\(\int \frac {\sqrt {e+f x^2}}{(a+b x^2) (c+d x^2)^{7/2}} \, dx\) [70]

Optimal result

Integrand size = 32, antiderivative size = 630 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=-\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {d} \left (a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b c \left (9 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c^3 (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

b^3*e^(3/2)*(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/c/(-a*d+b*c)^3/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/1
5*d*e^(3/2)*(b*c*(-11*c*f+9*d*e)-2*a*d*(-3*c*f+2*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))*f^(1/2)*(d*x^2+c)^(1/2)/c^3/(-a*d+b*c)^2/(-c*f+d*e)^2/(e*(d*
x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/5*d*x*(f*x^2+e)^(1/2)/c/(-a*d+b*c)/(d*x^2+c)^(5/2)-1/15*d*(b*c*(-8
*c*f+9*d*e)-a*d*(-3*c*f+4*d*e))*x*(f*x^2+e)^(1/2)/c^2/(-a*d+b*c)^2/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/15*(a*d*(3*c^2
*f^2-13*c*d*e*f+8*d^2*e^2)-2*b*c*(4*c^2*f^2-14*c*d*e*f+9*d^2*e^2))*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ell
ipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*d^(1/2)*(f*x^2+e)^(1/2)/c^(5/2)/(-a*d+b*c)^2/(-c
*f+d*e)^2/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)-b^2*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Elliptic
E(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*d^(1/2)*(f*x^2+e)^(1/2)/(-a*d+b*c)^3/c^(1/2)/(d*x^2+c
)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {560, 555, 553, 422, 540, 541, 539, 429} \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\frac {b^3 e^{3/2} \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{a c \sqrt {f} \sqrt {e+f x^2} (b c-a d)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} \sqrt {c+d x^2} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c^3 \sqrt {e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b c \left (4 c^2 f^2-14 c d e f+9 d^2 e^2\right )\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {d x \sqrt {e+f x^2} (b c (9 d e-8 c f)-a d (4 d e-3 c f))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2 (d e-c f)}-\frac {d x \sqrt {e+f x^2}}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)} \]

[In]

Int[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(7/2)),x]

[Out]

-1/5*(d*x*Sqrt[e + f*x^2])/(c*(b*c - a*d)*(c + d*x^2)^(5/2)) - (d*(b*c*(9*d*e - 8*c*f) - a*d*(4*d*e - 3*c*f))*
x*Sqrt[e + f*x^2])/(15*c^2*(b*c - a*d)^2*(d*e - c*f)*(c + d*x^2)^(3/2)) - (b^2*Sqrt[d]*Sqrt[e + f*x^2]*Ellipti
cE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/
(e*(c + d*x^2))]) + (Sqrt[d]*(a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*c*(9*d^2*e^2 - 14*c*d*e*f + 4*c^2
*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*(b*c - a*d)^2*(d*e
- c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (d*e^(3/2)*Sqrt[f]*(b*c*(9*d*e - 11*c*f) -
2*a*d*(2*d*e - 3*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(b*c -
a*d)^2*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^3*e^(3/2)*Sqrt[c + d*x^2]*El
lipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^3*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 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(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]

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 541

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 + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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 555

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

Rule 560

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^2}-\frac {d \int \frac {\left (2 b c-a d+b d x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx}{(b c-a d)^2} \\ & = -\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {b^3 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^3}-\frac {\left (b^2 d\right ) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^3}+\frac {\int \frac {-d (9 b c-4 a d) e-d (8 b c-3 a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c (b c-a d)^2} \\ & = -\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a 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 {\int \frac {d e (b c (18 d e-19 c f)-a d (8 d e-9 c f))+d f (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)} \\ & = -\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a 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 {(d e f (b c (9 d e-11 c f)-2 a d (2 d e-3 c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)^2}+\frac {\left (d \left (a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b c \left (9 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)^2} \\ & = -\frac {d x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {d (b c (9 d e-8 c f)-a d (4 d e-3 c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {b^2 \sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {d} \left (a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b c \left (9 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d e^{3/2} \sqrt {f} (b c (9 d e-11 c f)-2 a d (2 d e-3 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 c^3 (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a 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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.80 (sec) , antiderivative size = 584, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\frac {-a d \sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)^2+c (b c-a d) (-d e+c f) (a d (4 d e-3 c f)+b c (-9 d e+8 c f)) \left (c+d x^2\right )+\left (a b c d \left (-26 d^2 e^2+41 c d e f-11 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (33 d^2 e^2-58 c d e f+23 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )-i \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (a d e \left (a b c d \left (-26 d^2 e^2+41 c d e f-11 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (33 d^2 e^2-58 c d e f+23 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-(d e-c f) \left (-a \left (2 a b c d^2 e (13 d e-14 c f)+a^2 d^3 e (-8 d e+9 c f)+b^2 c^2 \left (-33 d^2 e^2+49 c d e f-15 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+15 b^2 c^3 (b e-a f) (-d e+c f) \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )\right )}{15 a c^3 \sqrt {\frac {d}{c}} (b c-a d)^3 (d e-c f)^2 \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \]

[In]

Integrate[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(7/2)),x]

[Out]

(-(a*d*Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 + c*(b*c - a*d)*(-(d*e) + c*f)*(a*d*(4*d*e -
3*c*f) + b*c*(-9*d*e + 8*c*f))*(c + d*x^2) + (a*b*c*d*(-26*d^2*e^2 + 41*c*d*e*f - 11*c^2*f^2) + a^2*d^2*(8*d^
2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(33*d^2*e^2 - 58*c*d*e*f + 23*c^2*f^2))*(c + d*x^2)^2)) - I*(c + d*x
^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*d*e*(a*b*c*d*(-26*d^2*e^2 + 41*c*d*e*f - 11*c^2*f^2) + a^2*d^
2*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(33*d^2*e^2 - 58*c*d*e*f + 23*c^2*f^2))*EllipticE[I*ArcSinh[S
qrt[d/c]*x], (c*f)/(d*e)] - (d*e - c*f)*(-(a*(2*a*b*c*d^2*e*(13*d*e - 14*c*f) + a^2*d^3*e*(-8*d*e + 9*c*f) + b
^2*c^2*(-33*d^2*e^2 + 49*c*d*e*f - 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]) + 15*b^2*c^3*(
b*e - a*f)*(-(d*e) + c*f)*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*a*c^3*Sqrt[d/c]*
(b*c - a*d)^3*(d*e - c*f)^2*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3137\) vs. \(2(714)=1428\).

Time = 6.96 (sec) , antiderivative size = 3138, normalized size of antiderivative = 4.98

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3138\)
default	\(\text {Expression too large to display}\)	\(6245\)

[In]

int((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(7/2),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)*(-4/15/(-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))*f/(c*f-d*e)/c^2/(
a*d-b*c)^2*a*e*d^2+3/5/(-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)*El
lipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*f/(c*f-d*e)/c/(a*d-b*c)^2*b*d*e+13/15/(-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*f-d*e)/c^2/(a*d-b*c)^3*a^2*d^3*e*f+26/15/(-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*f-d*e)/c^2/(a*d-b*c)^3*a*b*d^3*e^2
-8/15/(-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))*f^2/(c*f-d*e)/(a*d-b*c)^2*b-b^3/(a*d-b*c)^3/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))*e+1/15*(d*f*x^2+d*e)/c^3/(c*f-d*e)^2*x*(3*a^2*c^2*d^2*f^2-13*a^2*c*d^3*e*f+8*a^2*d^4*e^2-11*a*b*c^3*d*f^2+
41*a*b*c^2*d^2*e*f-26*a*b*c*d^3*e^2+23*b^2*c^4*f^2-58*b^2*c^3*d*e*f+33*b^2*c^2*d^2*e^2)/(a*d-b*c)^3/((x^2+c/d)
*(d*f*x^2+d*e))^(1/2)+1/5/d^2/c*x/(a*d-b*c)*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(x^2+c/d)^3-23/15/(-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*f-d*e)*c/(a*d-b*c)^3*b^2*f^2-41/15/(-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*f-d*e)/c/(a*d-b*c)^3*a*b*d^2*
e*f+11/15*d^2/(c*f-d*e)^2/(a*d-b*c)^3*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)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*a*b*f^2-41/15*d^3/(c*f-d*e)^2/c/(a*d-b*c)^3*
e^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)*EllipticE(x*(-d/c)^(1
/2),(-1+(c*f+d*e)/e/d)^(1/2))*a*b*f+b^2/(a*d-b*c)^3/(-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))*f+1/15*(3*a*c*d*f-4*a*
d^2*e-8*b*c^2*f+9*b*c*d*e)/d/(c*f-d*e)/c^2/(a*d-b*c)^2*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(x^2+c/d)^2+1/5/(
-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))*f^2/(c*f-d*e)/c/(a*d-b*c)^2*a*d-1/5/(-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*f-d*e)/c/(a*d-b*c)^3
*a^2*d^2*f^2-8/15/(-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)*Ellipti
cF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))/(c*f-d*e)/c^3/(a*d-b*c)^3*a^2*d^4*e^2+11/15/(-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*f-d*e)/(a*d-b*c)^3*a*b*d*f^2+58/15/(-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*f-d*e)/(a*d-b*c)^3*b^2*d*e*f-11/5/(-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*f-d*e)/c/(a*d-b*c)^3*b^2*d^2*e^2-1/5*d^3/(c*f-d*e)^2/c/(a*d-b*c)^3*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)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d
)^(1/2))*a^2*f^2+13/15*d^4/(c*f-d*e)^2/c^2/(a*d-b*c)^3*e^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)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*a^2*f+26/15*d^4/(c*f-d*e)
^2/c^2/(a*d-b*c)^3*e^3/(-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)*El
lipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*a*b-23/15*d/(c*f-d*e)^2*c/(a*d-b*c)^3*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)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1
/2))*b^2*f^2+58/15*d^2/(c*f-d*e)^2/(a*d-b*c)^3*e^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)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*b^2*f-8/15*d^5/(c*f-d*e)^2/c^3/(a
*d-b*c)^3*e^3/(-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)*EllipticE(x
*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*a^2-11/5*d^3/(c*f-d*e)^2/c/(a*d-b*c)^3*e^3/(-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)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*b
^2)

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {e + f x^{2}}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((f*x**2+e)**(1/2)/(b*x**2+a)/(d*x**2+c)**(7/2),x)

[Out]

Integral(sqrt(e + f*x**2)/((a + b*x**2)*(c + d*x**2)**(7/2)), x)

Maxima [F]

\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(f*x^2 + e)/((b*x^2 + a)*(d*x^2 + c)^(7/2)), x)

Giac [F]

\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(f*x^2 + e)/((b*x^2 + a)*(d*x^2 + c)^(7/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {f\,x^2+e}}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{7/2}} \,d x \]

[In]

int((e + f*x^2)^(1/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x)

[Out]

int((e + f*x^2)^(1/2)/((a + b*x^2)*(c + d*x^2)^(7/2)), x)