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

Optimal. Leaf size=543 \[ \frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]

[Out]

((7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*x*
Sqrt[c + d*x^2])/(105*d^3*f*Sqrt[e + f*x^2]) + ((14*a*d*f*(3*d*e - c*f) + b*(3*d^2*e^2 - 15*c*d*e*f + 8*c^2*f^
2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d^2*f) + ((3*b*d*e - 4*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e
+ f*x^2])/(35*d^2) + (b*x*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(7*d) - (Sqrt[e]*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e
*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan
[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2
]) + (e^(3/2)*(7*a*d*f*(9*d*e - c*f) - b*(3*d^2*e^2 + 9*c*d*e*f - 4*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan
[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2
])

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Rubi [A]  time = 0.64488, antiderivative size = 543, 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} \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt{e+f x^2}}+\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*x*
Sqrt[c + d*x^2])/(105*d^3*f*Sqrt[e + f*x^2]) + ((14*a*d*f*(3*d*e - c*f) + b*(3*d^2*e^2 - 15*c*d*e*f + 8*c^2*f^
2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d^2*f) + ((3*b*d*e - 4*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e
+ f*x^2])/(35*d^2) + (b*x*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(7*d) - (Sqrt[e]*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e
*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan
[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2
]) + (e^(3/2)*(7*a*d*f*(9*d*e - c*f) - b*(3*d^2*e^2 + 9*c*d*e*f - 4*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan
[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(3/2)*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 ) \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2} \, dx &=\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\int \sqrt{c+d x^2} \sqrt{e+f x^2} \left (-(b c-7 a d) e+(3 b d e-4 b c f+7 a d f) x^2\right ) \, dx}{7 d}\\ &=\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\int \frac{\sqrt{c+d x^2} \left (-e (4 b c (2 d e-c f)-7 a d (5 d e-c f))+\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x^2\right )}{\sqrt{e+f x^2}} \, dx}{35 d^2}\\ &=\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\int \frac{c e \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right )+\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d^2 f}\\ &=\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\left (c e \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d^2 f}+\frac{\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d^2 f}\\ &=\frac{\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d^3 f \sqrt{e+f x^2}}+\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{e^{3/2} \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 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^2 f^{3/2} \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 (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 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^3 f}\\ &=\frac{\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d^3 f \sqrt{e+f x^2}}+\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}-\frac{\sqrt{e} \left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 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^3 f^{3/2} \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 (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 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^2 f^{3/2} \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 = 1.14135, size = 372, normalized size = 0.69 \[ \frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (b \left (4 c^2 f^2-6 c d e f+6 d^2 e^2\right )-7 a d f (c f+3 d e)\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+f x \left (-\sqrt{\frac{d}{c}}\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-7 a d f \left (c f+6 d e+3 d f x^2\right )+4 b c^2 f^2-3 b c d f \left (3 e+f x^2\right )-3 b d^2 \left (e^2+8 e f x^2+5 f^2 x^4\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )+b \left (-19 c^2 d e f^2+8 c^3 f^3+9 c d^2 e^2 f-6 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 c^2 f^2 \left (\frac{d}{c}\right )^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(4*b*c^2*f^2 - 3*b*c*d*f*(3*e + f*x^2) - 7*a*d*f*(6*d*e + c*f + 3*d*f
*x^2) - 3*b*d^2*(e^2 + 8*e*f*x^2 + 5*f^2*x^4))) - I*e*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) + b*(-6*d^3
*e^3 + 9*c*d^2*e^2*f - 19*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSin
h[Sqrt[d/c]*x], (c*f)/(d*e)] + I*e*(-(d*e) + c*f)*(-7*a*d*f*(3*d*e + c*f) + b*(6*d^2*e^2 - 6*c*d*e*f + 4*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*c^2*(d/c)^(5
/2)*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.016, 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)^(1/2)*(f*x^2+e)^(3/2),x)

[Out]

1/105*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(-(-d/c)^(1/2)*x^5*b*c^2*d*f^4+8*((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^3-21*((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+3*(-d/c)^(1/2)*x*b*c*d^2*e^3*f-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^2*d*e^2*f^2+15*(-d/c)^(1/2)*x^9*b*d^3*f^4+10*((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^2-12*((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-14*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*El
lipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d*e*f^3+7*((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^3-4*((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+21*((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+28*(-d/c)^(1/2)*x^5*a*c*d^2*f^4+63*(-d/c)^(1/2)*x^5*a*d^3*e*f^3+6*((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^4-6*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellip
ticE(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+8*(-d/c)^(1/2)*x^3*b*c^2*d*e*f^3+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*d^2*e^3*f+21*(-d/c)^(1/2)*x^7*a*d^3*f^4-4*(-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+36*(-d/c)^(1/2)*x^3*b*c*d^2*e^2*f^2+7*(-d/c)^(1/2)*x*a*c^2*d*e*f^3+42*(-d/c)^(1/2)*x*a*c*d^2*e^2*f^2+2
7*(-d/c)^(1/2)*x^5*b*d^3*e^2*f^2+7*(-d/c)^(1/2)*x^3*a*c^2*d*f^4+42*(-d/c)^(1/2)*x^3*a*d^3*e^2*f^2+3*(-d/c)^(1/
2)*x^3*b*d^3*e^3*f-4*(-d/c)^(1/2)*x*b*c^3*e*f^3+18*(-d/c)^(1/2)*x^7*b*c*d^2*f^4+39*(-d/c)^(1/2)*x^7*b*d^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*c*d^2*e^2*f^2)/f^2/(d
*f*x^4+c*f*x^2+d*e*x^2+c*e)/d^2/(-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 )} \sqrt{d x^{2} + c}{\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((b*x^2+a)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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