3.33 \(\int \frac{(a+b x^2) (e+f x^2)^{3/2}}{(c+d x^2)^{7/2}} \, dx\)

Optimal. Leaf size=376 \[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 c^3 d^2 \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{5/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{x \sqrt{e+f x^2} (d e (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.423039, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {526, 525, 418, 411} \[ \frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{5/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 d^2 \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} (d e (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]

[Out]

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

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]

Rule 525

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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2}} \, dx &=-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac{\int \frac{\sqrt{e+f x^2} \left (-(b c+4 a d) e-(4 b c+a d) f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx}{5 c d}\\ &=\frac{(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt{e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{\int \frac{e (a d (8 d e+c f)+2 b c (d e+2 c f))+f (2 a d (2 d e+c f)+b c (d e+8 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}} \, dx}{15 c^2 d^2}\\ &=\frac{(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt{e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac{(e f (b c (d e-4 c f)+a d (4 d e-c f))) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 c^2 d^2 (d e-c f)}+\frac{\left (b c \left (2 d^2 e^2+3 c d e f-8 c^2 f^2\right )+a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right ) \int \frac{\sqrt{e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 d^2 (d e-c f)}\\ &=\frac{(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt{e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{\left (b c \left (2 d^2 e^2+3 c d e f-8 c^2 f^2\right )+a d \left (8 d^2 e^2-3 c d e f-2 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} d^{5/2} (d e-c f) \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{e^{3/2} \sqrt{f} (b c (d e-4 c f)+a d (4 d e-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 d^2 (d e-c 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]  time = 1.30004, size = 382, normalized size = 1.02 \[ \frac{\sqrt{\frac{d}{c}} \left (-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a d \left (2 c^2 f^2+3 c d e f-8 d^2 e^2\right )+b c \left (8 c^2 f^2-3 c d e f-2 d^2 e^2\right )\right )+3 c^2 (b c-a d) (d e-c f)^2-c \left (c+d x^2\right ) (d e-c f) (2 a d (c f+2 d e)+b c (d e-7 c f))\right )-i e \sqrt{\frac{d x^2}{c}+1} \left (c+d x^2\right )^2 \sqrt{\frac{f x^2}{e}+1} \left ((d e-c f) (a d (c f+8 d e)+2 b c (2 c f+d e)) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+\left (a d \left (2 c^2 f^2+3 c d e f-8 d^2 e^2\right )+b c \left (8 c^2 f^2-3 c d e f-2 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )\right )}{15 c^2 d^3 \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]

[Out]

(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e - c*f)*(b*c*(d*e - 7*c*f) + 2*
a*d*(2*d*e + c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c
^2*f^2))*(c + d*x^2)^2)) - I*e*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((a*d*(-8*d^2*e^2 + 3*c*d
*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] +
(d*e - c*f)*(a*d*(8*d*e + c*f) + 2*b*c*(d*e + 2*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*c
^2*d^3*(d*e - c*f)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.037, size = 2860, normalized size = 7.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(7/2),x)

[Out]

-1/15*(-3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+16*E
llipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticF(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-16*EllipticE(x*(-d/c)
^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((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))*x^4*b*c*d^4*e^3*((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))*x^4*b*c*d^4*e^3*((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))*b*c^4*
d*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*c^4*d*e*f^2*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*d^2*e^2*f*((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*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)-7*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*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))*x^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*El
lipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*Elliptic
E(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*x^5*a*d^5*e^3*(-
d/c)^(1/2)-6*x^5*a*c^3*d^2*f^3*(-d/c)^(1/2)-8*x^7*b*c^3*d^2*f^3*(-d/c)^(1/2)-4*EllipticE(x*(-d/c)^(1/2),(c*f/d
/e)^(1/2))*x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-9*x^5*b*c^4*d*f^3*(-d/c)^(1/2)+2*x^5*b*c*
d^4*e^3*(-d/c)^(1/2)-x^3*a*c^4*d*f^3*(-d/c)^(1/2)+20*x^3*a*c*d^4*e^3*(-d/c)^(1/2)+5*x^3*b*c^2*d^3*e^3*(-d/c)^(
1/2)+15*x*a*c^2*d^3*e^3*(-d/c)^(1/2)-4*x*b*c^5*e*f^2*(-d/c)^(1/2)-2*x^7*a*c^2*d^3*f^3*(-d/c)^(1/2)+8*x^7*a*d^5
*e^2*f*(-d/c)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*d^2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2)-8*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*El
lipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^5*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))*b*c^3*d^2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticF(x*(-d/c)^(1/2),(c*
f/d/e)^(1/2))*x^2*b*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(
1/2))*x^2*a*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*
x^2*a*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^5*
e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*x^7*a*c*d^4*e*f^2*(-d/c)^(1/2)+3*x^7*b*c^2*d^3*e*f^2*(-d/c)^(1
/2)+2*x^7*b*c*d^4*e^2*f*(-d/c)^(1/2)-10*x^5*a*c^2*d^3*e*f^2*(-d/c)^(1/2)+17*x^5*a*c*d^4*e^2*f*(-d/c)^(1/2)-10*
x^5*b*c^3*d^2*e*f^2*(-d/c)^(1/2)+8*x^5*b*c^2*d^3*e^2*f*(-d/c)^(1/2)-17*x^3*a*c^3*d^2*e*f^2*(-d/c)^(1/2)-x*a*c^
4*d*e*f^2*(-d/c)^(1/2)-11*x*a*c^3*d^2*e^2*f*(-d/c)^(1/2)+x*b*c^4*d*e^2*f*(-d/c)^(1/2)+8*EllipticF(x*(-d/c)^(1/
2),(c*f/d/e)^(1/2))*x^4*a*d^5*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+7*x^3*a*c^2*d^3*e^2*f*(-d/c)^(1/2)-8
*x^3*b*c^4*d*e*f^2*(-d/c)^(1/2)-2*x^3*b*c^3*d^2*e^2*f*(-d/c)^(1/2)-4*x^3*b*c^5*f^3*(-d/c)^(1/2)-8*EllipticE(x*
(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*d^5*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticF(x*(-d/c)^(1/2)
,(c*f/d/e)^(1/2))*a*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(
1/2))*x^4*b*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*
x^4*b*c^2*d^3*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))*x^2*a*
c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-14*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^2*d
^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+16*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^4*d*e*f^
2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((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))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c
)^(1/2)*((f*x^2+e)/e)^(1/2)-7*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*e^2*f*((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)-4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^
2+e)/e)^(1/2)+4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2))/(f*x^2+e)^(1/2)/c^3/(c*f-d*e)/(-d/c)^(1/2)/(d*x^2+c)^(5/2)/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b f x^{4} +{\left (b e + a f\right )} x^{2} + a e\right )} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{d^{4} x^{8} + 4 \, c d^{3} x^{6} + 6 \, c^{2} d^{2} x^{4} + 4 \, c^{3} d x^{2} + c^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral((b*f*x^4 + (b*e + a*f)*x^2 + a*e)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^4*x^8 + 4*c*d^3*x^6 + 6*c^2*d^2*
x^4 + 4*c^3*d*x^2 + c^4), x)

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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((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2), x)