Integrand size = 19, antiderivative size = 307 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {e^3}{2 c \sqrt {a+c x^4}}+\frac {x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a \sqrt {a+c x^4}}-\frac {3 d e^2 x \sqrt {a+c x^4}}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {3 d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}+\frac {d \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt {a+c x^4}} \] Output:
-1/2*e^3/c/(c*x^4+a)^(1/2)+1/2*x*(3*d*e^2*x^2+3*d^2*e*x+d^3)/a/(c*x^4+a)^( 1/2)-3/2*d*e^2*x*(c*x^4+a)^(1/2)/a/c^(1/2)/(a^(1/2)+c^(1/2)*x^2)+3/2*d*e^2 *(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE (sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/c^(3/4)/(c*x^4+a)^( 1/2)+1/4*d*(c^(1/2)*d^2-3*a^(1/2)*e^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a ^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1 /2*2^(1/2))/a^(5/4)/c^(3/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.41 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {-a e^3+c d^3 x+3 c d^2 e x^2+c d^3 x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 c d e^2 x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{2 a c \sqrt {a+c x^4}} \] Input:
Integrate[(d + e*x)^3/(a + c*x^4)^(3/2),x]
Output:
(-(a*e^3) + c*d^3*x + 3*c*d^2*e*x^2 + c*d^3*x*Sqrt[1 + (c*x^4)/a]*Hypergeo metric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*c*d*e^2*x^3*Sqrt[1 + (c*x^4)/a] *Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(2*a*c*Sqrt[a + c*x^4])
Time = 0.64 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2393, 25, 27, 1512, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle -\frac {\int -\frac {d \left (d^2-3 e^2 x^2\right )}{\sqrt {c x^4+a}}dx}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d \left (d^2-3 e^2 x^2\right )}{\sqrt {c x^4+a}}dx}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \frac {d^2-3 e^2 x^2}{\sqrt {c x^4+a}}dx}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {d \left (\left (d^2-\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+a}}dx+\frac {3 \sqrt {a} e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{\sqrt {c}}\right )}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\left (d^2-\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+a}}dx+\frac {3 e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {d \left (\frac {3 e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (d^2-\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}\right )}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {d \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (d^2-\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {3 e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{2 a}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt {a+c x^4}}\) |
Input:
Int[(d + e*x)^3/(a + c*x^4)^(3/2),x]
Output:
-1/2*(a*e^3 - c*x*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2))/(a*c*Sqrt[a + c*x^4]) + (d*((3*e^2*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sq rt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE [2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])))/Sqrt[c] + ((d^2 - (3*Sqrt[a]*e^2)/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4 )/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2] )/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4])))/(2*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.80
| method | result | size |
| elliptic | \(-\frac {2 c \left (-\frac {3 d \,e^{2} x^{3}}{4 a c}-\frac {3 d^{2} e \,x^{2}}{4 c a}-\frac {d^{3} x}{4 a c}+\frac {e^{3}}{4 c^{2}}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 i d \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(245\) |
| default | \(d^{3} \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )-\frac {e^{3}}{2 c \sqrt {c \,x^{4}+a}}+3 d \,e^{2} \left (\frac {x^{3}}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+\frac {3 d^{2} e \,x^{2}}{2 \sqrt {c \,x^{4}+a}\, a}\) | \(261\) |
Input:
int((e*x+d)^3/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(-3/4*d*e^2/a/c*x^3-3/4*d^2*e/c/a*x^2-1/4*d^3/a/c*x+1/4*e^3/c^2)/(c*( a/c+x^4))^(1/2)+1/2*d^3/a/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/ 2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c ^(1/2)/a^(1/2))^(1/2),I)-3/2*I*d*e^2/a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1- I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/ 2)/c^(1/2)*(EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2 )/a^(1/2))^(1/2),I))
Time = 0.12 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (c d e^{2} x^{4} + a d e^{2}\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (c d^{3} + 3 \, c d e^{2}\right )} x^{4} + a d^{3} + 3 \, a d e^{2}\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, c d e^{2} x^{3} + 3 \, c d^{2} e x^{2} + c d^{3} x - a e^{3}\right )} \sqrt {c x^{4} + a}}{2 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} \] Input:
integrate((e*x+d)^3/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/2*(3*(c*d*e^2*x^4 + a*d*e^2)*sqrt(a)*(-c/a)^(3/4)*elliptic_e(arcsin(x*(- c/a)^(1/4)), -1) - ((c*d^3 + 3*c*d*e^2)*x^4 + a*d^3 + 3*a*d*e^2)*sqrt(a)*( -c/a)^(3/4)*elliptic_f(arcsin(x*(-c/a)^(1/4)), -1) + (3*c*d*e^2*x^3 + 3*c* d^2*e*x^2 + c*d^3*x - a*e^3)*sqrt(c*x^4 + a))/(a*c^2*x^4 + a^2*c)
\[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (a + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**3/(c*x**4+a)**(3/2),x)
Output:
Integral((d + e*x)**3/(a + c*x**4)**(3/2), x)
\[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x+d)^3/(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x + d)^3/(c*x^4 + a)^(3/2), x)
\[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x+d)^3/(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x + d)^3/(c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \] Input:
int((d + e*x)^3/(a + c*x^4)^(3/2),x)
Output:
int((d + e*x)^3/(a + c*x^4)^(3/2), x)
\[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{2} e^{3}+9 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a c \,d^{2} e \,x^{2}-2 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a c \,e^{3} x^{4}+12 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c^{2} d^{2} e \,x^{6}+4 \sqrt {c \,x^{4}+a}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c^{2} d^{3} x^{2}+4 \sqrt {c \,x^{4}+a}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{3} d^{3} x^{6}+12 \sqrt {c \,x^{4}+a}\, \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c^{2} d \,e^{2} x^{2}+12 \sqrt {c \,x^{4}+a}\, \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{3} d \,e^{2} x^{6}+2 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{3} c \,d^{3}+6 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c^{2} d^{3} x^{4}+4 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{3} d^{3} x^{8}+6 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{3} c d \,e^{2}+18 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c^{2} d \,e^{2} x^{4}+12 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{3} d \,e^{2} x^{8}+3 a^{2} c \,d^{2} e -2 a^{2} c \,e^{3} x^{2}+15 a \,c^{2} d^{2} e \,x^{4}-2 a \,c^{2} e^{3} x^{6}+12 c^{3} d^{2} e \,x^{8}}{2 a c \left (2 \sqrt {c \,x^{4}+a}\, a c \,x^{2}+2 \sqrt {c \,x^{4}+a}\, c^{2} x^{6}+\sqrt {c}\, a^{2}+3 \sqrt {c}\, a c \,x^{4}+2 \sqrt {c}\, c^{2} x^{8}\right )} \] Input:
int((e*x+d)^3/(c*x^4+a)^(3/2),x)
Output:
( - sqrt(c)*sqrt(a + c*x**4)*a**2*e**3 + 9*sqrt(c)*sqrt(a + c*x**4)*a*c*d* *2*e*x**2 - 2*sqrt(c)*sqrt(a + c*x**4)*a*c*e**3*x**4 + 12*sqrt(c)*sqrt(a + c*x**4)*c**2*d**2*e*x**6 + 4*sqrt(a + c*x**4)*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d**3*x**2 + 4*sqrt(a + c*x**4)*int( sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**3*d**3*x**6 + 12* sqrt(a + c*x**4)*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x** 8),x)*a**2*c**2*d*e**2*x**2 + 12*sqrt(a + c*x**4)*int((sqrt(a + c*x**4)*x* *2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**3*d*e**2*x**6 + 2*sqrt(c)*int( sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**3*c*d**3 + 6*sqrt(c )*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d**3*x **4 + 4*sqrt(c)*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a* c**3*d**3*x**8 + 6*sqrt(c)*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**3*c*d*e**2 + 18*sqrt(c)*int((sqrt(a + c*x**4)*x**2)/(a* *2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d*e**2*x**4 + 12*sqrt(c)*int((sq rt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**3*d*e**2*x**8 + 3*a**2*c*d**2*e - 2*a**2*c*e**3*x**2 + 15*a*c**2*d**2*e*x**4 - 2*a*c**2 *e**3*x**6 + 12*c**3*d**2*e*x**8)/(2*a*c*(2*sqrt(a + c*x**4)*a*c*x**2 + 2* sqrt(a + c*x**4)*c**2*x**6 + sqrt(c)*a**2 + 3*sqrt(c)*a*c*x**4 + 2*sqrt(c) *c**2*x**8))