\(\int \frac {(a+c x^4)^2}{\sqrt {d+e x^2}} \, dx\) [375]

Optimal result

Integrand size = 21, antiderivative size = 174 \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=-\frac {c d \left (35 c d^2+96 a e^2\right ) x \sqrt {d+e x^2}}{128 e^4}+\frac {c \left (35 c d^2+96 a e^2\right ) x^3 \sqrt {d+e x^2}}{192 e^3}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{48 e^2}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}+\frac {\left (35 c^2 d^4+96 a c d^2 e^2+128 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{128 e^{9/2}} \] Output:

-1/128*c*d*(96*a*e^2+35*c*d^2)*x*(e*x^2+d)^(1/2)/e^4+1/192*c*(96*a*e^2+35* 
c*d^2)*x^3*(e*x^2+d)^(1/2)/e^3-7/48*c^2*d*x^5*(e*x^2+d)^(1/2)/e^2+1/8*c^2* 
x^7*(e*x^2+d)^(1/2)/e+1/128*(128*a^2*e^4+96*a*c*d^2*e^2+35*c^2*d^4)*arctan 
h(e^(1/2)*x/(e*x^2+d)^(1/2))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\frac {c x \sqrt {d+e x^2} \left (-105 c d^3-288 a d e^2+70 c d^2 e x^2+192 a e^3 x^2-56 c d e^2 x^4+48 c e^3 x^6\right )}{384 e^4}+\frac {\left (-35 c^2 d^4-96 a c d^2 e^2-128 a^2 e^4\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{128 e^{9/2}} \] Input:

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

Output:

(c*x*Sqrt[d + e*x^2]*(-105*c*d^3 - 288*a*d*e^2 + 70*c*d^2*e*x^2 + 192*a*e^ 
3*x^2 - 56*c*d*e^2*x^4 + 48*c*e^3*x^6))/(384*e^4) + ((-35*c^2*d^4 - 96*a*c 
*d^2*e^2 - 128*a^2*e^4)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/(128*e^(9/2))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1474, 2346, 1474, 27, 299, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\int \frac {-7 c^2 d x^6+16 a c e x^4+8 a^2 e}{\sqrt {e x^2+d}}dx}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\int \frac {c \left (35 c d^2+96 a e^2\right ) x^4+48 a^2 e^2}{\sqrt {e x^2+d}}dx}{6 e}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{6 e}}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (64 a^2 e^3-c d \left (35 c d^2+96 a e^2\right ) x^2\right )}{\sqrt {e x^2+d}}dx}{4 e}+\frac {c x^3 \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{4 e}}{6 e}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{6 e}}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {64 a^2 e^3-c d \left (35 c d^2+96 a e^2\right ) x^2}{\sqrt {e x^2+d}}dx}{4 e}+\frac {c x^3 \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{4 e}}{6 e}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{6 e}}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (128 a^2 e^4+96 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{2 e}-\frac {c d x \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{2 e}\right )}{4 e}+\frac {c x^3 \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{4 e}}{6 e}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{6 e}}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (128 a^2 e^4+96 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 e}-\frac {c d x \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{2 e}\right )}{4 e}+\frac {c x^3 \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{4 e}}{6 e}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{6 e}}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (128 a^2 e^4+96 a c d^2 e^2+35 c^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}-\frac {c d x \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{2 e}\right )}{4 e}+\frac {c x^3 \sqrt {d+e x^2} \left (96 a e^2+35 c d^2\right )}{4 e}}{6 e}-\frac {7 c^2 d x^5 \sqrt {d+e x^2}}{6 e}}{8 e}+\frac {c^2 x^7 \sqrt {d+e x^2}}{8 e}\)

Input:

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

Output:

(c^2*x^7*Sqrt[d + e*x^2])/(8*e) + ((-7*c^2*d*x^5*Sqrt[d + e*x^2])/(6*e) + 
((c*(35*c*d^2 + 96*a*e^2)*x^3*Sqrt[d + e*x^2])/(4*e) + (3*(-1/2*(c*d*(35*c 
*d^2 + 96*a*e^2)*x*Sqrt[d + e*x^2])/e + ((35*c^2*d^4 + 96*a*c*d^2*e^2 + 12 
8*a^2*e^4)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*e^(3/2))))/(4*e))/(6*e 
))/(8*e)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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

rule 224

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1474

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

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {\left (a^{2} e^{4}+\frac {3}{4} a c \,d^{2} e^{2}+\frac {35}{128} c^{2} d^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )-\frac {3 x \left (d \left (\frac {7 c \,x^{4}}{36}+a \right ) e^{\frac {5}{2}}-\frac {2 x^{2} \left (\frac {c \,x^{4}}{4}+a \right ) e^{\frac {7}{2}}}{3}+\frac {35 d^{2} c \left (-\frac {2 e^{\frac {3}{2}} x^{2}}{3}+\sqrt {e}\, d \right )}{96}\right ) \sqrt {e \,x^{2}+d}\, c}{4}}{e^{\frac {9}{2}}}\)	\(113\)
risch	\(-\frac {c x \left (-48 e^{3} c \,x^{6}+56 c d \,e^{2} x^{4}-192 a \,e^{3} x^{2}-70 c \,d^{2} e \,x^{2}+288 d \,e^{2} a +105 d^{3} c \right ) \sqrt {e \,x^{2}+d}}{384 e^{4}}+\frac {\left (128 a^{2} e^{4}+96 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{128 e^{\frac {9}{2}}}\)	\(118\)
default	\(\frac {a^{2} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{\sqrt {e}}+c^{2} \left (\frac {x^{7} \sqrt {e \,x^{2}+d}}{8 e}-\frac {7 d \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{8 e}\right )+2 a c \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )\)	\(205\)

Input:

int((c*x^4+a)^2/(e*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/e^(9/2)*((a^2*e^4+3/4*a*c*d^2*e^2+35/128*c^2*d^4)*arctanh((e*x^2+d)^(1/2 
)/x/e^(1/2))-3/4*x*(d*(7/36*c*x^4+a)*e^(5/2)-2/3*x^2*(1/4*c*x^4+a)*e^(7/2) 
+35/96*d^2*c*(-2/3*e^(3/2)*x^2+e^(1/2)*d))*(e*x^2+d)^(1/2)*c)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\left [\frac {3 \, {\left (35 \, c^{2} d^{4} + 96 \, a c d^{2} e^{2} + 128 \, a^{2} e^{4}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (48 \, c^{2} e^{4} x^{7} - 56 \, c^{2} d e^{3} x^{5} + 2 \, {\left (35 \, c^{2} d^{2} e^{2} + 96 \, a c e^{4}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{3} e + 96 \, a c d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{768 \, e^{5}}, -\frac {3 \, {\left (35 \, c^{2} d^{4} + 96 \, a c d^{2} e^{2} + 128 \, a^{2} e^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (48 \, c^{2} e^{4} x^{7} - 56 \, c^{2} d e^{3} x^{5} + 2 \, {\left (35 \, c^{2} d^{2} e^{2} + 96 \, a c e^{4}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{3} e + 96 \, a c d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{384 \, e^{5}}\right ] \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

Output:

[1/768*(3*(35*c^2*d^4 + 96*a*c*d^2*e^2 + 128*a^2*e^4)*sqrt(e)*log(-2*e*x^2 
 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(48*c^2*e^4*x^7 - 56*c^2*d*e^3*x^5 
 + 2*(35*c^2*d^2*e^2 + 96*a*c*e^4)*x^3 - 3*(35*c^2*d^3*e + 96*a*c*d*e^3)*x 
)*sqrt(e*x^2 + d))/e^5, -1/384*(3*(35*c^2*d^4 + 96*a*c*d^2*e^2 + 128*a^2*e 
^4)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (48*c^2*e^4*x^7 - 56*c^2 
*d*e^3*x^5 + 2*(35*c^2*d^2*e^2 + 96*a*c*e^4)*x^3 - 3*(35*c^2*d^3*e + 96*a* 
c*d*e^3)*x)*sqrt(e*x^2 + d))/e^5]
 

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\begin {cases} \left (a^{2} + \frac {3 d^{2} \cdot \left (2 a c + \frac {35 c^{2} d^{2}}{48 e^{2}}\right )}{8 e^{2}}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {e} \sqrt {d + e x^{2}} + 2 e x \right )}}{\sqrt {e}} & \text {for}\: d \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {e x^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {d + e x^{2}} \left (- \frac {7 c^{2} d x^{5}}{48 e^{2}} + \frac {c^{2} x^{7}}{8 e} - \frac {3 d x \left (2 a c + \frac {35 c^{2} d^{2}}{48 e^{2}}\right )}{8 e^{2}} + \frac {x^{3} \cdot \left (2 a c + \frac {35 c^{2} d^{2}}{48 e^{2}}\right )}{4 e}\right ) & \text {for}\: e \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{5}}{5} + \frac {c^{2} x^{9}}{9}}{\sqrt {d}} & \text {otherwise} \end {cases} \] Input:

integrate((c*x**4+a)**2/(e*x**2+d)**(1/2),x)
 

Output:

Piecewise(((a**2 + 3*d**2*(2*a*c + 35*c**2*d**2/(48*e**2))/(8*e**2))*Piece 
wise((log(2*sqrt(e)*sqrt(d + e*x**2) + 2*e*x)/sqrt(e), Ne(d, 0)), (x*log(x 
)/sqrt(e*x**2), True)) + sqrt(d + e*x**2)*(-7*c**2*d*x**5/(48*e**2) + c**2 
*x**7/(8*e) - 3*d*x*(2*a*c + 35*c**2*d**2/(48*e**2))/(8*e**2) + x**3*(2*a* 
c + 35*c**2*d**2/(48*e**2))/(4*e)), Ne(e, 0)), ((a**2*x + 2*a*c*x**5/5 + c 
**2*x**9/9)/sqrt(d), True))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, c^{2} x^{2}}{e} - \frac {7 \, c^{2} d}{e^{2}}\right )} x^{2} + \frac {35 \, c^{2} d^{2} e^{4} + 96 \, a c e^{6}}{e^{7}}\right )} x^{2} - \frac {3 \, {\left (35 \, c^{2} d^{3} e^{3} + 96 \, a c d e^{5}\right )}}{e^{7}}\right )} \sqrt {e x^{2} + d} x - \frac {{\left (35 \, c^{2} d^{4} + 96 \, a c d^{2} e^{2} + 128 \, a^{2} e^{4}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{128 \, e^{\frac {9}{2}}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

1/384*(2*(4*(6*c^2*x^2/e - 7*c^2*d/e^2)*x^2 + (35*c^2*d^2*e^4 + 96*a*c*e^6 
)/e^7)*x^2 - 3*(35*c^2*d^3*e^3 + 96*a*c*d*e^5)/e^7)*sqrt(e*x^2 + d)*x - 1/ 
128*(35*c^2*d^4 + 96*a*c*d^2*e^2 + 128*a^2*e^4)*log(abs(-sqrt(e)*x + sqrt( 
e*x^2 + d)))/e^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\int \frac {{\left (c\,x^4+a\right )}^2}{\sqrt {e\,x^2+d}} \,d x \] Input:

int((a + c*x^4)^2/(d + e*x^2)^(1/2),x)
 

Output:

int((a + c*x^4)^2/(d + e*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+c x^4\right )^2}{\sqrt {d+e x^2}} \, dx=\frac {-288 \sqrt {e \,x^{2}+d}\, a c d \,e^{3} x +192 \sqrt {e \,x^{2}+d}\, a c \,e^{4} x^{3}-105 \sqrt {e \,x^{2}+d}\, c^{2} d^{3} e x +70 \sqrt {e \,x^{2}+d}\, c^{2} d^{2} e^{2} x^{3}-56 \sqrt {e \,x^{2}+d}\, c^{2} d \,e^{3} x^{5}+48 \sqrt {e \,x^{2}+d}\, c^{2} e^{4} x^{7}+384 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a^{2} e^{4}+288 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a c \,d^{2} e^{2}+105 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{4}}{384 e^{5}} \] Input:

int((c*x^4+a)^2/(e*x^2+d)^(1/2),x)
 

Output:

( - 288*sqrt(d + e*x**2)*a*c*d*e**3*x + 192*sqrt(d + e*x**2)*a*c*e**4*x**3 
 - 105*sqrt(d + e*x**2)*c**2*d**3*e*x + 70*sqrt(d + e*x**2)*c**2*d**2*e**2 
*x**3 - 56*sqrt(d + e*x**2)*c**2*d*e**3*x**5 + 48*sqrt(d + e*x**2)*c**2*e* 
*4*x**7 + 384*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*a**2*e** 
4 + 288*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*a*c*d**2*e**2 
+ 105*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*c**2*d**4)/(384* 
e**5)