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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.11334, antiderivative size = 980, normalized size of antiderivative = 1., 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 = {544, 543, 539, 528, 531, 418, 492, 411, 526} \[ \frac{e^{3/2} \sqrt{d x^2+c} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)^3}{a b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} (b d e+4 b c f-3 a d f) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)}{3 b \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d (5 b c-3 a d) e^{3/2} \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)}{3 b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d x \sqrt{d x^2+c} \sqrt{f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac{(b d e+4 b c f-3 a d f) x \sqrt{d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt{f x^2+e}}-\frac{\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 \sqrt{e} f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d f e-3 c^2 f^2\right )\right ) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt{d x^2+c} \sqrt{f x^2+e}}{3 e f (b e-a f)^2}+\frac{(d e-c f) x \left (d x^2+c\right )^{3/2}}{e (b e-a f) \sqrt{f x^2+e}}+\frac{\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) x \sqrt{d x^2+c}}{3 e f (b e-a f)^2 \sqrt{f x^2+e}} \]

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 544

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]

Rule 543

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 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 528

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 531

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 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 526

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]

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*}

Mathematica [C]  time = 1.56713, size = 352, normalized size = 0.36 \[ \frac{-i a d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (-a d f+3 b c f-2 b d e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-f \left (a b^2 x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) (d e-c f)^2+i e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b \left (c^2 f^2-2 c d e f+2 d^2 e^2\right )-a d^2 e f\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a b^2 e f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)} \]

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 = 0.032, size = 1063, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

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

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

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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)