3.277 \(\int (d+e x^2)^{3/2} (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=175 \[ \frac {x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac {d x \sqrt {d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}-\frac {x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]

[Out]

1/192*(48*a*e^2-8*b*d*e+3*c*d^2)*x*(e*x^2+d)^(3/2)/e^2-1/48*(-8*b*e+3*c*d)*x*(e*x^2+d)^(5/2)/e^2+1/8*c*x^3*(e*
x^2+d)^(5/2)/e+1/128*d^2*(48*a*e^2-8*b*d*e+3*c*d^2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(5/2)+1/128*d*(48*a*e
^2-8*b*d*e+3*c*d^2)*x*(e*x^2+d)^(1/2)/e^2

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Rubi [A]  time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1159, 388, 195, 217, 206} \[ \frac {x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac {d x \sqrt {d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}-\frac {x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x]

[Out]

(d*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*Sqrt[d + e*x^2])/(128*e^2) + ((3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*(d + e*x^2)
^(3/2))/(192*e^2) - ((3*c*d - 8*b*e)*x*(d + e*x^2)^(5/2))/(48*e^2) + (c*x^3*(d + e*x^2)^(5/2))/(8*e) + (d^2*(3
*c*d^2 - 8*b*d*e + 48*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(5/2))

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 1159

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(c^p*x^(4*p - 1)*
(d + e*x^2)^(q + 1))/(e*(4*p + 2*q + 1)), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*
p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /
; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[
q, -1]

Rubi steps

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

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Mathematica [A]  time = 0.32, size = 157, normalized size = 0.90 \[ \frac {\sqrt {d+e x^2} \left (\frac {3 d^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (8 e (6 a e-b d)+3 c d^2\right )}{\sqrt {\frac {e x^2}{d}+1}}+\sqrt {e} x \left (8 e \left (6 a e \left (5 d+2 e x^2\right )+b \left (3 d^2+14 d e x^2+8 e^2 x^4\right )\right )+c \left (-9 d^3+6 d^2 e x^2+72 d e^2 x^4+48 e^3 x^6\right )\right )\right )}{384 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x]

[Out]

(Sqrt[d + e*x^2]*(Sqrt[e]*x*(c*(-9*d^3 + 6*d^2*e*x^2 + 72*d*e^2*x^4 + 48*e^3*x^6) + 8*e*(6*a*e*(5*d + 2*e*x^2)
+ b*(3*d^2 + 14*d*e*x^2 + 8*e^2*x^4))) + (3*d^(3/2)*(3*c*d^2 + 8*e*(-(b*d) + 6*a*e))*ArcSinh[(Sqrt[e]*x)/Sqrt
[d]])/Sqrt[1 + (e*x^2)/d]))/(384*e^(5/2))

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fricas [A]  time = 1.13, size = 304, normalized size = 1.74 \[ \left [\frac {3 \, {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (48 \, c e^{4} x^{7} + 8 \, {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} x^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{768 \, e^{3}}, -\frac {3 \, {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (48 \, c e^{4} x^{7} + 8 \, {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} x^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{384 \, e^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[1/768*(3*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(48
*c*e^4*x^7 + 8*(9*c*d*e^3 + 8*b*e^4)*x^5 + 2*(3*c*d^2*e^2 + 56*b*d*e^3 + 48*a*e^4)*x^3 - 3*(3*c*d^3*e - 8*b*d^
2*e^2 - 80*a*d*e^3)*x)*sqrt(e*x^2 + d))/e^3, -1/384*(3*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*sqrt(-e)*arctan(sq
rt(-e)*x/sqrt(e*x^2 + d)) - (48*c*e^4*x^7 + 8*(9*c*d*e^3 + 8*b*e^4)*x^5 + 2*(3*c*d^2*e^2 + 56*b*d*e^3 + 48*a*e
^4)*x^3 - 3*(3*c*d^3*e - 8*b*d^2*e^2 - 80*a*d*e^3)*x)*sqrt(e*x^2 + d))/e^3]

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giac [A]  time = 0.22, size = 145, normalized size = 0.83 \[ -\frac {1}{128} \, {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -x e^{\frac {1}{2}} + \sqrt {x^{2} e + d} \right |}\right ) + \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, c x^{2} e + {\left (9 \, c d e^{6} + 8 \, b e^{7}\right )} e^{\left (-6\right )}\right )} x^{2} + {\left (3 \, c d^{2} e^{5} + 56 \, b d e^{6} + 48 \, a e^{7}\right )} e^{\left (-6\right )}\right )} x^{2} - 3 \, {\left (3 \, c d^{3} e^{4} - 8 \, b d^{2} e^{5} - 80 \, a d e^{6}\right )} e^{\left (-6\right )}\right )} \sqrt {x^{2} e + d} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/128*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*e^(-5/2)*log(abs(-x*e^(1/2) + sqrt(x^2*e + d))) + 1/384*(2*(4*(6*c
*x^2*e + (9*c*d*e^6 + 8*b*e^7)*e^(-6))*x^2 + (3*c*d^2*e^5 + 56*b*d*e^6 + 48*a*e^7)*e^(-6))*x^2 - 3*(3*c*d^3*e^
4 - 8*b*d^2*e^5 - 80*a*d*e^6)*e^(-6))*sqrt(x^2*e + d)*x

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maple [A]  time = 0.01, size = 229, normalized size = 1.31 \[ \frac {3 a \,d^{2} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{8 \sqrt {e}}-\frac {b \,d^{3} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{16 e^{\frac {3}{2}}}+\frac {3 c \,d^{4} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{128 e^{\frac {5}{2}}}+\frac {3 \sqrt {e \,x^{2}+d}\, a d x}{8}-\frac {\sqrt {e \,x^{2}+d}\, b \,d^{2} x}{16 e}+\frac {3 \sqrt {e \,x^{2}+d}\, c \,d^{3} x}{128 e^{2}}+\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} c \,x^{3}}{8 e}+\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} a x}{4}-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} b d x}{24 e}+\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} c \,d^{2} x}{64 e^{2}}+\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} b x}{6 e}-\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} c d x}{16 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x)

[Out]

1/8*c*x^3*(e*x^2+d)^(5/2)/e-1/16*c*d/e^2*x*(e*x^2+d)^(5/2)+1/64*c*d^2/e^2*x*(e*x^2+d)^(3/2)+3/128*c*d^3/e^2*x*
(e*x^2+d)^(1/2)+3/128*c*d^4/e^(5/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))+1/6*b*x*(e*x^2+d)^(5/2)/e-1/24*b*d/e*x*(e*x^
2+d)^(3/2)-1/16*b*d^2/e*x*(e*x^2+d)^(1/2)-1/16*b*d^3/e^(3/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))+1/4*a*x*(e*x^2+d)^(
3/2)+3/8*a*d*x*(e*x^2+d)^(1/2)+3/8*a*d^2/e^(1/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))

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maxima [A]  time = 1.02, size = 207, normalized size = 1.18 \[ \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} c x^{3}}{8 \, e} + \frac {1}{4} \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} a x + \frac {3}{8} \, \sqrt {e x^{2} + d} a d x - \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} c d x}{16 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} c d^{2} x}{64 \, e^{2}} + \frac {3 \, \sqrt {e x^{2} + d} c d^{3} x}{128 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} b x}{6 \, e} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} b d x}{24 \, e} - \frac {\sqrt {e x^{2} + d} b d^{2} x}{16 \, e} + \frac {3 \, c d^{4} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{128 \, e^{\frac {5}{2}}} - \frac {b d^{3} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{16 \, e^{\frac {3}{2}}} + \frac {3 \, a d^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

1/8*(e*x^2 + d)^(5/2)*c*x^3/e + 1/4*(e*x^2 + d)^(3/2)*a*x + 3/8*sqrt(e*x^2 + d)*a*d*x - 1/16*(e*x^2 + d)^(5/2)
*c*d*x/e^2 + 1/64*(e*x^2 + d)^(3/2)*c*d^2*x/e^2 + 3/128*sqrt(e*x^2 + d)*c*d^3*x/e^2 + 1/6*(e*x^2 + d)^(5/2)*b*
x/e - 1/24*(e*x^2 + d)^(3/2)*b*d*x/e - 1/16*sqrt(e*x^2 + d)*b*d^2*x/e + 3/128*c*d^4*arcsinh(e*x/sqrt(d*e))/e^(
5/2) - 1/16*b*d^3*arcsinh(e*x/sqrt(d*e))/e^(3/2) + 3/8*a*d^2*arcsinh(e*x/sqrt(d*e))/sqrt(e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x)

[Out]

int((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4), x)

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sympy [B]  time = 31.10, size = 413, normalized size = 2.36 \[ \frac {a d^{\frac {3}{2}} x \sqrt {1 + \frac {e x^{2}}{d}}}{2} + \frac {a d^{\frac {3}{2}} x}{8 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 a \sqrt {d} e x^{3}}{8 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{8 \sqrt {e}} + \frac {a e^{2} x^{5}}{4 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {b d^{\frac {5}{2}} x}{16 e \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {17 b d^{\frac {3}{2}} x^{3}}{48 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {11 b \sqrt {d} e x^{5}}{24 \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {b d^{3} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{16 e^{\frac {3}{2}}} + \frac {b e^{2} x^{7}}{6 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {3 c d^{\frac {7}{2}} x}{128 e^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {c d^{\frac {5}{2}} x^{3}}{128 e \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {13 c d^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {5 c \sqrt {d} e x^{7}}{16 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 c d^{4} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{128 e^{\frac {5}{2}}} + \frac {c e^{2} x^{9}}{8 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(c*x**4+b*x**2+a),x)

[Out]

a*d**(3/2)*x*sqrt(1 + e*x**2/d)/2 + a*d**(3/2)*x/(8*sqrt(1 + e*x**2/d)) + 3*a*sqrt(d)*e*x**3/(8*sqrt(1 + e*x**
2/d)) + 3*a*d**2*asinh(sqrt(e)*x/sqrt(d))/(8*sqrt(e)) + a*e**2*x**5/(4*sqrt(d)*sqrt(1 + e*x**2/d)) + b*d**(5/2
)*x/(16*e*sqrt(1 + e*x**2/d)) + 17*b*d**(3/2)*x**3/(48*sqrt(1 + e*x**2/d)) + 11*b*sqrt(d)*e*x**5/(24*sqrt(1 +
e*x**2/d)) - b*d**3*asinh(sqrt(e)*x/sqrt(d))/(16*e**(3/2)) + b*e**2*x**7/(6*sqrt(d)*sqrt(1 + e*x**2/d)) - 3*c*
d**(7/2)*x/(128*e**2*sqrt(1 + e*x**2/d)) - c*d**(5/2)*x**3/(128*e*sqrt(1 + e*x**2/d)) + 13*c*d**(3/2)*x**5/(64
*sqrt(1 + e*x**2/d)) + 5*c*sqrt(d)*e*x**7/(16*sqrt(1 + e*x**2/d)) + 3*c*d**4*asinh(sqrt(e)*x/sqrt(d))/(128*e**
(5/2)) + c*e**2*x**9/(8*sqrt(d)*sqrt(1 + e*x**2/d))

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