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

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

[Out]

-5/192*a*c^2*(-a*e^2+8*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^4/(e*x+d)^4-1/48*c*(-a*e^2+8*c*d^2)*(
-c*d*x+a*e)*(c*x^2+a)^(5/2)/(a*e^2+c*d^2)^3/(e*x+d)^6-1/8*e*(c*x^2+a)^(7/2)/(a*e^2+c*d^2)/(e*x+d)^8-9/56*c*d*e
*(c*x^2+a)^(7/2)/(a*e^2+c*d^2)^2/(e*x+d)^7-5/128*a^3*c^4*(-a*e^2+8*c*d^2)*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(
1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(11/2)-5/128*a^2*c^3*(-a*e^2+8*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+
c*d^2)^5/(e*x+d)^2

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Rubi [A]  time = 0.26, antiderivative size = 332, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 1.17, size = 489, normalized size = 1.47 \[ \frac {\frac {105 a^3 c^4 \left (a e^2-8 c d^2\right ) \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{11/2}}+\frac {105 a^3 c^4 \left (8 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{11/2}}-\frac {\sqrt {a+c x^2} \left (2 c^2 (d+e x)^4 \left (413 a^2 e^4+880 a c d^2 e^2+440 c^2 d^4\right ) \left (a e^2+c d^2\right )^3-2 c^3 d (d+e x)^5 \left (87 a^2 e^4+32 a c d^2 e^2+8 c^2 d^4\right ) \left (a e^2+c d^2\right )^2-c^3 (d+e x)^6 \left (-105 a^3 e^6+282 a^2 c d^2 e^4+88 a c^2 d^4 e^2+16 c^3 d^6\right ) \left (a e^2+c d^2\right )-c^4 d (d+e x)^7 \left (-663 a^3 e^6+370 a^2 c d^2 e^4+104 a c^2 d^4 e^2+16 c^3 d^6\right )-8 c^2 d (d+e x)^3 \left (307 a e^2+310 c d^2\right ) \left (a e^2+c d^2\right )^4-1584 c d (d+e x) \left (a e^2+c d^2\right )^6+8 c (d+e x)^2 \left (119 a e^2+362 c d^2\right ) \left (a e^2+c d^2\right )^5+336 \left (a e^2+c d^2\right )^7\right )}{(d+e x)^8 \left (a e^3+c d^2 e\right )^5}}{2688} \]

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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fricas [B]  time = 153.98, size = 3693, normalized size = 11.12 \[ \text {result too large to display} \]

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]

[1/5376*(105*(8*a^3*c^5*d^10 - a^4*c^4*d^8*e^2 + (8*a^3*c^5*d^2*e^8 - a^4*c^4*e^10)*x^8 + 8*(8*a^3*c^5*d^3*e^7
- a^4*c^4*d*e^9)*x^7 + 28*(8*a^3*c^5*d^4*e^6 - a^4*c^4*d^2*e^8)*x^6 + 56*(8*a^3*c^5*d^5*e^5 - a^4*c^4*d^3*e^7
)*x^5 + 70*(8*a^3*c^5*d^6*e^4 - a^4*c^4*d^4*e^6)*x^4 + 56*(8*a^3*c^5*d^7*e^3 - a^4*c^4*d^5*e^5)*x^3 + 28*(8*a^
3*c^5*d^8*e^2 - a^4*c^4*d^6*e^4)*x^2 + 8*(8*a^3*c^5*d^9*e - a^4*c^4*d^7*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c
*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)
)/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(2616*a^3*c^5*d^10*e + 6481*a^4*c^4*d^8*e^3 + 7443*a^5*c^3*d^6*e^5 + 5298*a^6
*c^2*d^4*e^7 + 2056*a^7*c*d^2*e^9 + 336*a^8*e^11 - (16*c^8*d^9*e^2 + 120*a*c^7*d^7*e^4 + 474*a^2*c^6*d^5*e^6 -
293*a^3*c^5*d^3*e^8 - 663*a^4*c^4*d*e^10)*x^7 - (128*c^8*d^10*e + 960*a*c^7*d^8*e^3 + 3792*a^2*c^6*d^6*e^5 -
1504*a^3*c^5*d^4*e^7 - 4569*a^4*c^4*d^2*e^9 - 105*a^5*c^3*e^11)*x^6 - (448*c^8*d^11 + 3352*a*c^7*d^9*e^2 + 132
12*a^2*c^6*d^7*e^4 - 2141*a^3*c^5*d^5*e^6 - 12905*a^4*c^4*d^3*e^8 - 456*a^5*c^3*d*e^10)*x^5 - (1280*a*c^7*d^10
*e + 12624*a^2*c^6*d^8*e^3 - 15784*a^3*c^5*d^6*e^5 - 32071*a^4*c^4*d^4*e^7 - 5769*a^5*c^3*d^2*e^9 - 826*a^6*c^
2*e^11)*x^4 - (1456*a*c^7*d^11 + 14706*a^2*c^6*d^9*e^2 - 12191*a^3*c^5*d^7*e^4 - 30449*a^4*c^4*d^5*e^6 - 5856*
a^5*c^3*d^3*e^8 - 848*a^6*c^2*d*e^10)*x^3 - (4416*a^2*c^6*d^10*e - 19072*a^3*c^5*d^8*e^3 - 35839*a^4*c^4*d^6*e
^5 - 17595*a^5*c^3*d^4*e^7 - 6196*a^6*c^2*d^2*e^9 - 952*a^7*c*e^11)*x^2 - (1848*a^2*c^6*d^11 - 5535*a^3*c^5*d^
9*e^2 - 11423*a^4*c^4*d^7*e^4 - 5784*a^5*c^3*d^5*e^6 - 2064*a^6*c^2*d^3*e^8 - 320*a^7*c*d*e^10)*x)*sqrt(c*x^2
+ a))/(c^6*d^20 + 6*a*c^5*d^18*e^2 + 15*a^2*c^4*d^16*e^4 + 20*a^3*c^3*d^14*e^6 + 15*a^4*c^2*d^12*e^8 + 6*a^5*c
*d^10*e^10 + a^6*d^8*e^12 + (c^6*d^12*e^8 + 6*a*c^5*d^10*e^10 + 15*a^2*c^4*d^8*e^12 + 20*a^3*c^3*d^6*e^14 + 15
*a^4*c^2*d^4*e^16 + 6*a^5*c*d^2*e^18 + a^6*e^20)*x^8 + 8*(c^6*d^13*e^7 + 6*a*c^5*d^11*e^9 + 15*a^2*c^4*d^9*e^1
1 + 20*a^3*c^3*d^7*e^13 + 15*a^4*c^2*d^5*e^15 + 6*a^5*c*d^3*e^17 + a^6*d*e^19)*x^7 + 28*(c^6*d^14*e^6 + 6*a*c^
5*d^12*e^8 + 15*a^2*c^4*d^10*e^10 + 20*a^3*c^3*d^8*e^12 + 15*a^4*c^2*d^6*e^14 + 6*a^5*c*d^4*e^16 + a^6*d^2*e^1
8)*x^6 + 56*(c^6*d^15*e^5 + 6*a*c^5*d^13*e^7 + 15*a^2*c^4*d^11*e^9 + 20*a^3*c^3*d^9*e^11 + 15*a^4*c^2*d^7*e^13
+ 6*a^5*c*d^5*e^15 + a^6*d^3*e^17)*x^5 + 70*(c^6*d^16*e^4 + 6*a*c^5*d^14*e^6 + 15*a^2*c^4*d^12*e^8 + 20*a^3*c
^3*d^10*e^10 + 15*a^4*c^2*d^8*e^12 + 6*a^5*c*d^6*e^14 + a^6*d^4*e^16)*x^4 + 56*(c^6*d^17*e^3 + 6*a*c^5*d^15*e^
5 + 15*a^2*c^4*d^13*e^7 + 20*a^3*c^3*d^11*e^9 + 15*a^4*c^2*d^9*e^11 + 6*a^5*c*d^7*e^13 + a^6*d^5*e^15)*x^3 + 2
8*(c^6*d^18*e^2 + 6*a*c^5*d^16*e^4 + 15*a^2*c^4*d^14*e^6 + 20*a^3*c^3*d^12*e^8 + 15*a^4*c^2*d^10*e^10 + 6*a^5*
c*d^8*e^12 + a^6*d^6*e^14)*x^2 + 8*(c^6*d^19*e + 6*a*c^5*d^17*e^3 + 15*a^2*c^4*d^15*e^5 + 20*a^3*c^3*d^13*e^7
+ 15*a^4*c^2*d^11*e^9 + 6*a^5*c*d^9*e^11 + a^6*d^7*e^13)*x), -1/2688*(105*(8*a^3*c^5*d^10 - a^4*c^4*d^8*e^2 +
(8*a^3*c^5*d^2*e^8 - a^4*c^4*e^10)*x^8 + 8*(8*a^3*c^5*d^3*e^7 - a^4*c^4*d*e^9)*x^7 + 28*(8*a^3*c^5*d^4*e^6 - a
^4*c^4*d^2*e^8)*x^6 + 56*(8*a^3*c^5*d^5*e^5 - a^4*c^4*d^3*e^7)*x^5 + 70*(8*a^3*c^5*d^6*e^4 - a^4*c^4*d^4*e^6)*
x^4 + 56*(8*a^3*c^5*d^7*e^3 - a^4*c^4*d^5*e^5)*x^3 + 28*(8*a^3*c^5*d^8*e^2 - a^4*c^4*d^6*e^4)*x^2 + 8*(8*a^3*c
^5*d^9*e - a^4*c^4*d^7*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/
(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + (2616*a^3*c^5*d^10*e + 6481*a^4*c^4*d^8*e^3 + 7443*a^5*c^3*d^
6*e^5 + 5298*a^6*c^2*d^4*e^7 + 2056*a^7*c*d^2*e^9 + 336*a^8*e^11 - (16*c^8*d^9*e^2 + 120*a*c^7*d^7*e^4 + 474*a
^2*c^6*d^5*e^6 - 293*a^3*c^5*d^3*e^8 - 663*a^4*c^4*d*e^10)*x^7 - (128*c^8*d^10*e + 960*a*c^7*d^8*e^3 + 3792*a^
2*c^6*d^6*e^5 - 1504*a^3*c^5*d^4*e^7 - 4569*a^4*c^4*d^2*e^9 - 105*a^5*c^3*e^11)*x^6 - (448*c^8*d^11 + 3352*a*c
^7*d^9*e^2 + 13212*a^2*c^6*d^7*e^4 - 2141*a^3*c^5*d^5*e^6 - 12905*a^4*c^4*d^3*e^8 - 456*a^5*c^3*d*e^10)*x^5 -
(1280*a*c^7*d^10*e + 12624*a^2*c^6*d^8*e^3 - 15784*a^3*c^5*d^6*e^5 - 32071*a^4*c^4*d^4*e^7 - 5769*a^5*c^3*d^2*
e^9 - 826*a^6*c^2*e^11)*x^4 - (1456*a*c^7*d^11 + 14706*a^2*c^6*d^9*e^2 - 12191*a^3*c^5*d^7*e^4 - 30449*a^4*c^4
*d^5*e^6 - 5856*a^5*c^3*d^3*e^8 - 848*a^6*c^2*d*e^10)*x^3 - (4416*a^2*c^6*d^10*e - 19072*a^3*c^5*d^8*e^3 - 358
39*a^4*c^4*d^6*e^5 - 17595*a^5*c^3*d^4*e^7 - 6196*a^6*c^2*d^2*e^9 - 952*a^7*c*e^11)*x^2 - (1848*a^2*c^6*d^11 -
5535*a^3*c^5*d^9*e^2 - 11423*a^4*c^4*d^7*e^4 - 5784*a^5*c^3*d^5*e^6 - 2064*a^6*c^2*d^3*e^8 - 320*a^7*c*d*e^10
)*x)*sqrt(c*x^2 + a))/(c^6*d^20 + 6*a*c^5*d^18*e^2 + 15*a^2*c^4*d^16*e^4 + 20*a^3*c^3*d^14*e^6 + 15*a^4*c^2*d^
12*e^8 + 6*a^5*c*d^10*e^10 + a^6*d^8*e^12 + (c^6*d^12*e^8 + 6*a*c^5*d^10*e^10 + 15*a^2*c^4*d^8*e^12 + 20*a^3*c
^3*d^6*e^14 + 15*a^4*c^2*d^4*e^16 + 6*a^5*c*d^2*e^18 + a^6*e^20)*x^8 + 8*(c^6*d^13*e^7 + 6*a*c^5*d^11*e^9 + 15
*a^2*c^4*d^9*e^11 + 20*a^3*c^3*d^7*e^13 + 15*a^4*c^2*d^5*e^15 + 6*a^5*c*d^3*e^17 + a^6*d*e^19)*x^7 + 28*(c^6*d
^14*e^6 + 6*a*c^5*d^12*e^8 + 15*a^2*c^4*d^10*e^10 + 20*a^3*c^3*d^8*e^12 + 15*a^4*c^2*d^6*e^14 + 6*a^5*c*d^4*e^
16 + a^6*d^2*e^18)*x^6 + 56*(c^6*d^15*e^5 + 6*a*c^5*d^13*e^7 + 15*a^2*c^4*d^11*e^9 + 20*a^3*c^3*d^9*e^11 + 15*
a^4*c^2*d^7*e^13 + 6*a^5*c*d^5*e^15 + a^6*d^3*e^17)*x^5 + 70*(c^6*d^16*e^4 + 6*a*c^5*d^14*e^6 + 15*a^2*c^4*d^1
2*e^8 + 20*a^3*c^3*d^10*e^10 + 15*a^4*c^2*d^8*e^12 + 6*a^5*c*d^6*e^14 + a^6*d^4*e^16)*x^4 + 56*(c^6*d^17*e^3 +
6*a*c^5*d^15*e^5 + 15*a^2*c^4*d^13*e^7 + 20*a^3*c^3*d^11*e^9 + 15*a^4*c^2*d^9*e^11 + 6*a^5*c*d^7*e^13 + a^6*d
^5*e^15)*x^3 + 28*(c^6*d^18*e^2 + 6*a*c^5*d^16*e^4 + 15*a^2*c^4*d^14*e^6 + 20*a^3*c^3*d^12*e^8 + 15*a^4*c^2*d^
10*e^10 + 6*a^5*c*d^8*e^12 + a^6*d^6*e^14)*x^2 + 8*(c^6*d^19*e + 6*a*c^5*d^17*e^3 + 15*a^2*c^4*d^15*e^5 + 20*a
^3*c^3*d^13*e^7 + 15*a^4*c^2*d^11*e^9 + 6*a^5*c*d^9*e^11 + a^6*d^7*e^13)*x)]

________________________________________________________________________________________

giac [B]  time = 0.89, size = 3116, normalized size = 9.39 \[ \text {result too large to display} \]

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)

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maple [B]  time = 0.13, size = 9978, normalized size = 30.05 \[ \text {output too large to display} \]

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

________________________________________________________________________________________

maxima [B]  time = 8.53, size = 9514, normalized size = 28.66 \[ \text {result too large to display} \]

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]

-45/256*c^10*d^11*arcsinh(c*x/sqrt(a*c))/(c^(15/2)*d^14*e^6 + 7*a*c^(13/2)*d^12*e^8 + 21*a^2*c^(11/2)*d^10*e^1
0 + 35*a^3*c^(9/2)*d^8*e^12 + 35*a^4*c^(7/2)*d^6*e^14 + 21*a^5*c^(5/2)*d^4*e^16 + 7*a^6*c^(3/2)*d^2*e^18 + a^7
*sqrt(c)*e^20) - 45/256*a*c^9*d^9*arcsinh(c*x/sqrt(a*c))/(c^(15/2)*d^14*e^4 + 7*a*c^(13/2)*d^12*e^6 + 21*a^2*c
^(11/2)*d^10*e^8 + 35*a^3*c^(9/2)*d^8*e^10 + 35*a^4*c^(7/2)*d^6*e^12 + 21*a^5*c^(5/2)*d^4*e^14 + 7*a^6*c^(3/2)
*d^2*e^16 + a^7*sqrt(c)*e^18) + 45/256*sqrt(c*x^2 + a)*c^9*d^9*x/(c^7*d^14*e^4 + 7*a*c^6*d^12*e^6 + 21*a^2*c^5
*d^10*e^8 + 35*a^3*c^4*d^8*e^10 + 35*a^4*c^3*d^6*e^12 + 21*a^5*c^2*d^4*e^14 + 7*a^6*c*d^2*e^16 + a^7*e^18) + 2
75/256*c^9*d^9*arcsinh(c*x/sqrt(a*c))/(c^(13/2)*d^12*e^6 + 6*a*c^(11/2)*d^10*e^8 + 15*a^2*c^(9/2)*d^8*e^10 + 2
0*a^3*c^(7/2)*d^6*e^12 + 15*a^4*c^(5/2)*d^4*e^14 + 6*a^5*c^(3/2)*d^2*e^16 + a^6*sqrt(c)*e^18) - 15/128*(c*x^2
+ a)^(3/2)*c^8*d^8/(c^7*d^14*e^3 + 7*a*c^6*d^12*e^5 + 21*a^2*c^5*d^10*e^7 + 35*a^3*c^4*d^8*e^9 + 35*a^4*c^3*d^
6*e^11 + 21*a^5*c^2*d^4*e^13 + 7*a^6*c*d^2*e^15 + a^7*e^17) + 15/128*(c*x^2 + a)^(3/2)*c^8*d^7*x/(c^7*d^14*e^2
+ 7*a*c^6*d^12*e^4 + 21*a^2*c^5*d^10*e^6 + 35*a^3*c^4*d^8*e^8 + 35*a^4*c^3*d^6*e^10 + 21*a^5*c^2*d^4*e^12 + 7
*a^6*c*d^2*e^14 + a^7*e^16) + 45/256*sqrt(c*x^2 + a)*a*c^8*d^7*x/(c^7*d^14*e^2 + 7*a*c^6*d^12*e^4 + 21*a^2*c^5
*d^10*e^6 + 35*a^3*c^4*d^8*e^8 + 35*a^4*c^3*d^6*e^10 + 21*a^5*c^2*d^4*e^12 + 7*a^6*c*d^2*e^14 + a^7*e^16) + 11
5/128*a*c^8*d^7*arcsinh(c*x/sqrt(a*c))/(c^(13/2)*d^12*e^4 + 6*a*c^(11/2)*d^10*e^6 + 15*a^2*c^(9/2)*d^8*e^8 + 2
0*a^3*c^(7/2)*d^6*e^10 + 15*a^4*c^(5/2)*d^4*e^12 + 6*a^5*c^(3/2)*d^2*e^14 + a^6*sqrt(c)*e^16) - 9/128*(c*x^2 +
a)^(5/2)*c^7*d^7/(c^7*d^14*e^2*x + 7*a*c^6*d^12*e^4*x + 21*a^2*c^5*d^10*e^6*x + 35*a^3*c^4*d^8*e^8*x + 35*a^4
*c^3*d^6*e^10*x + 21*a^5*c^2*d^4*e^12*x + 7*a^6*c*d^2*e^14*x + a^7*e^16*x + c^7*d^15*e + 7*a*c^6*d^13*e^3 + 21
*a^2*c^5*d^11*e^5 + 35*a^3*c^4*d^9*e^7 + 35*a^4*c^3*d^7*e^9 + 21*a^5*c^2*d^5*e^11 + 7*a^6*c*d^3*e^13 + a^7*d*e
^15) - 45/128*sqrt(c*x^2 + a)*c^8*d^8/(c^6*d^12*e^5 + 6*a*c^5*d^10*e^7 + 15*a^2*c^4*d^8*e^9 + 20*a^3*c^3*d^6*e
^11 + 15*a^4*c^2*d^4*e^13 + 6*a^5*c*d^2*e^15 + a^6*e^17) - 35/64*sqrt(c*x^2 + a)*c^8*d^7*x/(c^6*d^12*e^4 + 6*a
*c^5*d^10*e^6 + 15*a^2*c^4*d^8*e^8 + 20*a^3*c^3*d^6*e^10 + 15*a^4*c^2*d^4*e^12 + 6*a^5*c*d^2*e^14 + a^6*e^16)
- 285/128*c^8*d^7*arcsinh(c*x/sqrt(a*c))/(c^(11/2)*d^10*e^6 + 5*a*c^(9/2)*d^8*e^8 + 10*a^2*c^(7/2)*d^6*e^10 +
10*a^3*c^(5/2)*d^4*e^12 + 5*a^4*c^(3/2)*d^2*e^14 + a^5*sqrt(c)*e^16) + 3/128*(c*x^2 + a)^(7/2)*c^6*d^6/(c^7*d^
14*e*x^2 + 7*a*c^6*d^12*e^3*x^2 + 21*a^2*c^5*d^10*e^5*x^2 + 35*a^3*c^4*d^8*e^7*x^2 + 35*a^4*c^3*d^6*e^9*x^2 +
21*a^5*c^2*d^4*e^11*x^2 + 7*a^6*c*d^2*e^13*x^2 + a^7*e^15*x^2 + 2*c^7*d^15*x + 14*a*c^6*d^13*e^2*x + 42*a^2*c^
5*d^11*e^4*x + 70*a^3*c^4*d^9*e^6*x + 70*a^4*c^3*d^7*e^8*x + 42*a^5*c^2*d^5*e^10*x + 14*a^6*c*d^3*e^12*x + 2*a
^7*d*e^14*x + c^7*d^16/e + 7*a*c^6*d^14*e + 21*a^2*c^5*d^12*e^3 + 35*a^3*c^4*d^10*e^5 + 35*a^4*c^3*d^8*e^7 + 2
1*a^5*c^2*d^6*e^9 + 7*a^6*c*d^4*e^11 + a^7*d^2*e^13) - 3/128*(c*x^2 + a)^(5/2)*c^7*d^6/(c^7*d^14*e + 7*a*c^6*d
^12*e^3 + 21*a^2*c^5*d^10*e^5 + 35*a^3*c^4*d^8*e^7 + 35*a^4*c^3*d^6*e^9 + 21*a^5*c^2*d^4*e^11 + 7*a^6*c*d^2*e^
13 + a^7*e^15) + 35/96*(c*x^2 + a)^(3/2)*c^7*d^6/(c^6*d^12*e^3 + 6*a*c^5*d^10*e^5 + 15*a^2*c^4*d^8*e^7 + 20*a^
3*c^3*d^6*e^9 + 15*a^4*c^2*d^4*e^11 + 6*a^5*c*d^2*e^13 + a^6*e^15) - 185/384*(c*x^2 + a)^(3/2)*c^7*d^5*x/(c^6*
d^12*e^2 + 6*a*c^5*d^10*e^4 + 15*a^2*c^4*d^8*e^6 + 20*a^3*c^3*d^6*e^8 + 15*a^4*c^2*d^4*e^10 + 6*a^5*c*d^2*e^12
+ a^6*e^14) - 185/256*sqrt(c*x^2 + a)*a*c^7*d^5*x/(c^6*d^12*e^2 + 6*a*c^5*d^10*e^4 + 15*a^2*c^4*d^8*e^6 + 20*
a^3*c^3*d^6*e^8 + 15*a^4*c^2*d^4*e^10 + 6*a^5*c*d^2*e^12 + a^6*e^14) - 85/64*a*c^7*d^5*arcsinh(c*x/sqrt(a*c))/
(c^(11/2)*d^10*e^4 + 5*a*c^(9/2)*d^8*e^6 + 10*a^2*c^(7/2)*d^6*e^8 + 10*a^3*c^(5/2)*d^4*e^10 + 5*a^4*c^(3/2)*d^
2*e^12 + a^5*sqrt(c)*e^14) - 3/64*(c*x^2 + a)^(7/2)*c^5*d^5/(c^6*d^12*e^2*x^3 + 6*a*c^5*d^10*e^4*x^3 + 15*a^2*
c^4*d^8*e^6*x^3 + 20*a^3*c^3*d^6*e^8*x^3 + 15*a^4*c^2*d^4*e^10*x^3 + 6*a^5*c*d^2*e^12*x^3 + a^6*e^14*x^3 + 3*c
^6*d^13*e*x^2 + 18*a*c^5*d^11*e^3*x^2 + 45*a^2*c^4*d^9*e^5*x^2 + 60*a^3*c^3*d^7*e^7*x^2 + 45*a^4*c^2*d^5*e^9*x
^2 + 18*a^5*c*d^3*e^11*x^2 + 3*a^6*d*e^13*x^2 + 3*c^6*d^14*x + 18*a*c^5*d^12*e^2*x + 45*a^2*c^4*d^10*e^4*x + 6
0*a^3*c^3*d^8*e^6*x + 45*a^4*c^2*d^6*e^8*x + 18*a^5*c*d^4*e^10*x + 3*a^6*d^2*e^12*x + c^6*d^15/e + 6*a*c^5*d^1
3*e + 15*a^2*c^4*d^11*e^3 + 20*a^3*c^3*d^9*e^5 + 15*a^4*c^2*d^7*e^7 + 6*a^5*c*d^5*e^9 + a^6*d^3*e^11) + 31/128
*(c*x^2 + a)^(5/2)*c^6*d^5/(c^6*d^12*e^2*x + 6*a*c^5*d^10*e^4*x + 15*a^2*c^4*d^8*e^6*x + 20*a^3*c^3*d^6*e^8*x
+ 15*a^4*c^2*d^4*e^10*x + 6*a^5*c*d^2*e^12*x + a^6*e^14*x + c^6*d^13*e + 6*a*c^5*d^11*e^3 + 15*a^2*c^4*d^9*e^5
+ 20*a^3*c^3*d^7*e^7 + 15*a^4*c^2*d^5*e^9 + 6*a^5*c*d^3*e^11 + a^6*d*e^13) + 35/32*sqrt(c*x^2 + a)*c^7*d^6/(c
^5*d^10*e^5 + 5*a*c^4*d^8*e^7 + 10*a^2*c^3*d^6*e^9 + 10*a^3*c^2*d^4*e^11 + 5*a^4*c*d^2*e^13 + a^5*e^15) + 75/1
28*sqrt(c*x^2 + a)*c^7*d^5*x/(c^5*d^10*e^4 + 5*a*c^4*d^8*e^6 + 10*a^2*c^3*d^6*e^8 + 10*a^3*c^2*d^4*e^10 + 5*a^
4*c*d^2*e^12 + a^5*e^14) + 255/128*c^7*d^5*arcsinh(c*x/sqrt(a*c))/(c^(9/2)*d^8*e^6 + 4*a*c^(7/2)*d^6*e^8 + 6*a
^2*c^(5/2)*d^4*e^10 + 4*a^3*c^(3/2)*d^2*e^12 + a^4*sqrt(c)*e^14) - 55/384*(c*x^2 + a)^(7/2)*c^5*d^4/(c^6*d^12*
e*x^2 + 6*a*c^5*d^10*e^3*x^2 + 15*a^2*c^4*d^8*e^5*x^2 + 20*a^3*c^3*d^6*e^7*x^2 + 15*a^4*c^2*d^4*e^9*x^2 + 6*a^
5*c*d^2*e^11*x^2 + a^6*e^13*x^2 + 2*c^6*d^13*x + 12*a*c^5*d^11*e^2*x + 30*a^2*c^4*d^9*e^4*x + 40*a^3*c^3*d^7*e
^6*x + 30*a^4*c^2*d^5*e^8*x + 12*a^5*c*d^3*e^10*x + 2*a^6*d*e^12*x + c^6*d^14/e + 6*a*c^5*d^12*e + 15*a^2*c^4*
d^10*e^3 + 20*a^3*c^3*d^8*e^5 + 15*a^4*c^2*d^6*e^7 + 6*a^5*c*d^4*e^9 + a^6*d^2*e^11) + 55/384*(c*x^2 + a)^(5/2
)*c^6*d^4/(c^6*d^12*e + 6*a*c^5*d^10*e^3 + 15*a^2*c^4*d^8*e^5 + 20*a^3*c^3*d^6*e^7 + 15*a^4*c^2*d^4*e^9 + 6*a^
5*c*d^2*e^11 + a^6*e^13) - 9/64*(c*x^2 + a)^(7/2)*c^4*d^4/(c^5*d^10*e^3*x^4 + 5*a*c^4*d^8*e^5*x^4 + 10*a^2*c^3
*d^6*e^7*x^4 + 10*a^3*c^2*d^4*e^9*x^4 + 5*a^4*c*d^2*e^11*x^4 + a^5*e^13*x^4 + 4*c^5*d^11*e^2*x^3 + 20*a*c^4*d^
9*e^4*x^3 + 40*a^2*c^3*d^7*e^6*x^3 + 40*a^3*c^2*d^5*e^8*x^3 + 20*a^4*c*d^3*e^10*x^3 + 4*a^5*d*e^12*x^3 + 6*c^5
*d^12*e*x^2 + 30*a*c^4*d^10*e^3*x^2 + 60*a^2*c^3*d^8*e^5*x^2 + 60*a^3*c^2*d^6*e^7*x^2 + 30*a^4*c*d^4*e^9*x^2 +
6*a^5*d^2*e^11*x^2 + 4*c^5*d^13*x + 20*a*c^4*d^11*e^2*x + 40*a^2*c^3*d^9*e^4*x + 40*a^3*c^2*d^7*e^6*x + 20*a^
4*c*d^5*e^8*x + 4*a^5*d^3*e^10*x + c^5*d^14/e + 5*a*c^4*d^12*e + 10*a^2*c^3*d^10*e^3 + 10*a^3*c^2*d^8*e^5 + 5*
a^4*c*d^6*e^7 + a^5*d^4*e^9) - 25/64*(c*x^2 + a)^(3/2)*c^6*d^4/(c^5*d^10*e^3 + 5*a*c^4*d^8*e^5 + 10*a^2*c^3*d^
6*e^7 + 10*a^3*c^2*d^4*e^9 + 5*a^4*c*d^2*e^11 + a^5*e^13) + 155/384*(c*x^2 + a)^(3/2)*c^6*d^3*x/(c^5*d^10*e^2
+ 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) + 155/256*sqrt(c*x^
2 + a)*a*c^6*d^3*x/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^1
0 + a^5*e^12) + 85/128*a*c^6*d^3*arcsinh(c*x/sqrt(a*c))/(c^(9/2)*d^8*e^4 + 4*a*c^(7/2)*d^6*e^6 + 6*a^2*c^(5/2)
*d^4*e^8 + 4*a^3*c^(3/2)*d^2*e^10 + a^4*sqrt(c)*e^12) - 13/96*(c*x^2 + a)^(7/2)*c^4*d^3/(c^5*d^10*e^2*x^3 + 5*
a*c^4*d^8*e^4*x^3 + 10*a^2*c^3*d^6*e^6*x^3 + 10*a^3*c^2*d^4*e^8*x^3 + 5*a^4*c*d^2*e^10*x^3 + a^5*e^12*x^3 + 3*
c^5*d^11*e*x^2 + 15*a*c^4*d^9*e^3*x^2 + 30*a^2*c^3*d^7*e^5*x^2 + 30*a^3*c^2*d^5*e^7*x^2 + 15*a^4*c*d^3*e^9*x^2
+ 3*a^5*d*e^11*x^2 + 3*c^5*d^12*x + 15*a*c^4*d^10*e^2*x + 30*a^2*c^3*d^8*e^4*x + 30*a^3*c^2*d^6*e^6*x + 15*a^
4*c*d^4*e^8*x + 3*a^5*d^2*e^10*x + c^5*d^13/e + 5*a*c^4*d^11*e + 10*a^2*c^3*d^9*e^3 + 10*a^3*c^2*d^7*e^5 + 5*a
^4*c*d^5*e^7 + a^5*d^3*e^9) - 145/384*(c*x^2 + a)^(5/2)*c^5*d^3/(c^5*d^10*e^2*x + 5*a*c^4*d^8*e^4*x + 10*a^2*c
^3*d^6*e^6*x + 10*a^3*c^2*d^4*e^8*x + 5*a^4*c*d^2*e^10*x + a^5*e^12*x + c^5*d^11*e + 5*a*c^4*d^9*e^3 + 10*a^2*
c^3*d^7*e^5 + 10*a^3*c^2*d^5*e^7 + 5*a^4*c*d^3*e^9 + a^5*d*e^11) - 75/64*sqrt(c*x^2 + a)*c^6*d^4/(c^4*d^8*e^5
+ 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13) - 15/64*sqrt(c*x^2 + a)*c^6*d^3*x/(c^4*d^
8*e^4 + 4*a*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12) - 185/256*c^6*d^3*arcsinh(c*x/sqrt(
a*c))/(c^(7/2)*d^6*e^6 + 3*a*c^(5/2)*d^4*e^8 + 3*a^2*c^(3/2)*d^2*e^10 + a^3*sqrt(c)*e^12) - 7/128*(c*x^2 + a)^
(7/2)*c^4*d^2/(c^5*d^10*e*x^2 + 5*a*c^4*d^8*e^3*x^2 + 10*a^2*c^3*d^6*e^5*x^2 + 10*a^3*c^2*d^4*e^7*x^2 + 5*a^4*
c*d^2*e^9*x^2 + a^5*e^11*x^2 + 2*c^5*d^11*x + 10*a*c^4*d^9*e^2*x + 20*a^2*c^3*d^7*e^4*x + 20*a^3*c^2*d^5*e^6*x
+ 10*a^4*c*d^3*e^8*x + 2*a^5*d*e^10*x + c^5*d^12/e + 5*a*c^4*d^10*e + 10*a^2*c^3*d^8*e^3 + 10*a^3*c^2*d^6*e^5
+ 5*a^4*c*d^4*e^7 + a^5*d^2*e^9) + 7/128*(c*x^2 + a)^(5/2)*c^5*d^2/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3
*d^6*e^5 + 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 + a^5*e^11) - 3/16*(c*x^2 + a)^(7/2)*c^3*d^3/(c^4*d^8*e^4*x^5
+ 4*a*c^3*d^6*e^6*x^5 + 6*a^2*c^2*d^4*e^8*x^5 + 4*a^3*c*d^2*e^10*x^5 + a^4*e^12*x^5 + 5*c^4*d^9*e^3*x^4 + 20*a
*c^3*d^7*e^5*x^4 + 30*a^2*c^2*d^5*e^7*x^4 + 20*a^3*c*d^3*e^9*x^4 + 5*a^4*d*e^11*x^4 + 10*c^4*d^10*e^2*x^3 + 40
*a*c^3*d^8*e^4*x^3 + 60*a^2*c^2*d^6*e^6*x^3 + 40*a^3*c*d^4*e^8*x^3 + 10*a^4*d^2*e^10*x^3 + 10*c^4*d^11*e*x^2 +
40*a*c^3*d^9*e^3*x^2 + 60*a^2*c^2*d^7*e^5*x^2 + 40*a^3*c*d^5*e^7*x^2 + 10*a^4*d^3*e^9*x^2 + 5*c^4*d^12*x + 20
*a*c^3*d^10*e^2*x + 30*a^2*c^2*d^8*e^4*x + 20*a^3*c*d^6*e^6*x + 5*a^4*d^4*e^8*x + c^4*d^13/e + 4*a*c^3*d^11*e
+ 6*a^2*c^2*d^9*e^3 + 4*a^3*c*d^7*e^5 + a^4*d^5*e^7) - 1/32*(c*x^2 + a)^(7/2)*c^3*d^2/(c^4*d^8*e^3*x^4 + 4*a*c
^3*d^6*e^5*x^4 + 6*a^2*c^2*d^4*e^7*x^4 + 4*a^3*c*d^2*e^9*x^4 + a^4*e^11*x^4 + 4*c^4*d^9*e^2*x^3 + 16*a*c^3*d^7
*e^4*x^3 + 24*a^2*c^2*d^5*e^6*x^3 + 16*a^3*c*d^3*e^8*x^3 + 4*a^4*d*e^10*x^3 + 6*c^4*d^10*e*x^2 + 24*a*c^3*d^8*
e^3*x^2 + 36*a^2*c^2*d^6*e^5*x^2 + 24*a^3*c*d^4*e^7*x^2 + 6*a^4*d^2*e^9*x^2 + 4*c^4*d^11*x + 16*a*c^3*d^9*e^2*
x + 24*a^2*c^2*d^7*e^4*x + 16*a^3*c*d^5*e^6*x + 4*a^4*d^3*e^8*x + c^4*d^12/e + 4*a*c^3*d^10*e + 6*a^2*c^2*d^8*
e^3 + 4*a^3*c*d^6*e^5 + a^4*d^4*e^7) + 5/32*(c*x^2 + a)^(3/2)*c^5*d^2/(c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c
^2*d^4*e^7 + 4*a^3*c*d^2*e^9 + a^4*e^11) - 5/128*(c*x^2 + a)^(3/2)*c^5*d*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^4 + 6*
a^2*c^2*d^4*e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10) - 15/256*sqrt(c*x^2 + a)*a*c^5*d*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^
4 + 6*a^2*c^2*d^4*e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10) - 15/256*a*c^5*d*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e^4 +
3*a*c^(5/2)*d^4*e^6 + 3*a^2*c^(3/2)*d^2*e^8 + a^3*sqrt(c)*e^10) + 1/64*(c*x^2 + a)^(7/2)*c^3*d/(c^4*d^8*e^2*x
^3 + 4*a*c^3*d^6*e^4*x^3 + 6*a^2*c^2*d^4*e^6*x^3 + 4*a^3*c*d^2*e^8*x^3 + a^4*e^10*x^3 + 3*c^4*d^9*e*x^2 + 12*a
*c^3*d^7*e^3*x^2 + 18*a^2*c^2*d^5*e^5*x^2 + 12*a^3*c*d^3*e^7*x^2 + 3*a^4*d*e^9*x^2 + 3*c^4*d^10*x + 12*a*c^3*d
^8*e^2*x + 18*a^2*c^2*d^6*e^4*x + 12*a^3*c*d^4*e^6*x + 3*a^4*d^2*e^8*x + c^4*d^11/e + 4*a*c^3*d^9*e + 6*a^2*c^
2*d^7*e^3 + 4*a^3*c*d^5*e^5 + a^4*d^3*e^7) + 5/128*(c*x^2 + a)^(5/2)*c^4*d/(c^4*d^8*e^2*x + 4*a*c^3*d^6*e^4*x
+ 6*a^2*c^2*d^4*e^6*x + 4*a^3*c*d^2*e^8*x + a^4*e^10*x + c^4*d^9*e + 4*a*c^3*d^7*e^3 + 6*a^2*c^2*d^5*e^5 + 4*a
^3*c*d^3*e^7 + a^4*d*e^9) - 3/16*(c*x^2 + a)^(7/2)*c^2*d^2/(c^3*d^6*e^5*x^6 + 3*a*c^2*d^4*e^7*x^6 + 3*a^2*c*d^
2*e^9*x^6 + a^3*e^11*x^6 + 6*c^3*d^7*e^4*x^5 + 18*a*c^2*d^5*e^6*x^5 + 18*a^2*c*d^3*e^8*x^5 + 6*a^3*d*e^10*x^5
+ 15*c^3*d^8*e^3*x^4 + 45*a*c^2*d^6*e^5*x^4 + 45*a^2*c*d^4*e^7*x^4 + 15*a^3*d^2*e^9*x^4 + 20*c^3*d^9*e^2*x^3 +
60*a*c^2*d^7*e^4*x^3 + 60*a^2*c*d^5*e^6*x^3 + 20*a^3*d^3*e^8*x^3 + 15*c^3*d^10*e*x^2 + 45*a*c^2*d^8*e^3*x^2 +
45*a^2*c*d^6*e^5*x^2 + 15*a^3*d^4*e^7*x^2 + 6*c^3*d^11*x + 18*a*c^2*d^9*e^2*x + 18*a^2*c*d^7*e^4*x + 6*a^3*d^
5*e^6*x + c^3*d^12/e + 3*a*c^2*d^10*e + 3*a^2*c*d^8*e^3 + a^3*d^6*e^5) + 15/32*sqrt(c*x^2 + a)*c^5*d^2/(c^3*d^
6*e^5 + 3*a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^9 + a^3*e^11) + 5/256*sqrt(c*x^2 + a)*c^5*d*x/(c^3*d^6*e^4 + 3*a*c^2*d
^4*e^6 + 3*a^2*c*d^2*e^8 + a^3*e^10) + 15/256*c^5*d*arcsinh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^6 + 2*a*c^(3/2)*d^2*
e^8 + a^2*sqrt(c)*e^10) + 1/128*(c*x^2 + a)^(7/2)*c^3/(c^4*d^8*e*x^2 + 4*a*c^3*d^6*e^3*x^2 + 6*a^2*c^2*d^4*e^5
*x^2 + 4*a^3*c*d^2*e^7*x^2 + a^4*e^9*x^2 + 2*c^4*d^9*x + 8*a*c^3*d^7*e^2*x + 12*a^2*c^2*d^5*e^4*x + 8*a^3*c*d^
3*e^6*x + 2*a^4*d*e^8*x + c^4*d^10/e + 4*a*c^3*d^8*e + 6*a^2*c^2*d^6*e^3 + 4*a^3*c*d^4*e^5 + a^4*d^2*e^7) - 1/
128*(c*x^2 + a)^(5/2)*c^4/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) + 1/48
*(c*x^2 + a)^(7/2)*c^2*d/(c^3*d^6*e^4*x^5 + 3*a*c^2*d^4*e^6*x^5 + 3*a^2*c*d^2*e^8*x^5 + a^3*e^10*x^5 + 5*c^3*d
^7*e^3*x^4 + 15*a*c^2*d^5*e^5*x^4 + 15*a^2*c*d^3*e^7*x^4 + 5*a^3*d*e^9*x^4 + 10*c^3*d^8*e^2*x^3 + 30*a*c^2*d^6
*e^4*x^3 + 30*a^2*c*d^4*e^6*x^3 + 10*a^3*d^2*e^8*x^3 + 10*c^3*d^9*e*x^2 + 30*a*c^2*d^7*e^3*x^2 + 30*a^2*c*d^5*
e^5*x^2 + 10*a^3*d^3*e^7*x^2 + 5*c^3*d^10*x + 15*a*c^2*d^8*e^2*x + 15*a^2*c*d^6*e^4*x + 5*a^3*d^4*e^6*x + c^3*
d^11/e + 3*a*c^2*d^9*e + 3*a^2*c*d^7*e^3 + a^3*d^5*e^5) + 1/192*(c*x^2 + a)^(7/2)*c^2/(c^3*d^6*e^3*x^4 + 3*a*c
^2*d^4*e^5*x^4 + 3*a^2*c*d^2*e^7*x^4 + a^3*e^9*x^4 + 4*c^3*d^7*e^2*x^3 + 12*a*c^2*d^5*e^4*x^3 + 12*a^2*c*d^3*e
^6*x^3 + 4*a^3*d*e^8*x^3 + 6*c^3*d^8*e*x^2 + 18*a*c^2*d^6*e^3*x^2 + 18*a^2*c*d^4*e^5*x^2 + 6*a^3*d^2*e^7*x^2 +
4*c^3*d^9*x + 12*a*c^2*d^7*e^2*x + 12*a^2*c*d^5*e^4*x + 4*a^3*d^3*e^6*x + c^3*d^10/e + 3*a*c^2*d^8*e + 3*a^2*
c*d^6*e^3 + a^3*d^4*e^5) - 5/384*(c*x^2 + a)^(3/2)*c^4/(c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*
e^9) - 9/56*(c*x^2 + a)^(7/2)*c*d/(c^2*d^4*e^6*x^7 + 2*a*c*d^2*e^8*x^7 + a^2*e^10*x^7 + 7*c^2*d^5*e^5*x^6 + 14
*a*c*d^3*e^7*x^6 + 7*a^2*d*e^9*x^6 + 21*c^2*d^6*e^4*x^5 + 42*a*c*d^4*e^6*x^5 + 21*a^2*d^2*e^8*x^5 + 35*c^2*d^7
*e^3*x^4 + 70*a*c*d^5*e^5*x^4 + 35*a^2*d^3*e^7*x^4 + 35*c^2*d^8*e^2*x^3 + 70*a*c*d^6*e^4*x^3 + 35*a^2*d^4*e^6*
x^3 + 21*c^2*d^9*e*x^2 + 42*a*c*d^7*e^3*x^2 + 21*a^2*d^5*e^5*x^2 + 7*c^2*d^10*x + 14*a*c*d^8*e^2*x + 7*a^2*d^6
*e^4*x + c^2*d^11/e + 2*a*c*d^9*e + a^2*d^7*e^3) + 1/48*(c*x^2 + a)^(7/2)*c/(c^2*d^4*e^5*x^6 + 2*a*c*d^2*e^7*x
^6 + a^2*e^9*x^6 + 6*c^2*d^5*e^4*x^5 + 12*a*c*d^3*e^6*x^5 + 6*a^2*d*e^8*x^5 + 15*c^2*d^6*e^3*x^4 + 30*a*c*d^4*
e^5*x^4 + 15*a^2*d^2*e^7*x^4 + 20*c^2*d^7*e^2*x^3 + 40*a*c*d^5*e^4*x^3 + 20*a^2*d^3*e^6*x^3 + 15*c^2*d^8*e*x^2
+ 30*a*c*d^6*e^3*x^2 + 15*a^2*d^4*e^5*x^2 + 6*c^2*d^9*x + 12*a*c*d^7*e^2*x + 6*a^2*d^5*e^4*x + c^2*d^10/e + 2
*a*c*d^8*e + a^2*d^6*e^3) - 5/128*sqrt(c*x^2 + a)*c^4/(c^2*d^4*e^5 + 2*a*c*d^2*e^7 + a^2*e^9) - 1/8*(c*x^2 + a
)^(7/2)/(c*d^2*e^7*x^8 + a*e^9*x^8 + 8*c*d^3*e^6*x^7 + 8*a*d*e^8*x^7 + 28*c*d^4*e^5*x^6 + 28*a*d^2*e^7*x^6 + 5
6*c*d^5*e^4*x^5 + 56*a*d^3*e^6*x^5 + 70*c*d^6*e^3*x^4 + 70*a*d^4*e^5*x^4 + 56*c*d^7*e^2*x^3 + 56*a*d^5*e^4*x^3
+ 28*c*d^8*e*x^2 + 28*a*d^6*e^3*x^2 + 8*c*d^9*x + 8*a*d^7*e^2*x + c*d^10/e + a*d^8*e) - 45/128*c^8*d^8*arcsin
h(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(11/2)*e^17) + 35/32*c^7*d^6
*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(9/2)*e^15) - 75/64*c
^6*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(7/2)*e^13) + 1
5/32*c^5*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(5/2)*e^1
1) - 5/128*c^4*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(3/2)*e
^9)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^9,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^9, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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