\(\int \frac {(d+e x)^{7/2}}{(a-c x^2)^2} \, dx\) [624]

Optimal result

Integrand size = 20, antiderivative size = 263 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}} \]

[Out]

1/2*d*e*(e*x+d)^(3/2)/a/c+1/2*(c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(-c*x^2+a)+1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^
(1/2)+d*c^(1/2))^(1/2))*(e*a^(1/2)+d*c^(1/2))^(5/2)*(-5*e*a^(1/2)+2*d*c^(1/2))/a^(3/2)/c^(9/4)-1/4*arctanh(c^(
1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(5/2)*(5*e*a^(1/2)+2*d*c^(1/2))/a^(3/2
)/c^(9/4)+1/2*e*(5*a*e^2+c*d^2)*(e*x+d)^(1/2)/a/c^2

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {753, 839, 841, 1180, 214} \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=-\frac {\left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac {e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{2 a c^2}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac {d e (d+e x)^{3/2}}{2 a c} \]

[In]

Int[(d + e*x)^(7/2)/(a - c*x^2)^2,x]

[Out]

(e*(c*d^2 + 5*a*e^2)*Sqrt[d + e*x])/(2*a*c^2) + (d*e*(d + e*x)^(3/2))/(2*a*c) + ((a*e + c*d*x)*(d + e*x)^(5/2)
)/(2*a*c*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e
*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^
(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4))

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 753

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

Rule 839

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(
c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 841

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (-2 c d^2+5 a e^2\right )+\frac {3}{2} c d e x\right )}{a-c x^2} \, dx}{2 a c} \\ & = \frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (c d \left (c d^2-4 a e^2\right )-\frac {1}{2} c e \left (c d^2+5 a e^2\right ) x\right )}{a-c x^2} \, dx}{2 a c^2} \\ & = \frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {-\frac {1}{2} c \left (c d^2-5 a e^2\right ) \left (2 c d^2+a e^2\right )-\frac {1}{2} c^2 d e \left (c d^2-13 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c^3} \\ & = \frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} c^2 d^2 e \left (c d^2-13 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) \left (2 c d^2+a e^2\right )-\frac {1}{2} c^2 d e \left (c d^2-13 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c^3} \\ & = \frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}+\frac {\left (\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^3\right ) \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c^{3/2}}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (2 \sqrt {c} d+5 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c^{3/2}} \\ & = \frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^2 e^3+c^2 d^3 x+a c e \left (3 d^2+3 d e x-4 e^2 x^2\right )\right )}{-a+c x^2}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2} c^2} \]

[In]

Integrate[(d + e*x)^(7/2)/(a - c*x^2)^2,x]

[Out]

((-2*Sqrt[a]*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(3*d^2 + 3*d*e*x - 4*e^2*x^2)))/(-a + c*x^2) + ((2*S
qrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqr
t[c]*d + Sqrt[a]*e)])/Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e] - ((Sqrt[c]*d - Sqrt[a]*e)^3*(2*Sqrt[c]*d + 5*Sqrt[a]*e
)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/Sqrt[-(c*d) + Sqrt[a]*Sqrt
[c]*e])/(4*a^(3/2)*c^2)

Maple [A] (verified)

Time = 2.56 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {5 \left (\left (-c \,x^{2}+a \right ) c \left (\frac {\left (-13 a d \,e^{2}+d^{3} c \right ) \sqrt {a c \,e^{2}}}{5}+a^{2} e^{4}+\frac {9 a c \,d^{2} e^{2}}{5}-\frac {2 c^{2} d^{4}}{5}\right ) e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (-c \,x^{2}+a \right ) \left (\frac {\left (13 a d \,e^{2}-d^{3} c \right ) \sqrt {a c \,e^{2}}}{5}+a^{2} e^{4}+\frac {9 a c \,d^{2} e^{2}}{5}-\frac {2 c^{2} d^{4}}{5}\right ) c e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )-2 \sqrt {a c \,e^{2}}\, \left (\frac {c^{2} d^{3} x}{5}+\frac {3 \left (-\frac {4}{3} x^{2} e^{2}+d e x +d^{2}\right ) e a c}{5}+a^{2} e^{3}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\right )\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, c^{2} \left (-c \,x^{2}+a \right ) a}\)	\(341\)
derivativedivides	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}-\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-5 a^{2} e^{4}-9 a c \,d^{2} e^{2}+2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}+9 a c \,d^{2} e^{2}-2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}}{c^{2}}\right )\)	\(345\)
default	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}-\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-5 a^{2} e^{4}-9 a c \,d^{2} e^{2}+2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}+9 a c \,d^{2} e^{2}-2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}}{c^{2}}\right )\)	\(345\)
risch	\(\frac {2 e^{3} \sqrt {e x +d}}{c^{2}}+\frac {2 e^{3} \left (\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}-\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{c \left (e x +d \right )^{2}-2 c d \left (e x +d \right )-e^{2} a +c \,d^{2}}+\frac {c \left (\frac {\left (-5 a^{2} e^{4}-9 a c \,d^{2} e^{2}+2 c^{2} d^{4}+13 \sqrt {a c \,e^{2}}\, a d \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (5 a^{2} e^{4}+9 a c \,d^{2} e^{2}-2 c^{2} d^{4}+13 \sqrt {a c \,e^{2}}\, a d \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 e^{2} a}\right )}{c^{2}}\)	\(348\)

[In]

int((e*x+d)^(7/2)/(-c*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-5/4/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c*e^2)^(1/2)*((-c*x^2+a)*c*(1/5*(-13*
a*d*e^2+c*d^3)*(a*c*e^2)^(1/2)+a^2*e^4+9/5*a*c*d^2*e^2-2/5*c^2*d^4)*e*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c
*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*((-c*x^2+a)*(1/5*(13*a*d*e^2
-c*d^3)*(a*c*e^2)^(1/2)+a^2*e^4+9/5*a*c*d^2*e^2-2/5*c^2*d^4)*c*e*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2)
)*c)^(1/2))-2*(a*c*e^2)^(1/2)*(1/5*c^2*d^3*x+3/5*(-4/3*x^2*e^2+d*e*x+d^2)*e*a*c+a^2*e^3)*((c*d+(a*c*e^2)^(1/2)
)*c)^(1/2)*(e*x+d)^(1/2)))/c^2/(-c*x^2+a)/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2073 vs. \(2 (200) = 400\).

Time = 0.53 (sec) , antiderivative size = 2073, normalized size of antiderivative = 7.88 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(7/2)/(-c*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*((a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqr
t((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*
c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2
500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*
e^8 - 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 - 4*a^4*c^7*d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 2196
6*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a
^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 +
7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^
2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^
2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4
*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x + d) - (
35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 - 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 - 4*a^4*c^7*
d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e
^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^
4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^
3*c^4))) + (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c
^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)
/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e
^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^
3*d^2*e^8 - 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 - 4*a^4*c^7*d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8
+ 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2
+ 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*
e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 3
5*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21
966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771
*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x +
d) - (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 - 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 - 4*a^
4*c^7*d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625
*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c^4*sqrt((1
225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)
))/(a^3*c^4))) + 4*(4*a*c*e^3*x^2 - 3*a*c*d^2*e - 5*a^2*e^3 - (c^2*d^3 + 3*a*c*d*e^2)*x)*sqrt(e*x + d))/(a*c^3
*x^2 - a^2*c^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(7/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} - a\right )}^{2}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(-c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 - a)^2, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (200) = 400\).

Time = 0.38 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.24 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e^{3}}{c^{2}} + \frac {{\left ({\left (\sqrt {a c} c d^{3} e - 13 \, \sqrt {a c} a d e^{3}\right )} a^{2} e^{2} {\left | c \right |} + {\left (a c^{2} d^{4} e + 4 \, a^{2} c d^{2} e^{3} - 5 \, a^{3} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} - {\left (2 \, \sqrt {a c} a c^{2} d^{5} e - 9 \, \sqrt {a c} a^{2} c d^{3} e^{3} - 5 \, \sqrt {a c} a^{3} d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d - \sqrt {a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left ({\left (c^{2} d^{3} e - 13 \, a c d e^{3}\right )} a^{2} e^{2} {\left | c \right |} - {\left (\sqrt {a c} c^{2} d^{4} e + 4 \, \sqrt {a c} a c d^{2} e^{3} - 5 \, \sqrt {a c} a^{2} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} - {\left (2 \, a c^{3} d^{5} e - 9 \, a^{2} c^{2} d^{3} e^{3} - 5 \, a^{3} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - \sqrt {e x + d} c^{2} d^{4} e + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d e^{3} + \sqrt {e x + d} a^{2} e^{5}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c^{2}} \]

[In]

integrate((e*x+d)^(7/2)/(-c*x^2+a)^2,x, algorithm="giac")

[Out]

2*sqrt(e*x + d)*e^3/c^2 + 1/4*((sqrt(a*c)*c*d^3*e - 13*sqrt(a*c)*a*d*e^3)*a^2*e^2*abs(c) + (a*c^2*d^4*e + 4*a^
2*c*d^2*e^3 - 5*a^3*e^5)*abs(a)*abs(c)*abs(e) - (2*sqrt(a*c)*a*c^2*d^5*e - 9*sqrt(a*c)*a^2*c*d^3*e^3 - 5*sqrt(
a*c)*a^3*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^3*d + sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c
^3))/(a*c^3)))/((a^2*c^3*d - sqrt(a*c)*a^2*c^2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/4*((c^2*d^3*
e - 13*a*c*d*e^3)*a^2*e^2*abs(c) - (sqrt(a*c)*c^2*d^4*e + 4*sqrt(a*c)*a*c*d^2*e^3 - 5*sqrt(a*c)*a^2*e^5)*abs(a
)*abs(c)*abs(e) - (2*a*c^3*d^5*e - 9*a^2*c^2*d^3*e^3 - 5*a^3*c*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^
3*d - sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*e + sqrt(a*c)*a*c^3*d)*sqrt(-c^
2*d + sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/2*((e*x + d)^(3/2)*c^2*d^3*e - sqrt(e*x + d)*c^2*d^4*e + 3*(e*x + d)^(
3/2)*a*c*d*e^3 + sqrt(e*x + d)*a^2*e^5)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)*a*c^2)

Mupad [B] (verification not implemented)

Time = 10.35 (sec) , antiderivative size = 4090, normalized size of antiderivative = 15.55 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(7/2)/(a - c*x^2)^2,x)

[Out]

atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)
/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^
3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((491*a*d^3*e^11)/(2*c) - (885*d^5*e^9)/2 + (329*c*d^7*e^7)/(2*a)
+ (50*a^2*d*e^13)/c^2 - (35*c^2*d^9*e^5)/(2*a^2) + (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^7) + (335*d^2*e^12*(a^9
*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^4)
+ (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^3)) - (d^3*e^7*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) +
d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9)
+ (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*a^3
*c^4*d^7*e^7)/2 - (35*a^2*c^5*d^9*e^5)/2 - (885*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 + 50*a^6*c*d*e^1
3 + (125*a^2*e^14*(a^9*c^9)^(1/2))/(4*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^2) - (204*d^4*e^10*(a^9*c^9)^(1
/2))/c^2 + (335*a*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a*c)) + (d^5*e^5*(a^9*c^9
)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^
2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*
c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/(50*a^7*d*e^13 + (491*a^6*c*d^3*e^11)/2 - (35*a^3*c^4*d^9*e^5)/2 + (329*a
^4*c^3*d^7*e^7)/2 - (885*a^5*c^2*d^5*e^9)/2 + (125*a^3*e^14*(a^9*c^9)^(1/2))/(4*c^5) - (7*d^6*e^8*(a^9*c^9)^(1
/2))/(2*c^2) - (204*a*d^4*e^10*(a^9*c^9)^(1/2))/c^3 + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a*c) + (335*a^2*d^2*e^12
*(a^9*c^9)^(1/2))/(2*c^4)) + (a*d^2*e^8*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/
(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/
(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*d^7*e^7)/(2*a) - (885*d^5*e^9)/(2*
c) + (491*a*d^3*e^11)/(2*c^2) - (35*c*d^9*e^5)/(2*a^2) + (50*a^2*d*e^13)/c^3 + (125*e^14*(a^9*c^9)^(1/2))/(4*a
^2*c^8) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8*(
a^9*c^9)^(1/2))/(2*a^5*c^5) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^4)) - (c*d^4*e^6*(d + e*x)^(1/2)*((105*d*e
^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))
/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*7
0i)/((329*d^7*e^7)/(2*a) - (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) - (35*c*d^9*e^5)/(2*a^2) + (50*a^2*d
*e^13)/c^3 + (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^8) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^
10*(a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^5) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*
c^4)) - (d*e^9*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3
) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c
^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/(50*a^5*c^2*d*e^13 - (35*a*c^6*d^9*e^5)/2 + (329*a
^2*c^5*d^7*e^7)/2 - (885*a^3*c^4*d^5*e^9)/2 + (491*a^4*c^3*d^3*e^11)/2 + (125*a*e^14*(a^9*c^9)^(1/2))/(4*c^3)
- (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^2) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^2) + (35*c*d^8*e^6*(a^9*c^9)^(1/2)
)/(4*a^3) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a*c)))*((25*a^2*e^7*(a^9*c^9)^(1/2) + 4*a^3*c^8*d^7 + 105*a^6*c^5*
d*e^6 - 35*a^4*c^7*d^5*e^2 + 70*a^5*c^6*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c^9)^(1/2) + 154*a*c*d^2*e^5*(a^9*c^9)^(
1/2))/(64*a^6*c^9))^(1/2)*2i - atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3
*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(
1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((885*d^5*e^9)/2 - (491*a*d^3*e^11)
/(2*c) - (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 + (35*c^2*d^9*e^5)/(2*a^2) + (125*e^14*(a^9*c^9)^(1/2))/(
4*a^2*c^7) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e^
8*(a^9*c^9)^(1/2))/(2*a^5*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^3)) + (d^3*e^7*(a^9*c^9)^(1/2)*(d + e*x
)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7
*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*
a^6*c^7))^(1/2)*308i)/((35*a^2*c^5*d^9*e^5)/2 - (329*a^3*c^4*d^7*e^7)/2 + (885*a^4*c^3*d^5*e^9)/2 - (491*a^5*c
^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 + (125*a^2*e^14*(a^9*c^9)^(1/2))/(4*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^
2) - (204*d^4*e^10*(a^9*c^9)^(1/2))/c^2 + (335*a*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2
))/(2*a*c)) + (d^5*e^5*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(
32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(1/2))/(
32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/(50*a^7*d*e^13 + (491*a^6*c*d^3*e^11)/2 -
(35*a^3*c^4*d^9*e^5)/2 + (329*a^4*c^3*d^7*e^7)/2 - (885*a^5*c^2*d^5*e^9)/2 - (125*a^3*e^14*(a^9*c^9)^(1/2))/(4
*c^5) + (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*c^2) + (204*a*d^4*e^10*(a^9*c^9)^(1/2))/c^3 - (35*d^8*e^6*(a^9*c^9)^(1/
2))/(4*a*c) - (335*a^2*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^4)) - (a*d^2*e^8*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) +
d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9)
- (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*d^7
*e^7)/(2*a) - (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) - (35*c*d^9*e^5)/(2*a^2) + (50*a^2*d*e^13)/c^3 -
(125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^7) + (204*d^4*e^10*(a^9*c^9)^
(1/2))/(a^4*c^6) + (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^4)) + (c*d^
4*e^6*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*
c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^
9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((329*d^7*e^7)/(2*a) - (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) - (35
*c*d^9*e^5)/(2*a^2) + (50*a^2*d*e^13)/c^3 - (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(a^9*c^9)^(
1/2))/(2*a^3*c^7) + (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^6) + (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d
^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^4)) + (d*e^9*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*
a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2
*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((35*a*c^6*d^9*e^5)
/2 - 50*a^5*c^2*d*e^13 - (329*a^2*c^5*d^7*e^7)/2 + (885*a^3*c^4*d^5*e^9)/2 - (491*a^4*c^3*d^3*e^11)/2 + (125*a
*e^14*(a^9*c^9)^(1/2))/(4*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^2) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^2)
+ (35*c*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^3) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a*c)))*((4*a^3*c^8*d^7 - 25*a^2*e^7
*(a^9*c^9)^(1/2) + 105*a^6*c^5*d*e^6 - 35*a^4*c^7*d^5*e^2 + 70*a^5*c^6*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c^9)^(1/2
) - 154*a*c*d^2*e^5*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*2i - (((a^2*e^5 - c^2*d^4*e)*(d + e*x)^(1/2))/(2*a) +
((c^2*d^3*e + 3*a*c*d*e^3)*(d + e*x)^(3/2))/(2*a))/(c^3*(d + e*x)^2 + c^3*d^2 - a*c^2*e^2 - 2*c^3*d*(d + e*x)
) + (2*e^3*(d + e*x)^(1/2))/c^2