Integrand size = 29, antiderivative size = 236 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x^2}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2} e}-\frac {d^3 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e \left (c d^2-b d e+a e^2\right )^{3/2}} \]
1/2*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(3/2)/e-1/2*d ^3*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x^2)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x ^4+b*x^2+a)^(1/2))/e/(a*e^2-b*d*e+c*d^2)^(3/2)+(a*(-a*b*e-2*a*c*d+b^2*d)+( 2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)*x^2)/c/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2) /(c*x^4+b*x^2+a)^(1/2)
Time = 0.94 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.06 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-b^3 d x^2+a b \left (-b d+3 c d x^2+b e x^2\right )+a^2 \left (b e+2 c \left (d-e x^2\right )\right )}{c \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+b x^2+c x^4}}-\frac {d^3 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{e \left (c d^2+e (-b d+a e)\right )^2}-\frac {\log \left (c e \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{2 c^{3/2} e} \]
(-(b^3*d*x^2) + a*b*(-(b*d) + 3*c*d*x^2 + b*e*x^2) + a^2*(b*e + 2*c*(d - e *x^2)))/(c*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + b*x^2 + c*x^ 4]) - (d^3*Sqrt[-(c*d^2) + b*d*e - a*e^2]*ArcTan[(Sqrt[c]*(d + e*x^2) - e* Sqrt[a + b*x^2 + c*x^4])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(e*(c*d^2 + e*(- (b*d) + a*e))^2) - Log[c*e*(b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4 ])]/(2*c^(3/2)*e)
Time = 0.56 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1578, 1264, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^6}{\left (e x^2+d\right ) \left (c x^4+b x^2+a\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 1264 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {\frac {\left (b^2-4 a c\right ) d (b d-a e)}{c d^2-b e d+a e^2}-\left (b^2-4 a c\right ) x^2}{2 c \left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{b^2-4 a c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {\frac {\left (b^2-4 a c\right ) d (b d-a e)}{c d^2-b e d+a e^2}-\left (b^2-4 a c\right ) x^2}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {c d^3 \left (b^2-4 a c\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{e}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {c d^3 \left (b^2-4 a c\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{e}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {c d^3 \left (b^2-4 a c\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c} e}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {2 c d^3 \left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-x^4}d\left (-\frac {(2 c d-b e) x^2+b d-2 a e}{\sqrt {c x^4+b x^2+a}}\right )}{e \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c} e}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {c d^3 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{e \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c} e}}{c \left (b^2-4 a c\right )}\right )\) |
((2*(a*(b^2*d - 2*a*c*d - a*b*e) + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c* e)*x^2))/(c*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (-(((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x ^4])])/(Sqrt[c]*e)) + (c*(b^2 - 4*a*c)*d^3*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(e*(c *d^2 - b*d*e + a*e^2)^(3/2)))/(c*(b^2 - 4*a*c)))/2
3.4.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) ^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x )^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ (d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S imp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) )/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.59 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.55
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (a c -\frac {b^{2}}{4}\right ) d^{3} c^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e +\left (b \,x^{2}+2 a \right ) e -d \left (2 c \,x^{2}+b \right )}{e \,x^{2}+d}\right )+e \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \left (\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (a c -\frac {b^{2}}{4}\right ) \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )-\ln \left (2\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (a c -\frac {b^{2}}{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+e \left (\left (-a e \,x^{2}+d \left (\frac {3 b \,x^{2}}{2}+a \right )\right ) a c +\frac {b \left (b \,x^{2}+a \right ) \left (a e -b d \right )}{2}\right ) c^{\frac {3}{2}}\right )}{2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (a c -\frac {b^{2}}{4}\right ) c^{\frac {5}{2}}}\) | \(365\) |
| elliptic | \(\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e \,c^{\frac {3}{2}}}-\frac {2 c \,d^{3} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (-2 a c \sqrt {-4 a c +b^{2}}+2 b^{2} \sqrt {-4 a c +b^{2}}+6 a b c -2 b^{3}\right ) \sqrt {c \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}+\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{2 c^{2} \left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (-4 a c +b^{2}\right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {\left (2 a c \sqrt {-4 a c +b^{2}}-2 b^{2} \sqrt {-4 a c +b^{2}}+6 a b c -2 b^{3}\right ) \sqrt {c \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{2 c^{2} \left (e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \left (-4 a c +b^{2}\right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\) | \(580\) |
| default | \(\frac {-\frac {x^{2}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{4 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}}{e}+\frac {d^{2} \left (2 c \,x^{2}+b \right )}{e^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {d \left (b \,x^{2}+2 a \right )}{e^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-\frac {d^{3} \left (-\frac {2 c \sqrt {c \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}+\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (-4 a c +b^{2}\right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {c \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \left (-4 a c +b^{2}\right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\) | \(671\) |
1/2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*((c*x^4+b*x^2+a) ^(1/2)*(a*c-1/4*b^2)*d^3*c^(5/2)*ln((2*(c*x^4+b*x^2+a)^(1/2)*((a*e^2-b*d*e +c*d^2)/e^2)^(1/2)*e+(b*x^2+2*a)*e-d*(2*c*x^2+b))/(e*x^2+d))+e*((a*e^2-b*d *e+c*d^2)/e^2)^(1/2)*((c*x^4+b*x^2+a)^(1/2)*(a*e^2-b*d*e+c*d^2)*c*(a*c-1/4 *b^2)*ln((2*c*x^2+2*(c*x^4+b*x^2+a)^(1/2)*c^(1/2)+b)/c^(1/2))-ln(2)*(a*e^2 -b*d*e+c*d^2)*c*(a*c-1/4*b^2)*(c*x^4+b*x^2+a)^(1/2)+e*((-a*e*x^2+d*(3/2*b* x^2+a))*a*c+1/2*b*(b*x^2+a)*(a*e-b*d))*c^(3/2)))/e^2/(a*e^2-b*d*e+c*d^2)/( a*c-1/4*b^2)/c^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (214) = 428\).
Time = 37.45 (sec) , antiderivative size = 4901, normalized size of antiderivative = 20.77 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/4*(((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^3*e + (a* b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d*e^3 + ( a^3*b^2 - 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4 - 2*(b^3*c^2 - 4*a*b*c^3 )*d^3*e + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^2 - 2*(a*b^3*c - 4*a^2*b *c^2)*d*e^3 + (a^2*b^2*c - 4*a^3*c^2)*e^4)*x^4 + ((b^3*c^2 - 4*a*b*c^3)*d^ 4 - 2*(b^4*c - 4*a*b^2*c^2)*d^3*e + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*e^ 2 - 2*(a*b^4 - 4*a^2*b^2*c)*d*e^3 + (a^2*b^3 - 4*a^3*b*c)*e^4)*x^2)*sqrt(c )*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + ((b^2*c^3 - 4*a*c^4)*d^3*x^4 + (b^3*c^2 - 4*a*b*c^3) *d^3*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^3)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-( (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2 - 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(c*d^2 - b*d*e + a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e))/(e^2*x^4 + 2*d*e*x^2 + d^2)) - 4*(a^3*b*c*e^4 - (a*b^2*c^2 - 2*a^2*c^3)*d^3*e + (a*b^3*c - a^2*b*c^2)*d^2*e^2 - 2*(a^2*b^2*c - a^3*c^ 2)*d*e^3 - ((b^3*c^2 - 3*a*b*c^3)*d^3*e - (b^4*c - 2*a*b^2*c^2 - 2*a^2*c^3 )*d^2*e^2 + (2*a*b^3*c - 5*a^2*b*c^2)*d*e^3 - (a^2*b^2*c - 2*a^3*c^2)*e^4) *x^2)*sqrt(c*x^4 + b*x^2 + a))/((a*b^2*c^4 - 4*a^2*c^5)*d^4*e - 2*(a*b^3*c ^3 - 4*a^2*b*c^4)*d^3*e^2 + (a*b^4*c^2 - 2*a^2*b^2*c^3 - 8*a^3*c^4)*d^2*e^ 3 - 2*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*e^4 + (a^3*b^2*c^2 - 4*a^4*c^3)*e^5...
\[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{7}}{\left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \]
Exception generated. \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^7}{\left (e\,x^2+d\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]