3.21.16 \(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2016]

Optimal. Leaf size=158 \[ -\frac {5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac {5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}+\frac {5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]

[Out]

-5/3*e/(-a*e^2+c*d^2)^2/(e*x+d)^(3/2)-1/(-a*e^2+c*d^2)/(c*d*x+a*e)/(e*x+d)^(3/2)+5*c^(3/2)*d^(3/2)*e*arctanh(c
^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/(-a*e^2+c*d^2)^(7/2)-5*c*d*e/(-a*e^2+c*d^2)^3/(e*x+d)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 44, 53, 65, 214} \begin {gather*} \frac {5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}-\frac {5 c d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )^3}-\frac {1}{(d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac {5 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]

[Out]

(-5*e)/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*(d + e*x)^(3/2)) - (5*c*d*e)/(
(c*d^2 - a*e^2)^3*Sqrt[d + e*x]) + (5*c^(3/2)*d^(3/2)*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a
*e^2]])/(c*d^2 - a*e^2)^(7/2)

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 640

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac {(5 e) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac {(5 c d e) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac {5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (5 c^2 d^2 e\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 \left (c d^2-a e^2\right )^3}\\ &=-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac {5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (5 c^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{\left (c d^2-a e^2\right )^3}\\ &=-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{3/2}}-\frac {5 c d e}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}+\frac {5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 156, normalized size = 0.99 \begin {gather*} \frac {2 a^2 e^4-2 a c d e^2 (7 d+5 e x)-c^2 d^2 \left (3 d^2+20 d e x+15 e^2 x^2\right )}{3 \left (c d^2-a e^2\right )^3 (a e+c d x) (d+e x)^{3/2}}+\frac {5 c^{3/2} d^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]

[Out]

(2*a^2*e^4 - 2*a*c*d*e^2*(7*d + 5*e*x) - c^2*d^2*(3*d^2 + 20*d*e*x + 15*e^2*x^2))/(3*(c*d^2 - a*e^2)^3*(a*e +
c*d*x)*(d + e*x)^(3/2)) + (5*c^(3/2)*d^(3/2)*e*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])
/(-(c*d^2) + a*e^2)^(7/2)

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Maple [A]
time = 0.88, size = 153, normalized size = 0.97

method	result	size
derivativedivides	\(2 e \left (\frac {c^{2} d^{2} \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {5 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c d}{\left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {e x +d}}\right )\)	\(153\)
default	\(2 e \left (\frac {c^{2} d^{2} \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {5 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c d}{\left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {e x +d}}\right )\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

2*e*(1/(a*e^2-c*d^2)^3*c^2*d^2*(1/2*(e*x+d)^(1/2)/(c*d*(e*x+d)+e^2*a-c*d^2)+5/2/((a*e^2-c*d^2)*c*d)^(1/2)*arct
an(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)))-1/3/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)+2/(a*e^2-c*d^2)^3*c*d/(e*x+
d)^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (139) = 278\).
time = 2.44, size = 875, normalized size = 5.54 \begin {gather*} \left [-\frac {15 \, {\left (c^{2} d^{4} x e + a c d x^{2} e^{4} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} + a c d^{3}\right )} e^{2}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d x e + 2 \, c d^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} - a e^{2}}{c d x + a e}\right ) + 2 \, {\left (20 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 10 \, a c d x e^{3} - 2 \, a^{2} e^{4} + {\left (15 \, c^{2} d^{2} x^{2} + 14 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{6 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )}}, \frac {15 \, {\left (c^{2} d^{4} x e + a c d x^{2} e^{4} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} + a c d^{3}\right )} e^{2}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d x e + c d^{2}}\right ) - {\left (20 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 10 \, a c d x e^{3} - 2 \, a^{2} e^{4} + {\left (15 \, c^{2} d^{2} x^{2} + 14 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

[-1/6*(15*(c^2*d^4*x*e + a*c*d*x^2*e^4 + (c^2*d^2*x^3 + 2*a*c*d^2*x)*e^3 + (2*c^2*d^3*x^2 + a*c*d^3)*e^2)*sqrt
(c*d/(c*d^2 - a*e^2))*log((c*d*x*e + 2*c*d^2 - 2*(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(c*d/(c*d^2 - a*e^2)) - a*e
^2)/(c*d*x + a*e)) + 2*(20*c^2*d^3*x*e + 3*c^2*d^4 + 10*a*c*d*x*e^3 - 2*a^2*e^4 + (15*c^2*d^2*x^2 + 14*a*c*d^2
)*e^2)*sqrt(x*e + d))/(c^4*d^9*x - a^4*x^2*e^9 - (a^3*c*d*x^3 + 2*a^4*d*x)*e^8 + (a^3*c*d^2*x^2 - a^4*d^2)*e^7
+ (3*a^2*c^2*d^3*x^3 + 5*a^3*c*d^3*x)*e^6 + 3*(a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^5 - 3*(a*c^3*d^5*x^3 + a^2*c^2*
d^5*x)*e^4 - (5*a*c^3*d^6*x^2 + 3*a^2*c^2*d^6)*e^3 + (c^4*d^7*x^3 - a*c^3*d^7*x)*e^2 + (2*c^4*d^8*x^2 + a*c^3*
d^8)*e), 1/3*(15*(c^2*d^4*x*e + a*c*d*x^2*e^4 + (c^2*d^2*x^3 + 2*a*c*d^2*x)*e^3 + (2*c^2*d^3*x^2 + a*c*d^3)*e^
2)*sqrt(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*x*e + c*d^
2)) - (20*c^2*d^3*x*e + 3*c^2*d^4 + 10*a*c*d*x*e^3 - 2*a^2*e^4 + (15*c^2*d^2*x^2 + 14*a*c*d^2)*e^2)*sqrt(x*e +
d))/(c^4*d^9*x - a^4*x^2*e^9 - (a^3*c*d*x^3 + 2*a^4*d*x)*e^8 + (a^3*c*d^2*x^2 - a^4*d^2)*e^7 + (3*a^2*c^2*d^3
*x^3 + 5*a^3*c*d^3*x)*e^6 + 3*(a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^5 - 3*(a*c^3*d^5*x^3 + a^2*c^2*d^5*x)*e^4 - (5*a
*c^3*d^6*x^2 + 3*a^2*c^2*d^6)*e^3 + (c^4*d^7*x^3 - a*c^3*d^7*x)*e^2 + (2*c^4*d^8*x^2 + a*c^3*d^8)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {5}{2}} \left (a e + c d x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((d + e*x)**(5/2)*(a*e + c*d*x)**2), x)

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Giac [A]
time = 1.15, size = 249, normalized size = 1.58 \begin {gather*} -\frac {5 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} - \frac {\sqrt {x e + d} c^{2} d^{2} e}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d e + c d^{2} e - a e^{3}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

-5*c^2*d^2*arctan(sqrt(x*e + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))*e/((c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^
4 - a^3*e^6)*sqrt(-c^2*d^3 + a*c*d*e^2)) - sqrt(x*e + d)*c^2*d^2*e/((c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e
^4 - a^3*e^6)*((x*e + d)*c*d - c*d^2 + a*e^2)) - 2/3*(6*(x*e + d)*c*d*e + c*d^2*e - a*e^3)/((c^3*d^6 - 3*a*c^2
*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(x*e + d)^(3/2))

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Mupad [B]
time = 0.78, size = 200, normalized size = 1.27 \begin {gather*} \frac {\frac {10\,c\,d\,e\,\left (d+e\,x\right )}{3\,{\left (a\,e^2-c\,d^2\right )}^2}-\frac {2\,e}{3\,\left (a\,e^2-c\,d^2\right )}+\frac {5\,c^2\,d^2\,e\,{\left (d+e\,x\right )}^2}{{\left (a\,e^2-c\,d^2\right )}^3}}{\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{3/2}+c\,d\,{\left (d+e\,x\right )}^{5/2}}+\frac {5\,c^{3/2}\,d^{3/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^{7/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2),x)

[Out]

((10*c*d*e*(d + e*x))/(3*(a*e^2 - c*d^2)^2) - (2*e)/(3*(a*e^2 - c*d^2)) + (5*c^2*d^2*e*(d + e*x)^2)/(a*e^2 - c
*d^2)^3)/((a*e^2 - c*d^2)*(d + e*x)^(3/2) + c*d*(d + e*x)^(5/2)) + (5*c^(3/2)*d^(3/2)*e*atan((c^(1/2)*d^(1/2)*
(d + e*x)^(1/2)*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4))/(a*e^2 - c*d^2)^(7/2)))/(a*e^2 - c*d^
2)^(7/2)

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