 \, a c^{2} d\right )} f^{4} + 3 \, {\left (b d^{3} e f^{3} + {\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} f^{4}\right )} x^{4} - {\left (4 \, b d^{3} e^{2} f^{2} - {\left (9 \, b c d^{2} + 7 \, a d^{3}\right )} e f^{3} - 3 \, {\left (b c^{2} d + 14 \, a c d^{2}\right )} f^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{105 \, d^{2} f^{4} x} \]

[In]

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

[Out]

-1/105*((8*b*d^3*e^4 - (19*b*c*d^2 + 14*a*d^3)*e^3*f + (9*b*c^2*d + 49*a*c*d^2)*e^2*f^2 - 3*(2*b*c^3 - 7*a*c^2
*d)*e*f^3)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (8*b*d^3*e^4 - (19*b*c*d^2 + 1
4*a*d^3)*e^3*f + (9*b*c^2*d + (49*a + 4*b)*c*d^2)*e^2*f^2 - (6*b*c^3 - 3*(7*a - 3*b)*c^2*d + 7*a*c*d^2)*e*f^3
- 3*(b*c^3 - 21*a*c^2*d)*f^4)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_f(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (15*b*d^3*f
^4*x^6 + 8*b*d^3*e^3*f - (19*b*c*d^2 + 14*a*d^3)*e^2*f^2 + (9*b*c^2*d + 49*a*c*d^2)*e*f^3 - 3*(2*b*c^3 - 7*a*c
^2*d)*f^4 + 3*(b*d^3*e*f^3 + (8*b*c*d^2 + 7*a*d^3)*f^4)*x^4 - (4*b*d^3*e^2*f^2 - (9*b*c*d^2 + 7*a*d^3)*e*f^3 -
3*(b*c^2*d + 14*a*c*d^2)*f^4)*x^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(d^2*f^4*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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