Integrand size = 28, antiderivative size = 161 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\frac {c \left (5 c^2+18 a c e^2+24 a^2 e^4\right ) x \sqrt {-1+e x} \sqrt {1+e x}}{16 e^6}+\frac {c^2 \left (5 c+18 a e^2\right ) x^3 \sqrt {-1+e x} \sqrt {1+e x}}{24 e^4}+\frac {c^3 x^5 \sqrt {-1+e x} \sqrt {1+e x}}{6 e^2}+\frac {\left (c+2 a e^2\right ) \left (5 c^2+8 a c e^2+8 a^2 e^4\right ) \text {arccosh}(e x)}{16 e^7} \] Output:
1/16*c*(24*a^2*e^4+18*a*c*e^2+5*c^2)*x*(e*x-1)^(1/2)*(e*x+1)^(1/2)/e^6+1/2 4*c^2*(18*a*e^2+5*c)*x^3*(e*x-1)^(1/2)*(e*x+1)^(1/2)/e^4+1/6*c^3*x^5*(e*x- 1)^(1/2)*(e*x+1)^(1/2)/e^2+1/16*(2*a*e^2+c)*(8*a^2*e^4+8*a*c*e^2+5*c^2)*ar ccosh(e*x)/e^7
Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\frac {c e x \sqrt {-1+e x} \sqrt {1+e x} \left (72 a^2 e^4+18 a c e^2 \left (3+2 e^2 x^2\right )+c^2 \left (15+10 e^2 x^2+8 e^4 x^4\right )\right )+6 \left (5 c^3+18 a c^2 e^2+24 a^2 c e^4+16 a^3 e^6\right ) \text {arctanh}\left (\sqrt {\frac {-1+e x}{1+e x}}\right )}{48 e^7} \] Input:
Integrate[(a + c*x^2)^3/(Sqrt[-1 + e*x]*Sqrt[1 + e*x]),x]
Output:
(c*e*x*Sqrt[-1 + e*x]*Sqrt[1 + e*x]*(72*a^2*e^4 + 18*a*c*e^2*(3 + 2*e^2*x^ 2) + c^2*(15 + 10*e^2*x^2 + 8*e^4*x^4)) + 6*(5*c^3 + 18*a*c^2*e^2 + 24*a^2 *c*e^4 + 16*a^3*e^6)*ArcTanh[Sqrt[(-1 + e*x)/(1 + e*x)]])/(48*e^7)
Time = 0.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.32, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {648, 318, 403, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3}{\sqrt {e x-1} \sqrt {e x+1}} \, dx\) |
| \(\Big \downarrow \) 648 |
| \(\displaystyle \frac {\sqrt {e^2 x^2-1} \int \frac {\left (c x^2+a\right )^3}{\sqrt {e^2 x^2-1}}dx}{\sqrt {e x-1} \sqrt {e x+1}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {\int \frac {\left (c x^2+a\right ) \left (5 c \left (2 a e^2+c\right ) x^2+a \left (6 a e^2+c\right )\right )}{\sqrt {e^2 x^2-1}}dx}{6 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (a+c x^2\right )^2}{6 e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {\frac {\int \frac {c \left (44 a^2 e^4+44 a c e^2+15 c^2\right ) x^2+a \left (24 a^2 e^4+14 a c e^2+5 c^2\right )}{\sqrt {e^2 x^2-1}}dx}{4 e^2}+\frac {5 c x \sqrt {e^2 x^2-1} \left (2 a e^2+c\right ) \left (a+c x^2\right )}{4 e^2}}{6 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (a+c x^2\right )^2}{6 e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {\frac {\frac {3 \left (2 a e^2+c\right ) \left (8 a^2 e^4+8 a c e^2+5 c^2\right ) \int \frac {1}{\sqrt {e^2 x^2-1}}dx}{2 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (44 a^2 e^4+44 a c e^2+15 c^2\right )}{2 e^2}}{4 e^2}+\frac {5 c x \sqrt {e^2 x^2-1} \left (2 a e^2+c\right ) \left (a+c x^2\right )}{4 e^2}}{6 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (a+c x^2\right )^2}{6 e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {\frac {\frac {3 \left (2 a e^2+c\right ) \left (8 a^2 e^4+8 a c e^2+5 c^2\right ) \int \frac {1}{1-\frac {e^2 x^2}{e^2 x^2-1}}d\frac {x}{\sqrt {e^2 x^2-1}}}{2 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (44 a^2 e^4+44 a c e^2+15 c^2\right )}{2 e^2}}{4 e^2}+\frac {5 c x \sqrt {e^2 x^2-1} \left (2 a e^2+c\right ) \left (a+c x^2\right )}{4 e^2}}{6 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (a+c x^2\right )^2}{6 e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {\frac {\frac {3 \left (2 a e^2+c\right ) \left (8 a^2 e^4+8 a c e^2+5 c^2\right ) \text {arctanh}\left (\frac {e x}{\sqrt {e^2 x^2-1}}\right )}{2 e^3}+\frac {c x \sqrt {e^2 x^2-1} \left (44 a^2 e^4+44 a c e^2+15 c^2\right )}{2 e^2}}{4 e^2}+\frac {5 c x \sqrt {e^2 x^2-1} \left (2 a e^2+c\right ) \left (a+c x^2\right )}{4 e^2}}{6 e^2}+\frac {c x \sqrt {e^2 x^2-1} \left (a+c x^2\right )^2}{6 e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\) |
Input:
Int[(a + c*x^2)^3/(Sqrt[-1 + e*x]*Sqrt[1 + e*x]),x]
Output:
(Sqrt[-1 + e^2*x^2]*((c*x*(a + c*x^2)^2*Sqrt[-1 + e^2*x^2])/(6*e^2) + ((5* c*(c + 2*a*e^2)*x*(a + c*x^2)*Sqrt[-1 + e^2*x^2])/(4*e^2) + ((c*(15*c^2 + 44*a*c*e^2 + 44*a^2*e^4)*x*Sqrt[-1 + e^2*x^2])/(2*e^2) + (3*(c + 2*a*e^2)* (5*c^2 + 8*a*c*e^2 + 8*a^2*e^4)*ArcTanh[(e*x)/Sqrt[-1 + e^2*x^2]])/(2*e^3) )/(4*e^2))/(6*e^2)))/(Sqrt[-1 + e*x]*Sqrt[1 + e*x])
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])
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]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) ^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c *e + d*f*x^2)^FracPart[m]) Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && !(EqQ[p, 2] && LtQ[m, -1])
Time = 0.77 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\frac {c x \left (8 c^{2} x^{4} e^{4}+36 x^{2} a c \,e^{4}+72 a^{2} e^{4}+10 c^{2} e^{2} x^{2}+54 a c \,e^{2}+15 c^{2}\right ) \sqrt {e x -1}\, \sqrt {e x +1}}{48 e^{6}}+\frac {\left (16 e^{6} a^{3}+24 e^{4} a^{2} c +18 c^{2} a \,e^{2}+5 c^{3}\right ) \ln \left (\frac {e^{2} x}{\sqrt {e^{2}}}+\sqrt {e^{2} x^{2}-1}\right ) \sqrt {\left (e x -1\right ) \left (e x +1\right )}}{16 e^{6} \sqrt {e^{2}}\, \sqrt {e x -1}\, \sqrt {e x +1}}\) | \(168\) |
| default | \(\frac {\sqrt {e x -1}\, \sqrt {e x +1}\, \left (8 \,\operatorname {csgn}\left (e \right ) c^{3} e^{5} x^{5} \sqrt {e^{2} x^{2}-1}+36 \,\operatorname {csgn}\left (e \right ) a \,c^{2} e^{5} x^{3} \sqrt {e^{2} x^{2}-1}+72 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e^{5} a^{2} c x +10 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e^{3} c^{3} x^{3}+48 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a^{3} e^{6}+54 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e^{3} a \,c^{2} x +72 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a^{2} c \,e^{4}+15 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e \,c^{3} x +54 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a \,c^{2} e^{2}+15 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) c^{3}\right ) \operatorname {csgn}\left (e \right )}{48 e^{7} \sqrt {e^{2} x^{2}-1}}\) | \(292\) |
Input:
int((c*x^2+a)^3/(e*x-1)^(1/2)/(e*x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/48*c*x*(8*c^2*e^4*x^4+36*a*c*e^4*x^2+72*a^2*e^4+10*c^2*e^2*x^2+54*a*c*e^ 2+15*c^2)*(e*x-1)^(1/2)*(e*x+1)^(1/2)/e^6+1/16*(16*a^3*e^6+24*a^2*c*e^4+18 *a*c^2*e^2+5*c^3)/e^6*ln(e^2*x/(e^2)^(1/2)+(e^2*x^2-1)^(1/2))/(e^2)^(1/2)* ((e*x-1)*(e*x+1))^(1/2)/(e*x-1)^(1/2)/(e*x+1)^(1/2)
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\frac {{\left (8 \, c^{3} e^{5} x^{5} + 2 \, {\left (18 \, a c^{2} e^{5} + 5 \, c^{3} e^{3}\right )} x^{3} + 3 \, {\left (24 \, a^{2} c e^{5} + 18 \, a c^{2} e^{3} + 5 \, c^{3} e\right )} x\right )} \sqrt {e x + 1} \sqrt {e x - 1} - 3 \, {\left (16 \, a^{3} e^{6} + 24 \, a^{2} c e^{4} + 18 \, a c^{2} e^{2} + 5 \, c^{3}\right )} \log \left (-e x + \sqrt {e x + 1} \sqrt {e x - 1}\right )}{48 \, e^{7}} \] Input:
integrate((c*x^2+a)^3/(e*x-1)^(1/2)/(e*x+1)^(1/2),x, algorithm="fricas")
Output:
1/48*((8*c^3*e^5*x^5 + 2*(18*a*c^2*e^5 + 5*c^3*e^3)*x^3 + 3*(24*a^2*c*e^5 + 18*a*c^2*e^3 + 5*c^3*e)*x)*sqrt(e*x + 1)*sqrt(e*x - 1) - 3*(16*a^3*e^6 + 24*a^2*c*e^4 + 18*a*c^2*e^2 + 5*c^3)*log(-e*x + sqrt(e*x + 1)*sqrt(e*x - 1)))/e^7
Timed out. \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\text {Timed out} \] Input:
integrate((c*x**2+a)**3/(e*x-1)**(1/2)/(e*x+1)**(1/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (141) = 282\).
Time = 0.04 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\frac {\sqrt {e^{2} x^{2} - 1} c^{3} x^{5}}{6 \, e^{2}} + \frac {3 \, \sqrt {e^{2} x^{2} - 1} a c^{2} x^{3}}{4 \, e^{2}} + \frac {a^{3} \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{\sqrt {e^{2}}} + \frac {3 \, \sqrt {e^{2} x^{2} - 1} a^{2} c x}{2 \, e^{2}} + \frac {5 \, \sqrt {e^{2} x^{2} - 1} c^{3} x^{3}}{24 \, e^{4}} + \frac {3 \, a^{2} c \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{2 \, \sqrt {e^{2}} e^{2}} + \frac {9 \, \sqrt {e^{2} x^{2} - 1} a c^{2} x}{8 \, e^{4}} + \frac {9 \, a c^{2} \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{8 \, \sqrt {e^{2}} e^{4}} + \frac {5 \, \sqrt {e^{2} x^{2} - 1} c^{3} x}{16 \, e^{6}} + \frac {5 \, c^{3} \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{16 \, \sqrt {e^{2}} e^{6}} \] Input:
integrate((c*x^2+a)^3/(e*x-1)^(1/2)/(e*x+1)^(1/2),x, algorithm="maxima")
Output:
1/6*sqrt(e^2*x^2 - 1)*c^3*x^5/e^2 + 3/4*sqrt(e^2*x^2 - 1)*a*c^2*x^3/e^2 + a^3*log(2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2))/sqrt(e^2) + 3/2*sqrt(e^2* x^2 - 1)*a^2*c*x/e^2 + 5/24*sqrt(e^2*x^2 - 1)*c^3*x^3/e^4 + 3/2*a^2*c*log( 2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2))/(sqrt(e^2)*e^2) + 9/8*sqrt(e^2*x^ 2 - 1)*a*c^2*x/e^4 + 9/8*a*c^2*log(2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2) )/(sqrt(e^2)*e^4) + 5/16*sqrt(e^2*x^2 - 1)*c^3*x/e^6 + 5/16*c^3*log(2*e^2* x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2))/(sqrt(e^2)*e^6)
Time = 0.13 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\frac {{\left ({\left (2 \, {\left ({\left (e x + 1\right )} {\left (4 \, {\left (e x + 1\right )} {\left (\frac {{\left (e x + 1\right )} c^{3}}{e^{6}} - \frac {5 \, c^{3}}{e^{6}}\right )} + \frac {9 \, {\left (2 \, a c^{2} e^{38} + 5 \, c^{3} e^{36}\right )}}{e^{42}}\right )} - \frac {54 \, a c^{2} e^{38} + 55 \, c^{3} e^{36}}{e^{42}}\right )} {\left (e x + 1\right )} + \frac {72 \, a^{2} c e^{40} + 162 \, a c^{2} e^{38} + 85 \, c^{3} e^{36}}{e^{42}}\right )} {\left (e x + 1\right )} - \frac {3 \, {\left (24 \, a^{2} c e^{40} + 30 \, a c^{2} e^{38} + 11 \, c^{3} e^{36}\right )}}{e^{42}}\right )} \sqrt {e x + 1} \sqrt {e x - 1} - \frac {6 \, {\left (16 \, a^{3} e^{6} + 24 \, a^{2} c e^{4} + 18 \, a c^{2} e^{2} + 5 \, c^{3}\right )} \log \left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}{e^{6}}}{48 \, e} \] Input:
integrate((c*x^2+a)^3/(e*x-1)^(1/2)/(e*x+1)^(1/2),x, algorithm="giac")
Output:
1/48*(((2*((e*x + 1)*(4*(e*x + 1)*((e*x + 1)*c^3/e^6 - 5*c^3/e^6) + 9*(2*a *c^2*e^38 + 5*c^3*e^36)/e^42) - (54*a*c^2*e^38 + 55*c^3*e^36)/e^42)*(e*x + 1) + (72*a^2*c*e^40 + 162*a*c^2*e^38 + 85*c^3*e^36)/e^42)*(e*x + 1) - 3*( 24*a^2*c*e^40 + 30*a*c^2*e^38 + 11*c^3*e^36)/e^42)*sqrt(e*x + 1)*sqrt(e*x - 1) - 6*(16*a^3*e^6 + 24*a^2*c*e^4 + 18*a*c^2*e^2 + 5*c^3)*log(sqrt(e*x + 1) - sqrt(e*x - 1))/e^6)/e
Time = 17.09 (sec) , antiderivative size = 980, normalized size of antiderivative = 6.09 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\text {Too large to display} \] Input:
int((a + c*x^2)^3/((e*x - 1)^(1/2)*(e*x + 1)^(1/2)),x)
Output:
(atanh(((e*x - 1)^(1/2) - 1i)/((e*x + 1)^(1/2) - 1))*(c + 2*a*e^2)*(5*c^2 + 8*a^2*e^4 + 8*a*c*e^2))/(4*e^7) - ((((e*x - 1)^(1/2) - 1i)*((5*c^3)/4 + (9*a*c^2*e^2)/2 + 6*a^2*c*e^4))/((e*x + 1)^(1/2) - 1) + (((e*x - 1)^(1/2) - 1i)^23*((5*c^3)/4 + (9*a*c^2*e^2)/2 + 6*a^2*c*e^4))/((e*x + 1)^(1/2) - 1 )^23 - (((e*x - 1)^(1/2) - 1i)^3*((175*c^3)/12 + (105*a*c^2*e^2)/2 + 6*a^2 *c*e^4))/((e*x + 1)^(1/2) - 1)^3 - (((e*x - 1)^(1/2) - 1i)^21*((175*c^3)/1 2 + (105*a*c^2*e^2)/2 + 6*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^21 - (((e*x - 1)^(1/2) - 1i)^5*((669*a*c^2*e^2)/2 - (311*c^3)/4 + 126*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^5 - (((e*x - 1)^(1/2) - 1i)^19*((669*a*c^2*e^2)/2 - (311*c ^3)/4 + 126*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^19 + (((e*x - 1)^(1/2) - 1i) ^7*((8361*c^3)/4 + (1533*a*c^2*e^2)/2 + 510*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^7 + (((e*x - 1)^(1/2) - 1i)^17*((8361*c^3)/4 + (1533*a*c^2*e^2)/2 + 51 0*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^17 + (((e*x - 1)^(1/2) - 1i)^11*((2529 5*c^3)/2 - 549*a*c^2*e^2 + 420*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^11 + (((e *x - 1)^(1/2) - 1i)^13*((25295*c^3)/2 - 549*a*c^2*e^2 + 420*a^2*c*e^4))/(( e*x + 1)^(1/2) - 1)^13 + (((e*x - 1)^(1/2) - 1i)^9*((42259*c^3)/6 + 165*a* c^2*e^2 - 804*a^2*c*e^4))/((e*x + 1)^(1/2) - 1)^9 + (((e*x - 1)^(1/2) - 1i )^15*((42259*c^3)/6 + 165*a*c^2*e^2 - 804*a^2*c*e^4))/((e*x + 1)^(1/2) - 1 )^15)/(e^7 - (12*e^7*((e*x - 1)^(1/2) - 1i)^2)/((e*x + 1)^(1/2) - 1)^2 + ( 66*e^7*((e*x - 1)^(1/2) - 1i)^4)/((e*x + 1)^(1/2) - 1)^4 - (220*e^7*((e...
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a+c x^2\right )^3}{\sqrt {-1+e x} \sqrt {1+e x}} \, dx=\frac {72 \sqrt {e x +1}\, \sqrt {e x -1}\, a^{2} c \,e^{5} x +36 \sqrt {e x +1}\, \sqrt {e x -1}\, a \,c^{2} e^{5} x^{3}+54 \sqrt {e x +1}\, \sqrt {e x -1}\, a \,c^{2} e^{3} x +8 \sqrt {e x +1}\, \sqrt {e x -1}\, c^{3} e^{5} x^{5}+10 \sqrt {e x +1}\, \sqrt {e x -1}\, c^{3} e^{3} x^{3}+15 \sqrt {e x +1}\, \sqrt {e x -1}\, c^{3} e x +96 \,\mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a^{3} e^{6}+144 \,\mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a^{2} c \,e^{4}+108 \,\mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a \,c^{2} e^{2}+30 \,\mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) c^{3}}{48 e^{7}} \] Input:
int((c*x^2+a)^3/(e*x-1)^(1/2)/(e*x+1)^(1/2),x)
Output:
(72*sqrt(e*x + 1)*sqrt(e*x - 1)*a**2*c*e**5*x + 36*sqrt(e*x + 1)*sqrt(e*x - 1)*a*c**2*e**5*x**3 + 54*sqrt(e*x + 1)*sqrt(e*x - 1)*a*c**2*e**3*x + 8*s qrt(e*x + 1)*sqrt(e*x - 1)*c**3*e**5*x**5 + 10*sqrt(e*x + 1)*sqrt(e*x - 1) *c**3*e**3*x**3 + 15*sqrt(e*x + 1)*sqrt(e*x - 1)*c**3*e*x + 96*log((sqrt(e *x - 1) + sqrt(e*x + 1))/sqrt(2))*a**3*e**6 + 144*log((sqrt(e*x - 1) + sqr t(e*x + 1))/sqrt(2))*a**2*c*e**4 + 108*log((sqrt(e*x - 1) + sqrt(e*x + 1)) /sqrt(2))*a*c**2*e**2 + 30*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*c* *3)/(48*e**7)