3.1.83 \(\int \frac {(c+d x^2)^{5/2}}{(a+b x^2) (e+f x^2)^{3/2}} \, dx\) [83]

Optimal. Leaf size=980 \[ \frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b (b e-a f)^2 \sqrt {e+f x^2}}+\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{3 e f (b e-a f)^2 \sqrt {e+f x^2}}+\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}-\frac {(b c-a d) \sqrt {e} (b d e+4 b c f-3 a d f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

(-c*f+d*e)*x*(d*x^2+c)^(3/2)/e/(-a*f+b*e)/(f*x^2+e)^(1/2)+1/3*(-a*d+b*c)*(-3*a*d*f+4*b*c*f+b*d*e)*x*(d*x^2+c)^
(1/2)/b/(-a*f+b*e)^2/(f*x^2+e)^(1/2)+1/3*(b*e*(-c^2*f^2-7*c*d*e*f+6*d^2*e^2)-a*f*(3*c^2*f^2-13*c*d*e*f+8*d^2*e
^2))*x*(d*x^2+c)^(1/2)/e/f/(-a*f+b*e)^2/(f*x^2+e)^(1/2)-1/3*(b*e*(-c^2*f^2-7*c*d*e*f+6*d^2*e^2)-a*f*(3*c^2*f^2
-13*c*d*e*f+8*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))*(d*x^2+c)^(1/2)/f^(3/2)/(-a*f+b*e)^2/e^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2
)-1/3*(2*a*d*f*(-3*c*f+2*d*e)-b*(-3*c^2*f^2-2*c*d*e*f+3*d^2*e^2))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*Elli
pticF(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^(3/2)/(-a*f+b*e)^2/(e*(
d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d*(-3*a*d+5*b*c)*(-a*d+b*c)*e^(3/2)*(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)/b/c/(-a*f+b*e)^
2/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)^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/b/c/(-a
*f+b*e)^2/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/3*(-a*d+b*c)*(-3*a*d*f+4*b*c*f+b*d*e)*(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/(-a*f+b*e)^2/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d*(-a*d+b*c)*x*(d*x^
2+c)^(1/2)*(f*x^2+e)^(1/2)/(-a*f+b*e)^2+1/3*d*(a*f*(-3*c*f+4*d*e)-b*e*(-2*c*f+3*d*e))*x*(d*x^2+c)^(1/2)*(f*x^2
+e)^(1/2)/e/f/(-a*f+b*e)^2

________________________________________________________________________________________

Rubi [A]
time = 0.75, antiderivative size = 980, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {558, 557, 553, 542, 545, 429, 506, 422, 540} \begin {gather*} \frac {e^{3/2} \sqrt {d x^2+c} \Pi \left (1-\frac {b e}{a f};\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) (b c-a d)^3}{a b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} (b d e+4 b c f-3 a d f) \sqrt {d x^2+c} E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) (b c-a d)}{3 b \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d (5 b c-3 a d) e^{3/2} \sqrt {d x^2+c} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) (b c-a d)}{3 b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d x \sqrt {d x^2+c} \sqrt {f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac {(b d e+4 b c f-3 a d f) x \sqrt {d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt {f x^2+e}}-\frac {\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) \sqrt {d x^2+c} E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d f e-3 c^2 f^2\right )\right ) \sqrt {d x^2+c} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {d x^2+c} \sqrt {f x^2+e}}{3 e f (b e-a f)^2}+\frac {(d e-c f) x \left (d x^2+c\right )^{3/2}}{e (b e-a f) \sqrt {f x^2+e}}+\frac {\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) x \sqrt {d x^2+c}}{3 e f (b e-a f)^2 \sqrt {f x^2+e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]

[Out]

((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b*(b*e - a*f)^2*Sqrt[e + f*x^2]) + ((b*e*(6*d^2
*e^2 - 7*c*d*e*f - c^2*f^2) - a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*x*Sqrt[c + d*x^2])/(3*e*f*(b*e - a*f)^
2*Sqrt[e + f*x^2]) + ((d*e - c*f)*x*(c + d*x^2)^(3/2))/(e*(b*e - a*f)*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*x*Sqrt
[c + d*x^2]*Sqrt[e + f*x^2])/(3*(b*e - a*f)^2) + (d*(a*f*(4*d*e - 3*c*f) - b*e*(3*d*e - 2*c*f))*x*Sqrt[c + d*x
^2]*Sqrt[e + f*x^2])/(3*e*f*(b*e - a*f)^2) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*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*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) - ((b*e*(6*d^2*e^2 - 7*c*d*e*f - c^2*f^2) - a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f
^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*Sqrt[e]*f^(3/2)*(b*e - a*f)^2
*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*(b*c - a*d)*e^(3/2)*Sqrt[c + d*x^
2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/
(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(2*a*d*f*(2*d*e - 3*c*f) - b*(3*d^2*e^2 - 2*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)])/(3*f^(3/2)*(b*e - a*f)^2*Sqrt[(e*(c
+ d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^3*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f
), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b*c*Sqrt[f]*(b*e - a*f)^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]

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 558

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

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (-b d e^2+2 b c e f-a c f^2+(b c-a d) f^2 x^2\right )}{\left (e+f x^2\right )^{3/2}} \, dx}{(b e-a f)^2}+\frac {(b (b c-a d)) \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{a+b x^2} \, dx}{(b e-a f)^2}\\ &=\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {(d (b c-a d)) \int \frac {\left (2 b c-a d+b d x^2\right ) \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx}{b (b e-a f)^2}+\frac {(b c-a d)^3 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b (b e-a f)^2}+\frac {\int \frac {\sqrt {c+d x^2} \left (-c (b c-a d) e f^2+d f (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x^2\right )}{\sqrt {e+f x^2}} \, dx}{e f (b e-a f)^2}\\ &=\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}+\frac {(b c-a d)^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 b c \sqrt {f} (b e-a 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 c-a d) \int \frac {d (5 b c-3 a d) e+d (b d e+4 b c f-3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b (b e-a f)^2}+\frac {\int \frac {-c e f \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\right )\right )+d f \left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f^2 (b e-a f)^2}\\ &=\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}+\frac {(b c-a d)^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 b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(d (5 b c-3 a d) (b c-a d) e) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b (b e-a f)^2}+\frac {(d (b c-a d) (b d e+4 b c f-3 a d f)) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b (b e-a f)^2}-\frac {\left (c \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 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}{3 f (b e-a f)^2}+\frac {\left (d \left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f (b e-a f)^2}\\ &=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b (b e-a f)^2 \sqrt {e+f x^2}}+\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{3 e f (b e-a f)^2 \sqrt {e+f x^2}}+\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \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 c \sqrt {f} (b e-a f)^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} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 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 )}{3 f^{3/2} (b e-a 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 c-a d)^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 b c \sqrt {f} (b e-a 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 c-a d) e (b d e+4 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 (b e-a f)^2}-\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 f (b e-a f)^2}\\ &=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b (b e-a f)^2 \sqrt {e+f x^2}}+\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{3 e f (b e-a f)^2 \sqrt {e+f x^2}}+\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}-\frac {(b c-a d) \sqrt {e} (b d e+4 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 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 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 )}{3 \sqrt {e} f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \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 c \sqrt {f} (b e-a f)^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} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 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 )}{3 f^{3/2} (b e-a 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 c-a d)^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 b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.09, size = 352, normalized size = 0.36 \begin {gather*} \frac {-i a b d e \left (-a d^2 e f+b \left (2 d^2 e^2-2 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 \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a d^2 e (b e-a f) (-2 b d e+3 b c f-a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-f \left (a b^2 \sqrt {\frac {d}{c}} (d e-c f)^2 x \left (c+d x^2\right )+i (b c-a d)^3 e f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{a b^2 \sqrt {\frac {d}{c}} e f^2 (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]

[Out]

((-I)*a*b*d*e*(-(a*d^2*e*f) + b*(2*d^2*e^2 - 2*c*d*e*f + c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ell
ipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d^2*e*(b*e - a*f)*(-2*b*d*e + 3*b*c*f - 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)] - f*(a*b^2*Sqrt[d/c]*(d*e - c*f)^2*x
*(c + d*x^2) + I*(b*c - a*d)^3*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[S
qrt[d/c]*x], (c*f)/(d*e)]))/(a*b^2*Sqrt[d/c]*e*f^2*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]
time = 0.16, size = 1063, normalized size = 1.08 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(5/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(5/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*(f*x^2 + e)^(3/2)), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(5/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**(5/2)/(b*x**2+a)/(f*x**2+e)**(3/2),x)

[Out]

Integral((c + d*x**2)**(5/2)/((a + b*x**2)*(e + f*x**2)**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(5/2)/(b*x^2+a)/(f*x^2+e)^(3/2),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*(f*x^2 + e)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^{5/2}}{\left (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x)

[Out]

int((c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)), x)

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