Integrand size = 29, antiderivative size = 208 \[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=-\frac {\left (4 c d-b e-2 c e x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c e^2}+\frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3}-\frac {d \sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^3} \]
1/16*(8*c^2*d^2-b^2*e^2-4*c*e*(-a*e+b*d))*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/ (c*x^4+b*x^2+a)^(1/2))/c^(3/2)/e^3-1/2*d*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))*(a*e^2-b*d*e+c*d^ 2)^(1/2)/e^3-1/8*(-2*c*e*x^2-b*e+4*c*d)*(c*x^4+b*x^2+a)^(1/2)/c/e^2
Time = 0.65 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=\frac {2 \sqrt {c} \left (e \left (-4 c d+b e+2 c e x^2\right ) \sqrt {a+b x^2+c x^4}-8 c d \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+b d e-a e^2}}\right )\right )+\left (8 c^2 d^2-b^2 e^2+4 c e (-b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3} \]
(2*Sqrt[c]*(e*(-4*c*d + b*e + 2*c*e*x^2)*Sqrt[a + b*x^2 + c*x^4] - 8*c*d*S qrt[-(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) + b*d*e - a*e^2]]) + (8*c^2*d^2 - b^2*e^2 + 4*c *e*(-(b*d) + a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4 ])])/(16*c^(3/2)*e^3)
Time = 0.49 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1578, 1231, 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.
| \(\displaystyle \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \sqrt {c x^4+b x^2+a}}{e x^2+d}dx^2\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) x^2+d \left (-e b^2+4 c d b-4 a c e\right )}{2 \left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{4 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) x^2+d \left (-e b^2+4 c d b-4 a c e\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{8 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{e}-\frac {8 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e}}{8 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{e}-\frac {8 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e}}{8 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}-\frac {8 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e}}{8 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {16 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-x^4}d\left (-\frac {(2 c d-b e) x^2+b d-2 a e}{\sqrt {c x^4+b x^2+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}-\frac {8 c d \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{4 c e^2}\right )\) |
(-1/4*((4*c*d - b*e - 2*c*e*x^2)*Sqrt[a + b*x^2 + c*x^4])/(c*e^2) + (((8*c ^2*d^2 - b^2*e^2 - 4*c*e*(b*d - a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqr t[a + b*x^2 + c*x^4])])/(Sqrt[c]*e) - (8*c*d*Sqrt[c*d^2 - b*d*e + a*e^2]*A rcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sq rt[a + b*x^2 + c*x^4])])/e)/(8*c*e^2))/2
3.4.12.3.1 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_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.46 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.38
| method | result | size |
| pseudoelliptic | \(\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d \,c^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e +\left (b \,x^{2}+2 a \right ) e -d \left (2 c \,x^{2}+b \right )}{e \,x^{2}+d}\right )-\left (-\frac {\left (2 c^{2} d^{2}+\left (a \,e^{2}-b d e \right ) c -\frac {b^{2} e^{2}}{4}\right ) c \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{2}+e \,c^{\frac {3}{2}} \left (\left (-\frac {e \,x^{2}}{2}+d \right ) c -\frac {b e}{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {\left (2 c^{2} d^{2}+\left (a \,e^{2}-b d e \right ) c -\frac {b^{2} e^{2}}{4}\right ) \ln \left (2\right ) c}{2}\right ) e \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, c^{\frac {5}{2}} e^{4}}\) | \(286\) |
| risch | \(\frac {\left (2 c \,x^{2} e +b e -4 c d \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c \,e^{2}}+\frac {\frac {\left (4 e^{2} a c -b^{2} e^{2}-4 b c d e +8 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e \sqrt {c}}+\frac {4 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{8 c \,e^{2}}\) | \(297\) |
| default | \(\frac {\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}}{e}-\frac {d \left (\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x^{2}+\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2}}\) | \(426\) |
| elliptic | \(\frac {\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}}{2 e}-\frac {d \left (\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x^{2}+\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2}}\) | \(427\) |
1/2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((a*e^2-b*d*e+c*d^2)*d*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))-(-1/2*(2*c^2*d^2+(a*e^2-b*d*e)*c-1/4*b^2*e^2)*c*ln((2 *c*x^2+2*(c*x^4+b*x^2+a)^(1/2)*c^(1/2)+b)/c^(1/2))+e*c^(3/2)*((-1/2*e*x^2+ d)*c-1/4*b*e)*(c*x^4+b*x^2+a)^(1/2)+1/2*(2*c^2*d^2+(a*e^2-b*d*e)*c-1/4*b^2 *e^2)*ln(2)*c)*e*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(5/2)/e^4
Time = 10.36 (sec) , antiderivative size = 1231, normalized size of antiderivative = 5.92 \[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=\text {Too large to display} \]
[1/32*(8*sqrt(c*d^2 - b*d*e + a*e^2)*c^2*d*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)) + (8*c^2*d^2 - 4*b*c*d*e - (b^2 - 4*a*c)*e^2)*sqrt(c)*l og(-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) + 4*(2*c^2*e^2*x^2 - 4*c^2*d*e + b*c*e^2)*sqrt(c*x^4 + b* x^2 + a))/(c^2*e^3), -1/32*(16*sqrt(-c*d^2 + b*d*e - a*e^2)*c^2*d*arctan(- 1/2*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)/((c^2*d^2 - b*c*d*e + a*c*e^2)*x^4 + a*c*d^2 - a*b*d*e + a^2*e^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x^2)) - (8*c^2*d^2 - 4*b*c*d*e - ( b^2 - 4*a*c)*e^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) - 4*(2*c^2*e^2*x^2 - 4*c^2*d*e + b*c*e^2)*sqrt(c*x^4 + b*x^2 + a))/(c^2*e^3), 1/16*(4*sqrt(c*d^2 - b*d*e + a*e^2)*c^2*d*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)) - (8*c^2 *d^2 - 4*b*c*d*e - (b^2 - 4*a*c)*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^4 + b*x ^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 + b*c*x^2 + a*c)) + 2*(2*c^2*e^...
\[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=\int \frac {x^{3} \sqrt {a + b x^{2} + c x^{4}}}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx=\int \frac {x^3\,\sqrt {c\,x^4+b\,x^2+a}}{e\,x^2+d} \,d x \]