3.587 \(\int \frac{(d+e x)^3}{(f+g x)^3 (d^2-e^2 x^2)^{7/2}} \, dx\)
Optimal. Leaf size=398 \[ \frac{e^2 \left (e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )+90 d^3 g^2\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^5}+\frac{e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt{e^2 f^2-d^2 g^2}}+\frac{3 e g^4 \sqrt{d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac{e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac{4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3} \]
[Out]
(4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) - (e^2*(5*d*(e*f - 5*d*g) - e*(e*f + 31*d*g)*x))/(
15*d*(e*f + d*g)^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x))/(1
5*d^3*(e*f + d*g)^5*Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)*(e*f + d*g)^4*(f + g*x)^2)
+ (3*e*g^4*(3*e*f - 2*d*g)*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)^2*(e*f + d*g)^5*(f + g*x)) + (e^2*g^3*(20*e^2*
f^2 - 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(d^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2])])/(2*(e*
f - d*g)^2*(e*f + d*g)^5*Sqrt[e^2*f^2 - d^2*g^2])
________________________________________________________________________________________
Rubi [A] time = 2.56842, antiderivative size = 398, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used =
{1647, 1651, 807, 725, 204} \[ \frac{e^2 \left (e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )+90 d^3 g^2\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^5}+\frac{e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt{e^2 f^2-d^2 g^2}}+\frac{3 e g^4 \sqrt{d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac{e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac{4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^3/((f + g*x)^3*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
(4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) - (e^2*(5*d*(e*f - 5*d*g) - e*(e*f + 31*d*g)*x))/(
15*d*(e*f + d*g)^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x))/(1
5*d^3*(e*f + d*g)^5*Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)*(e*f + d*g)^4*(f + g*x)^2)
+ (3*e*g^4*(3*e*f - 2*d*g)*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)^2*(e*f + d*g)^5*(f + g*x)) + (e^2*g^3*(20*e^2*
f^2 - 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(d^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2])])/(2*(e*
f - d*g)^2*(e*f + d*g)^5*Sqrt[e^2*f^2 - d^2*g^2])
Rule 1647
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]
Rule 1651
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
Rule 807
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]
Rule 725
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{\frac{d^3 e^2 \left (e^3 f^3+15 d e^2 f^2 g+15 d^2 e f g^2+5 d^3 g^3\right )}{(e f+d g)^3}-\frac{d^2 e^3 \left (5 e^3 f^3-33 d e^2 f^2 g-45 d^2 e f g^2-15 d^3 g^3\right ) x}{(e f+d g)^3}+\frac{4 d^3 e^4 g^2 (12 e f+5 d g) x^2}{(e f+d g)^3}+\frac{16 d^3 e^5 g^3 x^3}{(e f+d g)^3}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{\frac{d^3 e^4 \left (2 e^4 f^4+17 d e^3 f^3 g+90 d^2 e^2 f^2 g^2+60 d^3 e f g^3+15 d^4 g^4\right )}{(e f+d g)^4}+\frac{3 d^3 e^5 g \left (2 e^2 f^2+45 d e f g+15 d^2 g^2\right ) x}{(e f+d g)^3}+\frac{3 d^3 e^6 g^2 \left (2 e^2 f^2+57 d e f g+25 d^2 g^2\right ) x^2}{(e f+d g)^4}+\frac{2 d^3 e^7 g^3 (e f+31 d g) x^3}{(e f+d g)^4}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{\frac{15 d^6 e^6 g^3 \left (10 e^2 f^2+5 d e f g+d^2 g^2\right )}{(e f+d g)^5}+\frac{45 d^6 e^7 g^4 (5 e f+d g) x}{(e f+d g)^5}+\frac{90 d^6 e^8 g^5 x^2}{(e f+d g)^5}}{(f+g x)^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac{\int \frac{\frac{30 d^6 e^7 g^3 \left (10 e^2 f^2-5 d e f g-3 d^2 g^2\right )}{(e f+d g)^4}+\frac{15 d^6 e^8 g^4 (11 e f-13 d g) x}{(e f+d g)^4}}{(f+g x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^6 e^6 \left (e^2 f^2-d^2 g^2\right )}\\ &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac{3 e g^4 (3 e f-2 d g) \sqrt{d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac{\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\right ) \int \frac{1}{(f+g x) \sqrt{d^2-e^2 x^2}} \, dx}{2 (e f-d g)^2 (e f+d g)^5}\\ &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac{3 e g^4 (3 e f-2 d g) \sqrt{d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}-\frac{\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-e^2 f^2+d^2 g^2-x^2} \, dx,x,\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5}\\ &=\frac{4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac{3 e g^4 (3 e f-2 d g) \sqrt{d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac{e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{e^2 f^2-d^2 g^2} \sqrt{d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5 \sqrt{e^2 f^2-d^2 g^2}}\\ \end{align*}
Mathematica [C] time = 1.26057, size = 387, normalized size = 0.97 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\frac{2 e^2 \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )}{d^3 (d-e x)}+\frac{2 e^2 (d g+e f) (17 d g+2 e f)}{d^2 (d-e x)^2}+\frac{6 e^2 (d g+e f)^2}{d (d-e x)^3}+\frac{45 e g^4 (3 e f-2 d g)}{(f+g x) (e f-d g)^2}+\frac{15 g^4 (d g+e f)}{(f+g x)^2 (e f-d g)}\right )-\frac{15 i e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \log \left (\frac{4 (e f-d g)^2 (d g+e f)^5 \left (\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}+i d^2 g+i e^2 f x\right )}{e^2 g^2 (f+g x) \sqrt{e^2 f^2-d^2 g^2} \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right )}\right )}{(e f-d g)^2 \sqrt{e^2 f^2-d^2 g^2}}}{30 (d g+e f)^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^3/((f + g*x)^3*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
(Sqrt[d^2 - e^2*x^2]*((6*e^2*(e*f + d*g)^2)/(d*(d - e*x)^3) + (2*e^2*(e*f + d*g)*(2*e*f + 17*d*g))/(d^2*(d - e
*x)^2) + (2*e^2*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2))/(d^3*(d - e*x)) + (15*g^4*(e*f + d*g))/((e*f - d*g)*(f
+ g*x)^2) + (45*e*g^4*(3*e*f - 2*d*g))/((e*f - d*g)^2*(f + g*x))) - ((15*I)*e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g
+ 13*d^2*g^2)*Log[(4*(e*f - d*g)^2*(e*f + d*g)^5*(I*d^2*g + I*e^2*f*x + Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2
*x^2]))/(e^2*g^2*Sqrt[e^2*f^2 - d^2*g^2]*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*(f + g*x))])/((e*f - d*g)^2*Sq
rt[e^2*f^2 - d^2*g^2]))/(30*(e*f + d*g)^5)
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Maple [B] time = 0.258, size = 9593, normalized size = 24.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^3/(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x)
[Out]
result too large to display
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3/(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 25.1647, size = 11237, normalized size = 28.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3/(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
[Out]
[1/30*(14*d^3*e^8*f^10 + 60*d^4*e^7*f^9*g + 78*d^5*e^6*f^8*g^2 - 480*d^6*e^5*f^7*g^3 + 312*d^7*e^4*f^6*g^4 + 3
30*d^8*e^3*f^5*g^5 - 419*d^9*e^2*f^4*g^6 + 90*d^10*e*f^3*g^7 + 15*d^11*f^2*g^8 - (14*e^11*f^8*g^2 + 60*d*e^10*
f^7*g^3 + 78*d^2*e^9*f^6*g^4 - 480*d^3*e^8*f^5*g^5 + 312*d^4*e^7*f^4*g^6 + 330*d^5*e^6*f^3*g^7 - 419*d^6*e^5*f
^2*g^8 + 90*d^7*e^4*f*g^9 + 15*d^8*e^3*g^10)*x^5 - (28*e^11*f^9*g + 78*d*e^10*f^8*g^2 - 24*d^2*e^9*f^7*g^3 - 1
194*d^3*e^8*f^6*g^4 + 2064*d^4*e^7*f^5*g^5 - 276*d^5*e^6*f^4*g^6 - 1828*d^6*e^5*f^3*g^7 + 1437*d^7*e^4*f^2*g^8
- 240*d^8*e^3*f*g^9 - 45*d^9*e^2*g^10)*x^4 - (14*e^11*f^10 - 24*d*e^10*f^9*g - 240*d^2*e^9*f^8*g^2 - 768*d^3*
e^8*f^7*g^3 + 3426*d^4*e^7*f^6*g^4 - 2982*d^5*e^6*f^5*g^5 - 1463*d^6*e^5*f^4*g^6 + 3594*d^7*e^4*f^3*g^7 - 1782
*d^8*e^3*f^2*g^8 + 180*d^9*e^2*f*g^9 + 45*d^10*e*g^10)*x^3 + (42*d*e^10*f^10 + 96*d^2*e^9*f^9*g - 112*d^3*e^8*
f^8*g^2 - 1848*d^4*e^7*f^7*g^3 + 3894*d^5*e^6*f^6*g^4 - 1362*d^6*e^5*f^5*g^5 - 2925*d^7*e^4*f^4*g^6 + 3114*d^8
*e^3*f^3*g^7 - 914*d^9*e^2*f^2*g^8 + 15*d^11*g^10)*x^2 - 15*(20*d^6*e^4*f^6*g^3 - 30*d^7*e^3*f^5*g^4 + 13*d^8*
e^2*f^4*g^5 - (20*d^3*e^7*f^4*g^5 - 30*d^4*e^6*f^3*g^6 + 13*d^5*e^5*f^2*g^7)*x^5 - (40*d^3*e^7*f^5*g^4 - 120*d
^4*e^6*f^4*g^5 + 116*d^5*e^5*f^3*g^6 - 39*d^6*e^4*f^2*g^7)*x^4 - (20*d^3*e^7*f^6*g^3 - 150*d^4*e^6*f^5*g^4 + 2
53*d^5*e^5*f^4*g^5 - 168*d^6*e^4*f^3*g^6 + 39*d^7*e^3*f^2*g^7)*x^3 + (60*d^4*e^6*f^6*g^3 - 210*d^5*e^5*f^5*g^4
+ 239*d^6*e^4*f^4*g^5 - 108*d^7*e^3*f^3*g^6 + 13*d^8*e^2*f^2*g^7)*x^2 - (60*d^5*e^5*f^6*g^3 - 130*d^6*e^4*f^5
*g^4 + 99*d^7*e^3*f^4*g^5 - 26*d^8*e^2*f^3*g^6)*x)*sqrt(-e^2*f^2 + d^2*g^2)*log((d*e^2*f*g*x + d^3*g^2 - sqrt(
-e^2*f^2 + d^2*g^2)*(e^2*f*x + d^2*g + sqrt(-e^2*x^2 + d^2)*d*g) - (e^2*f^2 - d^2*g^2)*sqrt(-e^2*x^2 + d^2))/(
g*x + f)) - (42*d^2*e^9*f^10 + 152*d^3*e^8*f^9*g + 114*d^4*e^7*f^8*g^2 - 1596*d^5*e^6*f^7*g^3 + 1896*d^6*e^5*f
^6*g^4 + 366*d^7*e^4*f^5*g^5 - 1917*d^8*e^3*f^4*g^6 + 1108*d^9*e^2*f^3*g^7 - 135*d^10*e*f^2*g^8 - 30*d^11*f*g^
9)*x + (14*d^2*e^8*f^10 + 60*d^3*e^7*f^9*g + 78*d^4*e^6*f^8*g^2 - 480*d^5*e^5*f^7*g^3 + 312*d^6*e^4*f^6*g^4 +
330*d^7*e^3*f^5*g^5 - 419*d^8*e^2*f^4*g^6 + 90*d^9*e*f^3*g^7 + 15*d^10*f^2*g^8 + (4*e^10*f^8*g^2 + 30*d*e^9*f^
7*g^3 + 138*d^2*e^8*f^6*g^4 - 555*d^3*e^7*f^5*g^5 + 162*d^4*e^6*f^4*g^6 + 525*d^5*e^5*f^3*g^7 - 304*d^6*e^4*f^
2*g^8)*x^4 + (8*e^10*f^9*g + 48*d*e^9*f^8*g^2 + 186*d^2*e^8*f^7*g^3 - 1224*d^3*e^7*f^6*g^4 + 1539*d^4*e^6*f^5*
g^5 + 459*d^5*e^5*f^4*g^6 - 1733*d^6*e^4*f^3*g^7 + 717*d^7*e^3*f^2*g^8)*x^3 + (4*e^10*f^10 + 6*d*e^9*f^9*g - 2
8*d^2*e^8*f^8*g^2 - 828*d^3*e^7*f^7*g^3 + 2400*d^4*e^6*f^6*g^4 - 1197*d^5*e^5*f^5*g^5 - 1897*d^6*e^4*f^4*g^6 +
2019*d^7*e^3*f^3*g^7 - 479*d^8*e^2*f^2*g^8)*x^2 - (12*d*e^9*f^10 + 62*d^2*e^8*f^9*g + 114*d^3*e^7*f^8*g^2 - 1
056*d^4*e^6*f^7*g^3 + 1626*d^5*e^5*f^6*g^4 + 81*d^6*e^4*f^5*g^5 - 1707*d^7*e^3*f^4*g^6 + 913*d^8*e^2*f^3*g^7 -
45*d^9*e*f^2*g^8)*x)*sqrt(-e^2*x^2 + d^2))/(d^6*e^9*f^13 + 3*d^7*e^8*f^12*g - 8*d^9*e^6*f^10*g^3 - 6*d^10*e^5
*f^9*g^4 + 6*d^11*e^4*f^8*g^5 + 8*d^12*e^3*f^7*g^6 - 3*d^14*e*f^5*g^8 - d^15*f^4*g^9 - (d^3*e^12*f^11*g^2 + 3*
d^4*e^11*f^10*g^3 - 8*d^6*e^9*f^8*g^5 - 6*d^7*e^8*f^7*g^6 + 6*d^8*e^7*f^6*g^7 + 8*d^9*e^6*f^5*g^8 - 3*d^11*e^4
*f^3*g^10 - d^12*e^3*f^2*g^11)*x^5 - (2*d^3*e^12*f^12*g + 3*d^4*e^11*f^11*g^2 - 9*d^5*e^10*f^10*g^3 - 16*d^6*e
^9*f^9*g^4 + 12*d^7*e^8*f^8*g^5 + 30*d^8*e^7*f^7*g^6 - 2*d^9*e^6*f^6*g^7 - 24*d^10*e^5*f^5*g^8 - 6*d^11*e^4*f^
4*g^9 + 7*d^12*e^3*f^3*g^10 + 3*d^13*e^2*f^2*g^11)*x^4 - (d^3*e^12*f^13 - 3*d^4*e^11*f^12*g - 15*d^5*e^10*f^11
*g^2 + d^6*e^9*f^10*g^3 + 42*d^7*e^8*f^9*g^4 + 18*d^8*e^7*f^8*g^5 - 46*d^9*e^6*f^7*g^6 - 30*d^10*e^5*f^6*g^7 +
21*d^11*e^4*f^5*g^8 + 17*d^12*e^3*f^4*g^9 - 3*d^13*e^2*f^3*g^10 - 3*d^14*e*f^2*g^11)*x^3 + (3*d^4*e^11*f^13 +
3*d^5*e^10*f^12*g - 17*d^6*e^9*f^11*g^2 - 21*d^7*e^8*f^10*g^3 + 30*d^8*e^7*f^9*g^4 + 46*d^9*e^6*f^8*g^5 - 18*
d^10*e^5*f^7*g^6 - 42*d^11*e^4*f^6*g^7 - d^12*e^3*f^5*g^8 + 15*d^13*e^2*f^4*g^9 + 3*d^14*e*f^3*g^10 - d^15*f^2
*g^11)*x^2 - (3*d^5*e^10*f^13 + 7*d^6*e^9*f^12*g - 6*d^7*e^8*f^11*g^2 - 24*d^8*e^7*f^10*g^3 - 2*d^9*e^6*f^9*g^
4 + 30*d^10*e^5*f^8*g^5 + 12*d^11*e^4*f^7*g^6 - 16*d^12*e^3*f^6*g^7 - 9*d^13*e^2*f^5*g^8 + 3*d^14*e*f^4*g^9 +
2*d^15*f^3*g^10)*x), 1/30*(14*d^3*e^8*f^10 + 60*d^4*e^7*f^9*g + 78*d^5*e^6*f^8*g^2 - 480*d^6*e^5*f^7*g^3 + 312
*d^7*e^4*f^6*g^4 + 330*d^8*e^3*f^5*g^5 - 419*d^9*e^2*f^4*g^6 + 90*d^10*e*f^3*g^7 + 15*d^11*f^2*g^8 - (14*e^11*
f^8*g^2 + 60*d*e^10*f^7*g^3 + 78*d^2*e^9*f^6*g^4 - 480*d^3*e^8*f^5*g^5 + 312*d^4*e^7*f^4*g^6 + 330*d^5*e^6*f^3
*g^7 - 419*d^6*e^5*f^2*g^8 + 90*d^7*e^4*f*g^9 + 15*d^8*e^3*g^10)*x^5 - (28*e^11*f^9*g + 78*d*e^10*f^8*g^2 - 24
*d^2*e^9*f^7*g^3 - 1194*d^3*e^8*f^6*g^4 + 2064*d^4*e^7*f^5*g^5 - 276*d^5*e^6*f^4*g^6 - 1828*d^6*e^5*f^3*g^7 +
1437*d^7*e^4*f^2*g^8 - 240*d^8*e^3*f*g^9 - 45*d^9*e^2*g^10)*x^4 - (14*e^11*f^10 - 24*d*e^10*f^9*g - 240*d^2*e^
9*f^8*g^2 - 768*d^3*e^8*f^7*g^3 + 3426*d^4*e^7*f^6*g^4 - 2982*d^5*e^6*f^5*g^5 - 1463*d^6*e^5*f^4*g^6 + 3594*d^
7*e^4*f^3*g^7 - 1782*d^8*e^3*f^2*g^8 + 180*d^9*e^2*f*g^9 + 45*d^10*e*g^10)*x^3 + (42*d*e^10*f^10 + 96*d^2*e^9*
f^9*g - 112*d^3*e^8*f^8*g^2 - 1848*d^4*e^7*f^7*g^3 + 3894*d^5*e^6*f^6*g^4 - 1362*d^6*e^5*f^5*g^5 - 2925*d^7*e^
4*f^4*g^6 + 3114*d^8*e^3*f^3*g^7 - 914*d^9*e^2*f^2*g^8 + 15*d^11*g^10)*x^2 + 30*(20*d^6*e^4*f^6*g^3 - 30*d^7*e
^3*f^5*g^4 + 13*d^8*e^2*f^4*g^5 - (20*d^3*e^7*f^4*g^5 - 30*d^4*e^6*f^3*g^6 + 13*d^5*e^5*f^2*g^7)*x^5 - (40*d^3
*e^7*f^5*g^4 - 120*d^4*e^6*f^4*g^5 + 116*d^5*e^5*f^3*g^6 - 39*d^6*e^4*f^2*g^7)*x^4 - (20*d^3*e^7*f^6*g^3 - 150
*d^4*e^6*f^5*g^4 + 253*d^5*e^5*f^4*g^5 - 168*d^6*e^4*f^3*g^6 + 39*d^7*e^3*f^2*g^7)*x^3 + (60*d^4*e^6*f^6*g^3 -
210*d^5*e^5*f^5*g^4 + 239*d^6*e^4*f^4*g^5 - 108*d^7*e^3*f^3*g^6 + 13*d^8*e^2*f^2*g^7)*x^2 - (60*d^5*e^5*f^6*g
^3 - 130*d^6*e^4*f^5*g^4 + 99*d^7*e^3*f^4*g^5 - 26*d^8*e^2*f^3*g^6)*x)*sqrt(e^2*f^2 - d^2*g^2)*arctan((d*g*x +
d*f - sqrt(-e^2*x^2 + d^2)*f)/(sqrt(e^2*f^2 - d^2*g^2)*x)) - (42*d^2*e^9*f^10 + 152*d^3*e^8*f^9*g + 114*d^4*e
^7*f^8*g^2 - 1596*d^5*e^6*f^7*g^3 + 1896*d^6*e^5*f^6*g^4 + 366*d^7*e^4*f^5*g^5 - 1917*d^8*e^3*f^4*g^6 + 1108*d
^9*e^2*f^3*g^7 - 135*d^10*e*f^2*g^8 - 30*d^11*f*g^9)*x + (14*d^2*e^8*f^10 + 60*d^3*e^7*f^9*g + 78*d^4*e^6*f^8*
g^2 - 480*d^5*e^5*f^7*g^3 + 312*d^6*e^4*f^6*g^4 + 330*d^7*e^3*f^5*g^5 - 419*d^8*e^2*f^4*g^6 + 90*d^9*e*f^3*g^7
+ 15*d^10*f^2*g^8 + (4*e^10*f^8*g^2 + 30*d*e^9*f^7*g^3 + 138*d^2*e^8*f^6*g^4 - 555*d^3*e^7*f^5*g^5 + 162*d^4*
e^6*f^4*g^6 + 525*d^5*e^5*f^3*g^7 - 304*d^6*e^4*f^2*g^8)*x^4 + (8*e^10*f^9*g + 48*d*e^9*f^8*g^2 + 186*d^2*e^8*
f^7*g^3 - 1224*d^3*e^7*f^6*g^4 + 1539*d^4*e^6*f^5*g^5 + 459*d^5*e^5*f^4*g^6 - 1733*d^6*e^4*f^3*g^7 + 717*d^7*e
^3*f^2*g^8)*x^3 + (4*e^10*f^10 + 6*d*e^9*f^9*g - 28*d^2*e^8*f^8*g^2 - 828*d^3*e^7*f^7*g^3 + 2400*d^4*e^6*f^6*g
^4 - 1197*d^5*e^5*f^5*g^5 - 1897*d^6*e^4*f^4*g^6 + 2019*d^7*e^3*f^3*g^7 - 479*d^8*e^2*f^2*g^8)*x^2 - (12*d*e^9
*f^10 + 62*d^2*e^8*f^9*g + 114*d^3*e^7*f^8*g^2 - 1056*d^4*e^6*f^7*g^3 + 1626*d^5*e^5*f^6*g^4 + 81*d^6*e^4*f^5*
g^5 - 1707*d^7*e^3*f^4*g^6 + 913*d^8*e^2*f^3*g^7 - 45*d^9*e*f^2*g^8)*x)*sqrt(-e^2*x^2 + d^2))/(d^6*e^9*f^13 +
3*d^7*e^8*f^12*g - 8*d^9*e^6*f^10*g^3 - 6*d^10*e^5*f^9*g^4 + 6*d^11*e^4*f^8*g^5 + 8*d^12*e^3*f^7*g^6 - 3*d^14*
e*f^5*g^8 - d^15*f^4*g^9 - (d^3*e^12*f^11*g^2 + 3*d^4*e^11*f^10*g^3 - 8*d^6*e^9*f^8*g^5 - 6*d^7*e^8*f^7*g^6 +
6*d^8*e^7*f^6*g^7 + 8*d^9*e^6*f^5*g^8 - 3*d^11*e^4*f^3*g^10 - d^12*e^3*f^2*g^11)*x^5 - (2*d^3*e^12*f^12*g + 3*
d^4*e^11*f^11*g^2 - 9*d^5*e^10*f^10*g^3 - 16*d^6*e^9*f^9*g^4 + 12*d^7*e^8*f^8*g^5 + 30*d^8*e^7*f^7*g^6 - 2*d^9
*e^6*f^6*g^7 - 24*d^10*e^5*f^5*g^8 - 6*d^11*e^4*f^4*g^9 + 7*d^12*e^3*f^3*g^10 + 3*d^13*e^2*f^2*g^11)*x^4 - (d^
3*e^12*f^13 - 3*d^4*e^11*f^12*g - 15*d^5*e^10*f^11*g^2 + d^6*e^9*f^10*g^3 + 42*d^7*e^8*f^9*g^4 + 18*d^8*e^7*f^
8*g^5 - 46*d^9*e^6*f^7*g^6 - 30*d^10*e^5*f^6*g^7 + 21*d^11*e^4*f^5*g^8 + 17*d^12*e^3*f^4*g^9 - 3*d^13*e^2*f^3*
g^10 - 3*d^14*e*f^2*g^11)*x^3 + (3*d^4*e^11*f^13 + 3*d^5*e^10*f^12*g - 17*d^6*e^9*f^11*g^2 - 21*d^7*e^8*f^10*g
^3 + 30*d^8*e^7*f^9*g^4 + 46*d^9*e^6*f^8*g^5 - 18*d^10*e^5*f^7*g^6 - 42*d^11*e^4*f^6*g^7 - d^12*e^3*f^5*g^8 +
15*d^13*e^2*f^4*g^9 + 3*d^14*e*f^3*g^10 - d^15*f^2*g^11)*x^2 - (3*d^5*e^10*f^13 + 7*d^6*e^9*f^12*g - 6*d^7*e^8
*f^11*g^2 - 24*d^8*e^7*f^10*g^3 - 2*d^9*e^6*f^9*g^4 + 30*d^10*e^5*f^8*g^5 + 12*d^11*e^4*f^7*g^6 - 16*d^12*e^3*
f^6*g^7 - 9*d^13*e^2*f^5*g^8 + 3*d^14*e*f^4*g^9 + 2*d^15*f^3*g^10)*x)]
________________________________________________________________________________________
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((e*x+d)**3/(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 2.71614, size = 8123, normalized size = 20.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3/(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
[Out]
-(13*d^9*g^12*e^8 - 69*d^8*f*g^11*e^9 + 123*d^7*f^2*g^10*e^10 - 25*d^6*f^3*g^9*e^11 - 195*d^5*f^4*g^8*e^12 + 2
37*d^4*f^5*g^7*e^13 - 31*d^3*f^6*g^6*e^14 - 123*d^2*f^7*g^5*e^15 + 90*d*f^8*g^4*e^16 - 20*f^9*g^3*e^17)*arctan
((d*g*e + (d*e + sqrt(-x^2*e^2 + d^2)*e)*f/x)/sqrt(-d^2*g^2*e^2 + f^2*e^4))/((d^14*g^14*e^5 - 7*d^12*f^2*g^12*
e^7 + 21*d^10*f^4*g^10*e^9 - 35*d^8*f^6*g^8*e^11 + 35*d^6*f^8*g^6*e^13 - 21*d^4*f^10*g^4*e^15 + 7*d^2*f^12*g^2
*e^17 - f^14*e^19)*sqrt(-d^2*g^2*e^2 + f^2*e^4)) - 1/15*sqrt(-x^2*e^2 + d^2)*((((((107*d^28*g^27*e^11 + 2694*d
^27*f*g^26*e^12 + 32577*d^26*f^2*g^25*e^13 + 251850*d^25*f^3*g^24*e^14 + 1397850*d^24*f^4*g^23*e^15 + 5929860*
d^23*f^5*g^22*e^16 + 19984470*d^22*f^6*g^21*e^17 + 54906060*d^21*f^7*g^20*e^18 + 125216025*d^20*f^8*g^19*e^19
+ 240109650*d^19*f^9*g^18*e^20 + 390736995*d^18*f^10*g^17*e^21 + 543134190*d^17*f^11*g^16*e^22 + 647660220*d^1
6*f^12*g^15*e^23 + 664152600*d^15*f^13*g^14*e^24 + 586148100*d^14*f^14*g^13*e^25 + 444848520*d^13*f^15*g^12*e^
26 + 289619565*d^12*f^16*g^11*e^27 + 161082570*d^11*f^17*g^10*e^28 + 76070775*d^10*f^18*g^9*e^29 + 30246150*d^
9*f^19*g^8*e^30 + 10011210*d^8*f^20*g^7*e^31 + 2717220*d^7*f^21*g^6*e^32 + 592710*d^6*f^22*g^5*e^33 + 101100*d
^5*f^23*g^4*e^34 + 12975*d^4*f^24*g^3*e^35 + 1182*d^3*f^25*g^2*e^36 + 69*d^2*f^26*g*e^37 + 2*d*f^27*e^38)*x/(d
^34*g^30*e^4 + 30*d^33*f*g^29*e^5 + 435*d^32*f^2*g^28*e^6 + 4060*d^31*f^3*g^27*e^7 + 27405*d^30*f^4*g^26*e^8 +
142506*d^29*f^5*g^25*e^9 + 593775*d^28*f^6*g^24*e^10 + 2035800*d^27*f^7*g^23*e^11 + 5852925*d^26*f^8*g^22*e^1
2 + 14307150*d^25*f^9*g^21*e^13 + 30045015*d^24*f^10*g^20*e^14 + 54627300*d^23*f^11*g^19*e^15 + 86493225*d^22*
f^12*g^18*e^16 + 119759850*d^21*f^13*g^17*e^17 + 145422675*d^20*f^14*g^16*e^18 + 155117520*d^19*f^15*g^15*e^19
+ 145422675*d^18*f^16*g^14*e^20 + 119759850*d^17*f^17*g^13*e^21 + 86493225*d^16*f^18*g^12*e^22 + 54627300*d^1
5*f^19*g^11*e^23 + 30045015*d^14*f^20*g^10*e^24 + 14307150*d^13*f^21*g^9*e^25 + 5852925*d^12*f^22*g^8*e^26 + 2
035800*d^11*f^23*g^7*e^27 + 593775*d^10*f^24*g^6*e^28 + 142506*d^9*f^25*g^5*e^29 + 27405*d^8*f^26*g^4*e^30 + 4
060*d^7*f^27*g^3*e^31 + 435*d^6*f^28*g^2*e^32 + 30*d^5*f^29*g*e^33 + d^4*f^30*e^34) + 90*(d^29*g^27*e^10 + 25*
d^28*f*g^26*e^11 + 300*d^27*f^2*g^25*e^12 + 2300*d^26*f^3*g^24*e^13 + 12650*d^25*f^4*g^23*e^14 + 53130*d^24*f^
5*g^22*e^15 + 177100*d^23*f^6*g^21*e^16 + 480700*d^22*f^7*g^20*e^17 + 1081575*d^21*f^8*g^19*e^18 + 2042975*d^2
0*f^9*g^18*e^19 + 3268760*d^19*f^10*g^17*e^20 + 4457400*d^18*f^11*g^16*e^21 + 5200300*d^17*f^12*g^15*e^22 + 52
00300*d^16*f^13*g^14*e^23 + 4457400*d^15*f^14*g^13*e^24 + 3268760*d^14*f^15*g^12*e^25 + 2042975*d^13*f^16*g^11
*e^26 + 1081575*d^12*f^17*g^10*e^27 + 480700*d^11*f^18*g^9*e^28 + 177100*d^10*f^19*g^8*e^29 + 53130*d^9*f^20*g
^7*e^30 + 12650*d^8*f^21*g^6*e^31 + 2300*d^7*f^22*g^5*e^32 + 300*d^6*f^23*g^4*e^33 + 25*d^5*f^24*g^3*e^34 + d^
4*f^25*g^2*e^35)/(d^34*g^30*e^4 + 30*d^33*f*g^29*e^5 + 435*d^32*f^2*g^28*e^6 + 4060*d^31*f^3*g^27*e^7 + 27405*
d^30*f^4*g^26*e^8 + 142506*d^29*f^5*g^25*e^9 + 593775*d^28*f^6*g^24*e^10 + 2035800*d^27*f^7*g^23*e^11 + 585292
5*d^26*f^8*g^22*e^12 + 14307150*d^25*f^9*g^21*e^13 + 30045015*d^24*f^10*g^20*e^14 + 54627300*d^23*f^11*g^19*e^
15 + 86493225*d^22*f^12*g^18*e^16 + 119759850*d^21*f^13*g^17*e^17 + 145422675*d^20*f^14*g^16*e^18 + 155117520*
d^19*f^15*g^15*e^19 + 145422675*d^18*f^16*g^14*e^20 + 119759850*d^17*f^17*g^13*e^21 + 86493225*d^16*f^18*g^12*
e^22 + 54627300*d^15*f^19*g^11*e^23 + 30045015*d^14*f^20*g^10*e^24 + 14307150*d^13*f^21*g^9*e^25 + 5852925*d^1
2*f^22*g^8*e^26 + 2035800*d^11*f^23*g^7*e^27 + 593775*d^10*f^24*g^6*e^28 + 142506*d^9*f^25*g^5*e^29 + 27405*d^
8*f^26*g^4*e^30 + 4060*d^7*f^27*g^3*e^31 + 435*d^6*f^28*g^2*e^32 + 30*d^5*f^29*g*e^33 + d^4*f^30*e^34))*x - 5*
(49*d^30*g^27*e^9 + 1239*d^29*f*g^26*e^10 + 15051*d^28*f^2*g^25*e^11 + 116925*d^27*f^3*g^24*e^12 + 652350*d^26
*f^4*g^23*e^13 + 2782770*d^25*f^5*g^22*e^14 + 9434370*d^24*f^6*g^21*e^15 + 26086830*d^23*f^7*g^20*e^16 + 59904
075*d^22*f^8*g^19*e^17 + 115728525*d^21*f^9*g^18*e^18 + 189852465*d^20*f^10*g^17*e^19 + 266218215*d^19*f^11*g^
16*e^20 + 320487060*d^18*f^12*g^15*e^21 + 332076300*d^17*f^13*g^14*e^22 + 296417100*d^16*f^14*g^13*e^23 + 2277
73140*d^15*f^15*g^12*e^24 + 150325815*d^14*f^16*g^11*e^25 + 84867585*d^13*f^17*g^10*e^26 + 40739325*d^12*f^18*
g^9*e^27 + 16489275*d^11*f^19*g^8*e^28 + 5563470*d^10*f^20*g^7*e^29 + 1540770*d^9*f^21*g^6*e^30 + 342930*d^8*f
^22*g^5*e^31 + 59550*d^7*f^23*g^4*e^32 + 7725*d^6*f^24*g^3*e^33 + 699*d^5*f^25*g^2*e^34 + 39*d^4*f^26*g*e^35 +
d^3*f^27*e^36)/(d^34*g^30*e^4 + 30*d^33*f*g^29*e^5 + 435*d^32*f^2*g^28*e^6 + 4060*d^31*f^3*g^27*e^7 + 27405*d
^30*f^4*g^26*e^8 + 142506*d^29*f^5*g^25*e^9 + 593775*d^28*f^6*g^24*e^10 + 2035800*d^27*f^7*g^23*e^11 + 5852925
*d^26*f^8*g^22*e^12 + 14307150*d^25*f^9*g^21*e^13 + 30045015*d^24*f^10*g^20*e^14 + 54627300*d^23*f^11*g^19*e^1
5 + 86493225*d^22*f^12*g^18*e^16 + 119759850*d^21*f^13*g^17*e^17 + 145422675*d^20*f^14*g^16*e^18 + 155117520*d
^19*f^15*g^15*e^19 + 145422675*d^18*f^16*g^14*e^20 + 119759850*d^17*f^17*g^13*e^21 + 86493225*d^16*f^18*g^12*e
^22 + 54627300*d^15*f^19*g^11*e^23 + 30045015*d^14*f^20*g^10*e^24 + 14307150*d^13*f^21*g^9*e^25 + 5852925*d^12
*f^22*g^8*e^26 + 2035800*d^11*f^23*g^7*e^27 + 593775*d^10*f^24*g^6*e^28 + 142506*d^9*f^25*g^5*e^29 + 27405*d^8
*f^26*g^4*e^30 + 4060*d^7*f^27*g^3*e^31 + 435*d^6*f^28*g^2*e^32 + 30*d^5*f^29*g*e^33 + d^4*f^30*e^34))*x - 5*(
41*d^31*g^27*e^8 + 1029*d^30*f*g^26*e^9 + 12399*d^29*f^2*g^25*e^10 + 95475*d^28*f^3*g^24*e^11 + 527550*d^27*f^
4*g^23*e^12 + 2226630*d^26*f^5*g^22*e^13 + 7460970*d^25*f^6*g^21*e^14 + 20363970*d^24*f^7*g^20*e^15 + 46090275
*d^23*f^8*g^19*e^16 + 87607575*d^22*f^9*g^18*e^17 + 141109485*d^21*f^10*g^17*e^18 + 193785465*d^20*f^11*g^16*e
^19 + 227773140*d^19*f^12*g^15*e^20 + 229556100*d^18*f^13*g^14*e^21 + 198354300*d^17*f^14*g^13*e^22 + 14664846
0*d^16*f^15*g^12*e^23 + 92379615*d^15*f^16*g^11*e^24 + 49247715*d^14*f^17*g^10*e^25 + 21992025*d^13*f^18*g^9*e
^26 + 8102325*d^12*f^19*g^8*e^27 + 2406030*d^11*f^20*g^7*e^28 + 554070*d^10*f^21*g^6*e^29 + 91770*d^9*f^22*g^5
*e^30 + 8850*d^8*f^23*g^4*e^31 - 75*d^7*f^24*g^3*e^32 - 159*d^6*f^25*g^2*e^33 - 21*d^5*f^26*g*e^34 - d^4*f^27*
e^35)/(d^34*g^30*e^4 + 30*d^33*f*g^29*e^5 + 435*d^32*f^2*g^28*e^6 + 4060*d^31*f^3*g^27*e^7 + 27405*d^30*f^4*g^
26*e^8 + 142506*d^29*f^5*g^25*e^9 + 593775*d^28*f^6*g^24*e^10 + 2035800*d^27*f^7*g^23*e^11 + 5852925*d^26*f^8*
g^22*e^12 + 14307150*d^25*f^9*g^21*e^13 + 30045015*d^24*f^10*g^20*e^14 + 54627300*d^23*f^11*g^19*e^15 + 864932
25*d^22*f^12*g^18*e^16 + 119759850*d^21*f^13*g^17*e^17 + 145422675*d^20*f^14*g^16*e^18 + 155117520*d^19*f^15*g
^15*e^19 + 145422675*d^18*f^16*g^14*e^20 + 119759850*d^17*f^17*g^13*e^21 + 86493225*d^16*f^18*g^12*e^22 + 5462
7300*d^15*f^19*g^11*e^23 + 30045015*d^14*f^20*g^10*e^24 + 14307150*d^13*f^21*g^9*e^25 + 5852925*d^12*f^22*g^8*
e^26 + 2035800*d^11*f^23*g^7*e^27 + 593775*d^10*f^24*g^6*e^28 + 142506*d^9*f^25*g^5*e^29 + 27405*d^8*f^26*g^4*
e^30 + 4060*d^7*f^27*g^3*e^31 + 435*d^6*f^28*g^2*e^32 + 30*d^5*f^29*g*e^33 + d^4*f^30*e^34))*x + 15*(10*d^32*g
^27*e^7 + 255*d^31*f*g^26*e^8 + 3126*d^30*f^2*g^25*e^9 + 24525*d^29*f^3*g^24*e^10 + 138300*d^28*f^4*g^23*e^11
+ 596850*d^27*f^5*g^22*e^12 + 2049300*d^26*f^6*g^21*e^13 + 5745630*d^25*f^7*g^20*e^14 + 13396350*d^24*f^8*g^19
*e^15 + 26318325*d^23*f^9*g^18*e^16 + 43984050*d^22*f^10*g^17*e^17 + 62960775*d^21*f^11*g^16*e^18 + 77558760*d
^20*f^12*g^15*e^19 + 82461900*d^19*f^13*g^14*e^20 + 75775800*d^18*f^14*g^13*e^21 + 60174900*d^17*f^15*g^12*e^2
2 + 41230950*d^16*f^16*g^11*e^23 + 24299385*d^15*f^17*g^10*e^24 + 12257850*d^14*f^18*g^9*e^25 + 5256075*d^13*f
^19*g^8*e^26 + 1897500*d^12*f^20*g^7*e^27 + 569250*d^11*f^21*g^6*e^28 + 139380*d^10*f^22*g^5*e^29 + 27150*d^9*
f^23*g^4*e^30 + 4050*d^8*f^24*g^3*e^31 + 435*d^7*f^25*g^2*e^32 + 30*d^6*f^26*g*e^33 + d^5*f^27*e^34)/(d^34*g^3
0*e^4 + 30*d^33*f*g^29*e^5 + 435*d^32*f^2*g^28*e^6 + 4060*d^31*f^3*g^27*e^7 + 27405*d^30*f^4*g^26*e^8 + 142506
*d^29*f^5*g^25*e^9 + 593775*d^28*f^6*g^24*e^10 + 2035800*d^27*f^7*g^23*e^11 + 5852925*d^26*f^8*g^22*e^12 + 143
07150*d^25*f^9*g^21*e^13 + 30045015*d^24*f^10*g^20*e^14 + 54627300*d^23*f^11*g^19*e^15 + 86493225*d^22*f^12*g^
18*e^16 + 119759850*d^21*f^13*g^17*e^17 + 145422675*d^20*f^14*g^16*e^18 + 155117520*d^19*f^15*g^15*e^19 + 1454
22675*d^18*f^16*g^14*e^20 + 119759850*d^17*f^17*g^13*e^21 + 86493225*d^16*f^18*g^12*e^22 + 54627300*d^15*f^19*
g^11*e^23 + 30045015*d^14*f^20*g^10*e^24 + 14307150*d^13*f^21*g^9*e^25 + 5852925*d^12*f^22*g^8*e^26 + 2035800*
d^11*f^23*g^7*e^27 + 593775*d^10*f^24*g^6*e^28 + 142506*d^9*f^25*g^5*e^29 + 27405*d^8*f^26*g^4*e^30 + 4060*d^7
*f^27*g^3*e^31 + 435*d^6*f^28*g^2*e^32 + 30*d^5*f^29*g*e^33 + d^4*f^30*e^34))*x + (127*d^33*g^27*e^6 + 3219*d^
32*f*g^26*e^7 + 39207*d^31*f^2*g^25*e^8 + 305475*d^30*f^3*g^24*e^9 + 1709850*d^29*f^4*g^23*e^10 + 7320210*d^28
*f^5*g^22*e^11 + 24917970*d^27*f^6*g^21*e^12 + 69213210*d^26*f^7*g^20*e^13 + 159750525*d^25*f^8*g^19*e^14 + 31
0412025*d^24*f^9*g^18*e^15 + 512594445*d^23*f^10*g^17*e^16 + 724216065*d^22*f^11*g^16*e^17 + 879445020*d^21*f^
12*g^15*e^18 + 920453100*d^20*f^13*g^14*e^19 + 831305100*d^19*f^14*g^13*e^20 + 647660220*d^18*f^15*g^12*e^21 +
434485065*d^17*f^16*g^11*e^22 + 250132245*d^16*f^17*g^10*e^23 + 122939025*d^15*f^18*g^9*e^24 + 51213525*d^14*
f^19*g^8*e^25 + 17904810*d^13*f^20*g^7*e^26 + 5183970*d^12*f^21*g^6*e^27 + 1220610*d^11*f^22*g^5*e^28 + 227850
*d^10*f^23*g^4*e^29 + 32475*d^9*f^24*g^3*e^30 + 3327*d^8*f^25*g^2*e^31 + 219*d^7*f^26*g*e^32 + 7*d^6*f^27*e^33
)/(d^34*g^30*e^4 + 30*d^33*f*g^29*e^5 + 435*d^32*f^2*g^28*e^6 + 4060*d^31*f^3*g^27*e^7 + 27405*d^30*f^4*g^26*e
^8 + 142506*d^29*f^5*g^25*e^9 + 593775*d^28*f^6*g^24*e^10 + 2035800*d^27*f^7*g^23*e^11 + 5852925*d^26*f^8*g^22
*e^12 + 14307150*d^25*f^9*g^21*e^13 + 30045015*d^24*f^10*g^20*e^14 + 54627300*d^23*f^11*g^19*e^15 + 86493225*d
^22*f^12*g^18*e^16 + 119759850*d^21*f^13*g^17*e^17 + 145422675*d^20*f^14*g^16*e^18 + 155117520*d^19*f^15*g^15*
e^19 + 145422675*d^18*f^16*g^14*e^20 + 119759850*d^17*f^17*g^13*e^21 + 86493225*d^16*f^18*g^12*e^22 + 54627300
*d^15*f^19*g^11*e^23 + 30045015*d^14*f^20*g^10*e^24 + 14307150*d^13*f^21*g^9*e^25 + 5852925*d^12*f^22*g^8*e^26
+ 2035800*d^11*f^23*g^7*e^27 + 593775*d^10*f^24*g^6*e^28 + 142506*d^9*f^25*g^5*e^29 + 27405*d^8*f^26*g^4*e^30
+ 4060*d^7*f^27*g^3*e^31 + 435*d^6*f^28*g^2*e^32 + 30*d^5*f^29*g*e^33 + d^4*f^30*e^34))/(x^2*e^2 - d^2)^3 + (
2*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^10*g^13*e^3/x^2 + 2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^9*f*g^12*e^6/x + 6*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^9*f*g^12*e^4/x^2 + 2*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^9*f*g^12*e^2/x^3 + d
^8*f^2*g^11*e^9 + 12*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^8*f^2*g^11*e^7/x - 51*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d
^8*f^2*g^11*e^5/x^2 + 3*d^7*f^3*g^10*e^10 - 79*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^7*f^3*g^10*e^8/x + 91*(d*e + s
qrt(-x^2*e^2 + d^2)*e)^2*d^7*f^3*g^10*e^6/x^2 - 25*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^7*f^3*g^10*e^4/x^3 - 26*
d^6*f^4*g^9*e^11 + 127*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^6*f^4*g^9*e^9/x - 48*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*
d^6*f^4*g^9*e^7/x^2 + 49*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^6*f^4*g^9*e^5/x^3 + 44*d^5*f^5*g^8*e^12 - 28*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)*d^5*f^5*g^8*e^10/x - 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^5*f^5*g^8*e^8/x^2 - 16*(d
*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^5*f^5*g^8*e^6/x^3 - 11*d^4*f^6*g^7*e^13 - 110*(d*e + sqrt(-x^2*e^2 + d^2)*e)*
d^4*f^6*g^7*e^11/x + 61*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4*f^6*g^7*e^9/x^2 - 38*(d*e + sqrt(-x^2*e^2 + d^2)*
e)^3*d^4*f^6*g^7*e^7/x^3 - 37*d^3*f^7*g^6*e^14 + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*f^7*g^6*e^12/x - 57*(d
*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^3*f^7*g^6*e^10/x^2 + 39*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^3*f^7*g^6*e^8/x^3
+ 36*d^2*f^8*g^5*e^15 - 29*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*f^8*g^5*e^13/x + 36*(d*e + sqrt(-x^2*e^2 + d^2)*
e)^2*d^2*f^8*g^5*e^11/x^2 - 11*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2*f^8*g^5*e^9/x^3 - 10*d*f^9*g^4*e^16 - 10*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d*f^9*g^4*e^12/x^2)/((d^12*f^2*g^12*e^5 - 6*d^10*f^4*g^10*e^7 + 15*d^8*f^6*g^8
*e^9 - 20*d^6*f^8*g^6*e^11 + 15*d^4*f^10*g^4*e^13 - 6*d^2*f^12*g^2*e^15 + f^14*e^17)*(2*(d*e + sqrt(-x^2*e^2 +
d^2)*e)*d*g*e^(-1)/x + f*e^2 + (d*e + sqrt(-x^2*e^2 + d^2)*e)^2*f*e^(-2)/x^2)^2)