3.557 $$\int \frac{(a+c x^2)^{5/2}}{(d+e x)^9} \, dx$$

Optimal. Leaf size=332 $-\frac{5 a^2 c^3 \sqrt{a+c x^2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{128 (d+e x)^2 \left (a e^2+c d^2\right )^5}-\frac{5 a^3 c^4 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{128 \left (a e^2+c d^2\right )^{11/2}}-\frac{5 a c^2 \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{192 (d+e x)^4 \left (a e^2+c d^2\right )^4}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 (d+e x)^7 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{48 (d+e x)^6 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{7/2}}{8 (d+e x)^8 \left (a e^2+c d^2\right )}$

[Out]

(-5*a^2*c^3*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(128*(c*d^2 + a*e^2)^5*(d + e*x)^2) - (5*a*c^2*(8
*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(192*(c*d^2 + a*e^2)^4*(d + e*x)^4) - (c*(8*c*d^2 - a*e^2)*(a
*e - c*d*x)*(a + c*x^2)^(5/2))/(48*(c*d^2 + a*e^2)^3*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(8*(c*d^2 + a*e^2)*(
d + e*x)^8) - (9*c*d*e*(a + c*x^2)^(7/2))/(56*(c*d^2 + a*e^2)^2*(d + e*x)^7) - (5*a^3*c^4*(8*c*d^2 - a*e^2)*Ar
cTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(128*(c*d^2 + a*e^2)^(11/2))

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Rubi [A]  time = 0.257773, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.263, Rules used = {745, 807, 721, 725, 206} $-\frac{5 a^2 c^3 \sqrt{a+c x^2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{128 (d+e x)^2 \left (a e^2+c d^2\right )^5}-\frac{5 a^3 c^4 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{128 \left (a e^2+c d^2\right )^{11/2}}-\frac{5 a c^2 \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{192 (d+e x)^4 \left (a e^2+c d^2\right )^4}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 (d+e x)^7 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{48 (d+e x)^6 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{7/2}}{8 (d+e x)^8 \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(-5*a^2*c^3*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(128*(c*d^2 + a*e^2)^5*(d + e*x)^2) - (5*a*c^2*(8
*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(192*(c*d^2 + a*e^2)^4*(d + e*x)^4) - (c*(8*c*d^2 - a*e^2)*(a
*e - c*d*x)*(a + c*x^2)^(5/2))/(48*(c*d^2 + a*e^2)^3*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(8*(c*d^2 + a*e^2)*(
d + e*x)^8) - (9*c*d*e*(a + c*x^2)^(7/2))/(56*(c*d^2 + a*e^2)^2*(d + e*x)^7) - (5*a^3*c^4*(8*c*d^2 - a*e^2)*Ar
cTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(128*(c*d^2 + a*e^2)^(11/2))

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

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 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx &=-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{c \int \frac{(-8 d+e x) \left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx}{8 \left (c d^2+a e^2\right )}\\ &=-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac{\left (c \left (8 c d^2-a e^2\right )\right ) \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx}{8 \left (c d^2+a e^2\right )^2}\\ &=-\frac{c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac{\left (5 a c^2 \left (8 c d^2-a e^2\right )\right ) \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{48 \left (c d^2+a e^2\right )^3}\\ &=-\frac{5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac{c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac{\left (5 a^2 c^3 \left (8 c d^2-a e^2\right )\right ) \int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx}{64 \left (c d^2+a e^2\right )^4}\\ &=-\frac{5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac{5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac{c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac{\left (5 a^3 c^4 \left (8 c d^2-a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{128 \left (c d^2+a e^2\right )^5}\\ &=-\frac{5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac{5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac{c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}-\frac{\left (5 a^3 c^4 \left (8 c d^2-a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{128 \left (c d^2+a e^2\right )^5}\\ &=-\frac{5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac{5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac{c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}-\frac{5 a^3 c^4 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{128 \left (c d^2+a e^2\right )^{11/2}}\\ \end{align*}

Mathematica [A]  time = 1.05807, size = 489, normalized size = 1.47 $\frac{-\frac{\sqrt{a+c x^2} \left (2 c^2 (d+e x)^4 \left (413 a^2 e^4+880 a c d^2 e^2+440 c^2 d^4\right ) \left (a e^2+c d^2\right )^3-2 c^3 d (d+e x)^5 \left (87 a^2 e^4+32 a c d^2 e^2+8 c^2 d^4\right ) \left (a e^2+c d^2\right )^2-c^3 (d+e x)^6 \left (282 a^2 c d^2 e^4-105 a^3 e^6+88 a c^2 d^4 e^2+16 c^3 d^6\right ) \left (a e^2+c d^2\right )-c^4 d (d+e x)^7 \left (370 a^2 c d^2 e^4-663 a^3 e^6+104 a c^2 d^4 e^2+16 c^3 d^6\right )-8 c^2 d (d+e x)^3 \left (307 a e^2+310 c d^2\right ) \left (a e^2+c d^2\right )^4-1584 c d (d+e x) \left (a e^2+c d^2\right )^6+8 c (d+e x)^2 \left (119 a e^2+362 c d^2\right ) \left (a e^2+c d^2\right )^5+336 \left (a e^2+c d^2\right )^7\right )}{(d+e x)^8 \left (a e^3+c d^2 e\right )^5}+\frac{105 a^3 c^4 \left (a e^2-8 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{11/2}}+\frac{105 a^3 c^4 \left (8 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{11/2}}}{2688}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(-((Sqrt[a + c*x^2]*(336*(c*d^2 + a*e^2)^7 - 1584*c*d*(c*d^2 + a*e^2)^6*(d + e*x) + 8*c*(c*d^2 + a*e^2)^5*(362
*c*d^2 + 119*a*e^2)*(d + e*x)^2 - 8*c^2*d*(c*d^2 + a*e^2)^4*(310*c*d^2 + 307*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2
+ a*e^2)^3*(440*c^2*d^4 + 880*a*c*d^2*e^2 + 413*a^2*e^4)*(d + e*x)^4 - 2*c^3*d*(c*d^2 + a*e^2)^2*(8*c^2*d^4 +
32*a*c*d^2*e^2 + 87*a^2*e^4)*(d + e*x)^5 - c^3*(c*d^2 + a*e^2)*(16*c^3*d^6 + 88*a*c^2*d^4*e^2 + 282*a^2*c*d^2
*e^4 - 105*a^3*e^6)*(d + e*x)^6 - c^4*d*(16*c^3*d^6 + 104*a*c^2*d^4*e^2 + 370*a^2*c*d^2*e^4 - 663*a^3*e^6)*(d
+ e*x)^7))/((c*d^2*e + a*e^3)^5*(d + e*x)^8)) + (105*a^3*c^4*(8*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(
11/2) + (105*a^3*c^4*(-8*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2
)^(11/2))/2688

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Maple [B]  time = 0.239, size = 9978, normalized size = 30.1 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^9,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((c*x^2+a)^(5/2)/(e*x+d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((c*x^2+a)^(5/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

Timed out

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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((c*x**2+a)**(5/2)/(e*x+d)**9,x)

[Out]

Timed out

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Giac [B]  time = 3.04028, size = 4207, normalized size = 12.67 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

5/64*(8*a^3*c^5*d^2 - a^4*c^4*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))
/((c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10)*sqrt(-c*
d^2 - a*e^2)) + 1/1344*(8192*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^11*d^14*e + 2048*(sqrt(c)*x - sqrt(c*x^2 + a))^
8*c^(23/2)*d^15 + 14336*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(21/2)*d^13*e^2 + 14336*(sqrt(c)*x - sqrt(c*x^2 + a
))^11*c^10*d^12*e^3 - 8192*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^11*d^14*e + 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^
12*c^(19/2)*d^11*e^4 - 15360*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(21/2)*d^13*e^2 + 3584*(sqrt(c)*x - sqrt(c*x^
2 + a))^13*c^9*d^10*e^5 + 10240*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^10*d^12*e^3 + 57344*(sqrt(c)*x - sqrt(c*x^
2 + a))^10*a*c^(19/2)*d^11*e^4 + 14336*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(21/2)*d^13*e^2 + 75264*(sqrt(c)*
x - sqrt(c*x^2 + a))^11*a*c^9*d^10*e^5 - 10240*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^10*d^12*e^3 + 44800*(sqrt
(c)*x - sqrt(c*x^2 + a))^12*a*c^(17/2)*d^9*e^6 - 85248*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(19/2)*d^11*e^4 +
17920*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a*c^8*d^8*e^7 - 54272*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^9*d^10*e^5
- 14336*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^10*d^12*e^3 + 71680*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(17/
2)*d^9*e^6 + 57344*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(19/2)*d^11*e^4 + 161280*(sqrt(c)*x - sqrt(c*x^2 + a)
)^11*a^2*c^8*d^8*e^7 + 54272*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^9*d^10*e^5 + 89600*(sqrt(c)*x - sqrt(c*x^2
+ a))^12*a^2*c^(15/2)*d^7*e^8 - 416384*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(17/2)*d^9*e^6 + 8960*(sqrt(c)*x
- sqrt(c*x^2 + a))^4*a^4*c^(19/2)*d^11*e^4 + 35840*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^2*c^7*d^6*e^9 - 877056*(
sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^8*d^8*e^7 - 75264*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^9*d^10*e^5 - 9166
08*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(15/2)*d^7*e^8 + 152320*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(17/2)
*d^9*e^6 - 486528*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^7*d^6*e^9 + 1334016*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a
^4*c^8*d^8*e^7 - 3584*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^9*d^10*e^5 - 208880*(sqrt(c)*x - sqrt(c*x^2 + a))^
12*a^3*c^(13/2)*d^5*e^10 + 2315376*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(15/2)*d^7*e^8 + 60928*(sqrt(c)*x - s
qrt(c*x^2 + a))^4*a^5*c^(17/2)*d^9*e^6 - 45920*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^6*d^4*e^11 + 2366784*(sq
rt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^7*d^6*e^9 - 274176*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^8*d^8*e^7 - 12600*
(sqrt(c)*x - sqrt(c*x^2 + a))^14*a^3*c^(11/2)*d^3*e^12 + 1412880*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(13/2)
*d^5*e^10 - 1755264*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(15/2)*d^7*e^8 + 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^
2*a^6*c^(17/2)*d^9*e^6 - 840*(sqrt(c)*x - sqrt(c*x^2 + a))^15*a^3*c^5*d^2*e^13 + 650160*(sqrt(c)*x - sqrt(c*x^
2 + a))^11*a^4*c^6*d^4*e^11 - 2796864*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^7*d^6*e^9 - 26880*(sqrt(c)*x - sqr
t(c*x^2 + a))^3*a^6*c^8*d^8*e^7 + 165830*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(11/2)*d^3*e^12 - 2638440*(sqr
t(c)*x - sqrt(c*x^2 + a))^8*a^5*c^(13/2)*d^5*e^10 + 255360*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(15/2)*d^7*e^
8 + 34580*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^4*c^5*d^2*e^13 - 1325520*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^5*c^6*
d^4*e^11 + 1495424*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^7*d^6*e^9 - 256*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^8
*d^8*e^7 + 1575*(sqrt(c)*x - sqrt(c*x^2 + a))^14*a^4*c^(9/2)*d*e^14 - 464520*(sqrt(c)*x - sqrt(c*x^2 + a))^10*
a^5*c^(11/2)*d^3*e^12 + 2173136*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*c^(13/2)*d^5*e^10 + 11520*(sqrt(c)*x - sqr
t(c*x^2 + a))^2*a^7*c^(15/2)*d^7*e^8 + 105*(sqrt(c)*x - sqrt(c*x^2 + a))^15*a^4*c^4*e^15 - 46620*(sqrt(c)*x -
sqrt(c*x^2 + a))^11*a^5*c^5*d^2*e^13 + 1851920*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^6*c^6*d^4*e^11 - 118272*(sqrt
(c)*x - sqrt(c*x^2 + a))^3*a^7*c^7*d^6*e^9 - 1505*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^5*c^(9/2)*d*e^14 + 755510
*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^6*c^(11/2)*d^3*e^12 - 779408*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(13/2)*d
^5*e^10 + 16*a^8*c^(15/2)*d^7*e^8 + 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^5*c^4*e^15 + 229040*(sqrt(c)*x - s
qrt(c*x^2 + a))^9*a^6*c^5*d^2*e^13 - 959280*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^6*d^4*e^11 - 1664*(sqrt(c)*x
- sqrt(c*x^2 + a))*a^8*c^7*d^6*e^9 + 15155*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^6*c^(9/2)*d*e^14 - 670040*(sqrt
(c)*x - sqrt(c*x^2 + a))^6*a^7*c^(11/2)*d^3*e^12 + 40608*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(13/2)*d^5*e^10
+ 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^6*c^4*e^15 - 142240*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^7*c^5*d^2*e^1
3 + 292544*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^8*c^6*d^4*e^11 - 23205*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^7*c^(9/2
)*d*e^14 + 290066*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^8*c^(11/2)*d^3*e^12 + 104*a^9*c^(13/2)*d^5*e^10 + 12355*(s
qrt(c)*x - sqrt(c*x^2 + a))^9*a^7*c^4*e^15 + 176148*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^8*c^5*d^2*e^13 - 5920*(s
qrt(c)*x - sqrt(c*x^2 + a))*a^9*c^6*d^4*e^11 + 21973*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^8*c^(9/2)*d*e^14 - 6461
6*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^9*c^(11/2)*d^3*e^12 + 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^8*c^4*e^15 -
39676*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^9*c^5*d^2*e^13 - 17059*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^9*c^(9/2)*d*
e^14 + 370*a^10*c^(11/2)*d^3*e^12 + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^9*c^4*e^15 + 9768*(sqrt(c)*x - sqrt
(c*x^2 + a))*a^10*c^5*d^2*e^13 + 3729*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^10*c^(9/2)*d*e^14 + 2779*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*a^10*c^4*e^15 - 663*a^11*c^(9/2)*d*e^14 + 105*(sqrt(c)*x - sqrt(c*x^2 + a))*a^11*c^4*e^15)/
((c^5*d^10*e^6 + 5*a*c^4*d^8*e^8 + 10*a^2*c^3*d^6*e^10 + 10*a^3*c^2*d^4*e^12 + 5*a^4*c*d^2*e^14 + a^5*e^16)*((
sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^8)