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

Optimal. Leaf size=401 \[ -\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} \left (a \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )+b c e (d e-9 c f)\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (a d \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b c \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{x \sqrt{e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (d e-c f)^2}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c \left (c+d x^2\right )^{5/2} (d e-c f)} \]

[Out]

-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*(d*e - c*f)*(c + d*x^2)^(5/2)) + ((4*a*d*(d*e - 2*c*f) + b*c*(d*e + 3*c*
f))*x*Sqrt[e + f*x^2])/(15*c^2*(d*e - c*f)^2*(c + d*x^2)^(3/2)) + ((b*c*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) +
a*d*(8*d^2*e^2 - 23*c*d*e*f + 23*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)*Sqrt[d]*(d*e - c*f)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt
[f]*(b*c*e*(d*e - 9*c*f) + a*(4*d^2*e^2 - 11*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*
x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.413409, antiderivative size = 401, 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 = {527, 525, 418, 411} \[ -\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} \left (a \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )+b c e (d e-9 c f)\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (a d \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b c \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{x \sqrt{e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (d e-c f)^2}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c \left (c+d x^2\right )^{5/2} (d e-c f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*(d*e - c*f)*(c + d*x^2)^(5/2)) + ((4*a*d*(d*e - 2*c*f) + b*c*(d*e + 3*c*
f))*x*Sqrt[e + f*x^2])/(15*c^2*(d*e - c*f)^2*(c + d*x^2)^(3/2)) + ((b*c*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) +
a*d*(8*d^2*e^2 - 23*c*d*e*f + 23*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)*Sqrt[d]*(d*e - c*f)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt
[f]*(b*c*e*(d*e - 9*c*f) + a*(4*d^2*e^2 - 11*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*
x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

Rule 527

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 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{a+b x^2}{\left (c+d x^2\right )^{7/2} \sqrt{e+f x^2}} \, dx &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}-\frac{\int \frac{-b c e-4 a d e+5 a c f+3 (b c-a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}} \, dx}{5 c (d e-c f)}\\ &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac{(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt{e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac{\int \frac{2 b c e (d e-3 c f)+a \left (8 d^2 e^2-19 c d e f+15 c^2 f^2\right )+f (4 a d (d e-2 c f)+b c (d e+3 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}} \, dx}{15 c^2 (d e-c f)^2}\\ &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac{(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt{e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}-\frac{\left (f \left (b c e (d e-9 c f)+a \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 c^2 (d e-c f)^3}+\frac{\left (b c \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+a d \left (8 d^2 e^2-23 c d e f+23 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 e-c f)^3}\\ &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac{(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt{e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac{\left (b c \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+a d \left (8 d^2 e^2-23 c d e f+23 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} \sqrt{d} (d e-c f)^3 \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \left (b c e (d e-9 c f)+a \left (4 d^2 e^2-11 c d e f+15 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 )}{15 c^3 (d e-c f)^3 \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 = 1.29057, size = 393, normalized size = 0.98 \[ \frac{-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a d \left (-23 c^2 f^2+23 c d e f-8 d^2 e^2\right )+b c \left (3 c^2 f^2+7 c d e f-2 d^2 e^2\right )\right )+3 c^2 (b c-a d) (d e-c f)^2+c \left (c+d x^2\right ) (c f-d e) (4 a d (d e-2 c f)+b c (3 c f+d e))\right )-i \sqrt{\frac{d x^2}{c}+1} \left (c+d x^2\right )^2 \sqrt{\frac{f x^2}{e}+1} \left ((d e-c f) \left (a \left (15 c^2 f^2-19 c d e f+8 d^2 e^2\right )+2 b c e (d e-3 c f)\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+e \left (a d \left (-23 c^2 f^2+23 c d e f-8 d^2 e^2\right )+b c \left (3 c^2 f^2+7 c d e f-2 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 c^3 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (d e-c f)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(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)*(4*a*d*(d*e - 2*c*f) + b*c*(d*e
+ 3*c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 23*c*d*e*f - 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*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]*(e*(a*d*(-8*d^2*e^2 + 23*c*d*e*f
- 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*
e - c*f)*(2*b*c*e*(d*e - 3*c*f) + a*(8*d^2*e^2 - 19*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (
c*f)/(d*e)]))/(15*c^3*Sqrt[d/c]*(d*e - c*f)^3*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.042, size = 3039, 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)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)

[Out]

1/15*(30*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^4*d*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*E
llipticE(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*EllipticF(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)+8*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)+23*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)-23*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)-34*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)+27*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)+8*E
llipticF(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)+23*Ellipt
icE(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)-23*EllipticE(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)+15*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^5*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-54*x^5*
a*c^3*d^2*f^3*(-d/c)^(1/2)+3*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)-34*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)+9*x*b*c^5*e*f^2*(-d/c)^(1/2)-23*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*Ell
ipticE(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)-3*EllipticE(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)+15*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*x^4*a*c^3*d^2*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-12*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)+46*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)-46*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)-6*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)+23*x^7*a*c*d^4*e*f^2*(-d/c)^(1/2)+7*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)+35*x^5*a*c^2*d^3*e*f^2*(-d/c)^(1/2)+3*x^5*a*c*d^4*e^2*f*(-d/c)^(1/2)+15*x^5*b*c^3*d^2*e*f
^2*(-d/c)^(1/2)+2*x^5*b*c^2*d^3*e^2*f*(-d/c)^(1/2)-13*x^3*a*c^3*d^2*e*f^2*(-d/c)^(1/2)-34*x*a*c^4*d*e*f^2*(-d/
c)^(1/2)+41*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)+43*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)+12*x^3*b*c^3*d^2*e^2*f*(-d/c)^(1/2)+9*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)-3*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)-7*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)-68*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)+54*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)-6*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)-14*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)-34*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)+27*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)-6*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)+16*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*f-d*e)^3/(-d/c)^(1/2)/c^3/(d*x^2+c)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{d^{4} f x^{10} +{\left (d^{4} e + 4 \, c d^{3} f\right )} x^{8} + 2 \,{\left (2 \, c d^{3} e + 3 \, c^{2} d^{2} f\right )} x^{6} + c^{4} e + 2 \,{\left (3 \, c^{2} d^{2} e + 2 \, c^{3} d f\right )} x^{4} +{\left (4 \, c^{3} d e + c^{4} f\right )} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^4*f*x^10 + (d^4*e + 4*c*d^3*f)*x^8 + 2*(2*c*d^3*e + 3*
c^2*d^2*f)*x^6 + c^4*e + 2*(3*c^2*d^2*e + 2*c^3*d*f)*x^4 + (4*c^3*d*e + c^4*f)*x^2), 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)/(d*x**2+c)**(7/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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