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

Optimal. Leaf size=544 \[ -\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (d e-9 c f)-b \left (-3 c^2 f^2-9 c d e f+4 d^2 e^2\right )\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{105 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (7 a d f (3 c f+d e)-b \left (6 c^2 f^2-6 c d e f+4 d^2 e^2\right )\right )}{105 d f^2}-\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right )\right )}{105 d^2 f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d} \]

[Out]

-((7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*x
*Sqrt[c + d*x^2])/(105*d^2*f^2*Sqrt[e + f*x^2]) + ((7*a*d*f*(d*e + 3*c*f) - b*(4*d^2*e^2 - 6*c*d*e*f + 6*c^2*f
^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d*f^2) + ((b*d*e - 2*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e
+ f*x^2])/(35*d*f) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*d) + (Sqrt[e]*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f
- 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
- (e^(3/2)*(7*a*d*f*(d*e - 9*c*f) - b*(4*d^2*e^2 - 9*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.681249, antiderivative size = 544, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (7 a d f (3 c f+d e)-b \left (6 c^2 f^2-6 c d e f+4 d^2 e^2\right )\right )}{105 d f^2}-\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right )\right )}{105 d^2 f^2 \sqrt{e+f x^2}}-\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (d e-9 c f)-b \left (-3 c^2 f^2-9 c d e f+4 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )-b \left (9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*x
*Sqrt[c + d*x^2])/(105*d^2*f^2*Sqrt[e + f*x^2]) + ((7*a*d*f*(d*e + 3*c*f) - b*(4*d^2*e^2 - 6*c*d*e*f + 6*c^2*f
^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d*f^2) + ((b*d*e - 2*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e
+ f*x^2])/(35*d*f) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*d) + (Sqrt[e]*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f
- 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
- (e^(3/2)*(7*a*d*f*(d*e - 9*c*f) - b*(4*d^2*e^2 - 9*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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]

Rubi steps

\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} \, dx &=\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d}+\frac{\int \frac{\left (c+d x^2\right )^{3/2} \left (-(b c-7 a d) e+(b d e-2 b c f+7 a d f) x^2\right )}{\sqrt{e+f x^2}} \, dx}{7 d}\\ &=\frac{(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d}+\frac{\int \frac{\sqrt{c+d x^2} \left (-c e (b d e+3 b c f-28 a d f)+\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x^2\right )}{\sqrt{e+f x^2}} \, dx}{35 d f}\\ &=\frac{\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d f^2}+\frac{(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d}+\frac{\int \frac{-c e \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right )+\left (-7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d f^2}\\ &=\frac{\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d f^2}+\frac{(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d}-\frac{\left (c e \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 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}{105 d f^2}-\frac{\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d f^2}\\ &=-\frac{\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d^2 f^2 \sqrt{e+f x^2}}+\frac{\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d f^2}+\frac{(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d}-\frac{e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 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 )}{105 d f^{5/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{\left (e \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right )\right ) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{105 d^2 f^2}\\ &=-\frac{\left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d^2 f^2 \sqrt{e+f x^2}}+\frac{\left (7 a d f (d e+3 c f)-b \left (4 d^2 e^2-6 c d e f+6 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d f^2}+\frac{(b d e-2 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d f}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 d}+\frac{\sqrt{e} \left (7 a d f \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )-b \left (8 d^3 e^3-19 c d^2 e^2 f+9 c^2 d e f^2-6 c^3 f^3\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 )}{105 d^2 f^{5/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{e^{3/2} \left (7 a d f (d e-9 c f)-b \left (4 d^2 e^2-9 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 )}{105 d f^{5/2} \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 = 1.13472, size = 373, normalized size = 0.69 \[ \frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (b \left (3 c^2 f^2-15 c d e f+8 d^2 e^2\right )-14 a d f (d e-3 c f)\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (6 c f+d \left (e+3 f x^2\right )\right )+b \left (3 c^2 f^2+3 c d f \left (3 e+8 f x^2\right )+d^2 \left (-4 e^2+3 e f x^2+15 f^2 x^4\right )\right )\right )+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )+b \left (-9 c^2 d e f^2+6 c^3 f^3+19 c d^2 e^2 f-8 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 d f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(6*c*f + d*(e + 3*f*x^2)) + b*(3*c^2*f^2 + 3*c*d*f*(3*e + 8*f*
x^2) + d^2*(-4*e^2 + 3*e*f*x^2 + 15*f^2*x^4))) + I*e*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + b*(-8*d^3*
e^3 + 19*c*d^2*e^2*f - 9*c^2*d*e*f^2 + 6*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh
[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(-14*a*d*f*(d*e - 3*c*f) + b*(8*d^2*e^2 - 15*c*d*e*f + 3*c^2*
f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(105*d*Sqrt[d/c]
*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.039, size = 1332, normalized size = 2.5 \begin{align*} \text{result 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)^(3/2)*(f*x^2+e)^(1/2),x)

[Out]

1/105*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(27*(-d/c)^(1/2)*x^5*b*c^2*d*f^4-6*((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))*b*c^3*e*f^3+14*((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*d^3*e^3*f-4*(-d/c)^(1/2)*x*b*c*d^2*e^3*f+9*((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))*b*c^2*d*e^2*f^2+15*(-d/c)^(1/2)*x^9*b*d^3*f^4-18*((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^2*d*e^2*f^2+49*((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*c*d^2*e^2*f^2+23*((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*d^2*e^3*f+21*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*E
llipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d*e*f^3+42*((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^2*d*e*f^3+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))*b*c^3*e*f^3-14*((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*d^3*e^3*f+63*(-d/c)^(1/2)*x^5*a*c*d^2*f^4+28*(-d/c)^(1/2)*x^5*a*d^3*e*f^3-8*((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*d^3*e^4+8*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ell
ipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*d^3*e^4+51*(-d/c)^(1/2)*x^5*b*c*d^2*e*f^3+70*(-d/c)^(1/2)*x^3*a*c*d^2
*e*f^3+36*(-d/c)^(1/2)*x^3*b*c^2*d*e*f^3-19*((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))*b*c*d^2*e^3*f+21*(-d/c)^(1/2)*x^7*a*d^3*f^4+3*(-d/c)^(1/2)*x^3*b*c^3*f^4+9*(-d/c)^(1/2)*x*b*c^
2*d*e^2*f^2+8*(-d/c)^(1/2)*x^3*b*c*d^2*e^2*f^2+42*(-d/c)^(1/2)*x*a*c^2*d*e*f^3+7*(-d/c)^(1/2)*x*a*c*d^2*e^2*f^
2-(-d/c)^(1/2)*x^5*b*d^3*e^2*f^2+42*(-d/c)^(1/2)*x^3*a*c^2*d*f^4+7*(-d/c)^(1/2)*x^3*a*d^3*e^2*f^2-4*(-d/c)^(1/
2)*x^3*b*d^3*e^3*f+3*(-d/c)^(1/2)*x*b*c^3*e*f^3+39*(-d/c)^(1/2)*x^7*b*c*d^2*f^4+18*(-d/c)^(1/2)*x^7*b*d^3*e*f^
3-56*((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*d^2*e^2*f^2)/d/(d*f
*x^4+c*f*x^2+d*e*x^2+c*e)/f^3/(-d/c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{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)^(3/2)*(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^(3/2)*(f*x^2+e)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(d*x**2+c)**(3/2)*(f*x**2+e)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{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)^(3/2)*(f*x^2+e)^(1/2),x, algorithm="giac")

[Out]

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