3.28 \(\int \frac{(a+b x^2) \sqrt{e+f x^2}}{(c+d x^2)^{7/2}} \, dx\)
Optimal. Leaf size=385 \[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 c^3 d \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{3/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{x \sqrt{e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]
[Out]
-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*d*(c + d*x^2)^(5/2)) + ((a*d*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*x*Sqrt
[e + f*x^2])/(15*c^2*d*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2
- 13*c*d*e*f + 3*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^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(2*a*d*(2*d
*e - 3*c*f) + b*c*(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*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
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Rubi [A] time = 0.380205, antiderivative size = 385, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{526, 527, 525, 418, 411} \[ \frac{\sqrt{e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{3/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 d \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]
[Out]
-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*d*(c + d*x^2)^(5/2)) + ((a*d*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*x*Sqrt
[e + f*x^2])/(15*c^2*d*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2
- 13*c*d*e*f + 3*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^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(2*a*d*(2*d
*e - 3*c*f) + b*c*(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*(d*e - c*f)^2*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 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{\left (a+b x^2\right ) \sqrt{e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac{\int \frac{-(b c+4 a d) e-(2 b c+3 a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}} \, dx}{5 c d}\\ &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt{e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac{\int \frac{e (a d (8 d e-9 c f)+b c (2 d e-c f))+f (d (b c+4 a d) e-c (2 b c+3 a d) f) x^2}{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}} \, dx}{15 c^2 d (d e-c f)}\\ &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt{e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac{(e f (2 a d (2 d e-3 c f)+b c (d e+c f))) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 c^2 d (d e-c f)^2}+\frac{\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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 (d e-c f)^2}\\ &=-\frac{(b c-a d) x \sqrt{e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt{e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac{\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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^{3/2} (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{e^{3/2} \sqrt{f} (2 a d (2 d e-3 c f)+b c (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 (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}}\\ \end{align*}
Mathematica [C] time = 1.26903, size = 379, normalized size = 0.98 \[ \frac{-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (-\left (c+d x^2\right )^2 \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right )+3 c^2 (b c-a d) (d e-c f)^2-c \left (c+d x^2\right ) (d e-c f) (a d (4 d e-3 c f)+b c (d e-2 c f))\right )+i e \left (c+d x^2\right )^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (\left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-(c f-d e) (a d (9 c f-8 d e)+b c (c f-2 d e)) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )\right )}{15 c^4 \left (\frac{d}{c}\right )^{3/2} \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (d e-c f)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),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)*(a*d*(4*d*e - 3*c*f) + b*c*(d*e -
2*c*f))*(c + d*x^2) - (2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*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]*((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a
*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(b*c*
(-2*d*e + c*f) + a*d*(-8*d*e + 9*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*c^4*(d/c)^(3/2)*(d
*e - c*f)^2*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])
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Maple [B] time = 0.055, size = 2856, normalized size = 7.4 \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)^(1/2)/(d*x^2+c)^(7/2),x)
[Out]
1/15*(2*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*Ell
ipticF(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)-3*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)-3*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)+13*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)+9*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)-17*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)-3*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)-3*
EllipticE(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)+13*Ellip
ticE(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)+9*x^5*a*c^3*d^2*f^3*(-d/c)^(1/2)+2*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)+6*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)+9*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)+x*b*c^5*e*f^2*(-d/c)^(1/2)+3*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)-2
*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)+2*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)-6*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)+26*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)+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)-13*x^7*a*c*d^4*e*f^2*(-d/c)^(1/2)-2*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)-30*x^5*a*c^2*d^3*e*f^2*(-d/c)^(1/2)+7*x^5*a*c*d^4*e^2*f*(-d/c)^(1/2)-5
*x^5*b*c^3*d^2*e*f^2*(-d/c)^(1/2)+3*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)+9*x*a
*c^4*d*e*f^2*(-d/c)^(1/2)-26*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)-18*x^3*a*c^2*d^3*e^2*f*(-d/c)^(1/
2)+7*x^3*b*c^4*d*e*f^2*(-d/c)^(1/2)-7*x^3*b*c^3*d^2*e^2*f*(-d/c)^(1/2)+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)-2*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)+2*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)+18*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)-34*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)-4*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)+4*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)+9*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)-17*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)+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)-6*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)/(-d/c)^(1/2)/(c*f-d*e)^2/c^3/d/(d*x^2+c)^(5/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}{{\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)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="maxima")
[Out]
integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(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 x^{2} + a\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)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="fricas")
[Out]
integral((b*x^2 + a)*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)**(1/2)/(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 )} \sqrt{f x^{2} + e}}{{\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)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="giac")
[Out]
integrate((b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(7/2), x)