Integrand size = 29, antiderivative size = 491 \[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\frac {e x \sqrt {d+e x^2}}{2 c}+\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {d \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}+\frac {\sqrt {e} \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2}+\frac {\sqrt {e} \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2} \]
1/2*d*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))*e^(1/2)/c+1/2*arctanh(x*e^(1/2)/( e*x^2+d)^(1/2))*(c*d-b*e+(-2*a*c*e+b^2*e-b*c*d)/(-4*a*c+b^2)^(1/2))*e^(1/2 )/c^2+1/2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))*(c*d-b*e+(2*a*c*e-b^2*e+b*c*d )/(-4*a*c+b^2)^(1/2))*e^(1/2)/c^2+1/2*e*x*(e*x^2+d)^(1/2)/c+1/2*arctan(x*( 2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2 ))^(1/2))*(c*d-b*e+(-2*a*c*e+b^2*e-b*c*d)/(-4*a*c+b^2)^(1/2))*(2*c*d-e*(b- (-4*a*c+b^2)^(1/2)))^(1/2)/c^2/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/2*arctan(x*( 2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^(1/2 ))^(1/2))*(c*d-b*e+(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*(2*c*d-e*(b+( -4*a*c+b^2)^(1/2)))^(1/2)/c^2/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.38 (sec) , antiderivative size = 916, normalized size of antiderivative = 1.87 \[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\frac {2 c e x \sqrt {d+e x^2}+4 \sqrt {e} (3 c d-2 b e) \text {arctanh}\left (\frac {\sqrt {e} x}{-\sqrt {d}+\sqrt {d+e x^2}}\right )+\text {RootSum}\left [a e^4+4 b d e^2 \text {$\#$1}^2-4 a e^3 \text {$\#$1}^2+16 c d^2 \text {$\#$1}^4-8 b d e \text {$\#$1}^4+6 a e^2 \text {$\#$1}^4+4 b d \text {$\#$1}^6-4 a e \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {2 a c d e^4 \log (x)-a b e^5 \log (x)-2 a c d e^4 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )+a b e^5 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right )-4 c^2 d^3 e \log (x) \text {$\#$1}^2+8 b c d^2 e^2 \log (x) \text {$\#$1}^2-4 b^2 d e^3 \log (x) \text {$\#$1}^2-2 a c d e^3 \log (x) \text {$\#$1}^2+3 a b e^4 \log (x) \text {$\#$1}^2+4 c^2 d^3 e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-8 b c d^2 e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b^2 d e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a c d e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a b e^4 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 c^2 d^3 \log (x) \text {$\#$1}^4-8 b c d^2 e \log (x) \text {$\#$1}^4+4 b^2 d e^2 \log (x) \text {$\#$1}^4+2 a c d e^2 \log (x) \text {$\#$1}^4-3 a b e^3 \log (x) \text {$\#$1}^4-4 c^2 d^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+8 b c d^2 e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-4 b^2 d e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-2 a c d e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a b e^3 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-2 a c d e \log (x) \text {$\#$1}^6+a b e^2 \log (x) \text {$\#$1}^6+2 a c d e \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6-a b e^2 \log \left (-\sqrt {d}+\sqrt {d+e x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{b d e^2 \text {$\#$1}-a e^3 \text {$\#$1}+8 c d^2 \text {$\#$1}^3-4 b d e \text {$\#$1}^3+3 a e^2 \text {$\#$1}^3+3 b d \text {$\#$1}^5-3 a e \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 c^2} \]
(2*c*e*x*Sqrt[d + e*x^2] + 4*Sqrt[e]*(3*c*d - 2*b*e)*ArcTanh[(Sqrt[e]*x)/( -Sqrt[d] + Sqrt[d + e*x^2])] + RootSum[a*e^4 + 4*b*d*e^2*#1^2 - 4*a*e^3*#1 ^2 + 16*c*d^2*#1^4 - 8*b*d*e*#1^4 + 6*a*e^2*#1^4 + 4*b*d*#1^6 - 4*a*e*#1^6 + a*#1^8 & , (2*a*c*d*e^4*Log[x] - a*b*e^5*Log[x] - 2*a*c*d*e^4*Log[-Sqrt [d] + Sqrt[d + e*x^2] - x*#1] + a*b*e^5*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x *#1] - 4*c^2*d^3*e*Log[x]*#1^2 + 8*b*c*d^2*e^2*Log[x]*#1^2 - 4*b^2*d*e^3*L og[x]*#1^2 - 2*a*c*d*e^3*Log[x]*#1^2 + 3*a*b*e^4*Log[x]*#1^2 + 4*c^2*d^3*e *Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 - 8*b*c*d^2*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 + 4*b^2*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2 ] - x*#1]*#1^2 + 2*a*c*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 - 3*a*b*e^4*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^2 + 4*c^2*d^3*Log[x]* #1^4 - 8*b*c*d^2*e*Log[x]*#1^4 + 4*b^2*d*e^2*Log[x]*#1^4 + 2*a*c*d*e^2*Log [x]*#1^4 - 3*a*b*e^3*Log[x]*#1^4 - 4*c^2*d^3*Log[-Sqrt[d] + Sqrt[d + e*x^2 ] - x*#1]*#1^4 + 8*b*c*d^2*e*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 - 4*b^2*d*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 - 2*a*c*d*e^2*Log [-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 + 3*a*b*e^3*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^4 - 2*a*c*d*e*Log[x]*#1^6 + a*b*e^2*Log[x]*#1^6 + 2*a *c*d*e*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^6 - a*b*e^2*Log[-Sqrt[d] + Sqrt[d + e*x^2] - x*#1]*#1^6)/(b*d*e^2*#1 - a*e^3*#1 + 8*c*d^2*#1^3 - 4* b*d*e*#1^3 + 3*a*e^2*#1^3 + 3*b*d*#1^5 - 3*a*e*#1^5 + a*#1^7) & ])/(4*c...
Time = 1.18 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1616, 211, 224, 219, 2256, 2009}
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.
| \(\displaystyle \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1616 |
| \(\displaystyle \frac {e \int \sqrt {e x^2+d}dx}{c}-\frac {\int \frac {\sqrt {e x^2+d} \left (a e-(c d-b e) x^2\right )}{c x^4+b x^2+a}dx}{c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {e \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )}{c}-\frac {\int \frac {\sqrt {e x^2+d} \left (a e-(c d-b e) x^2\right )}{c x^4+b x^2+a}dx}{c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {e \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{c}-\frac {\int \frac {\sqrt {e x^2+d} \left (a e-(c d-b e) x^2\right )}{c x^4+b x^2+a}dx}{c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {e \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{c}-\frac {\int \frac {\sqrt {e x^2+d} \left (a e-(c d-b e) x^2\right )}{c x^4+b x^2+a}dx}{c}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {e \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{c}-\frac {\int \left (\frac {\sqrt {e x^2+d} \left (-c d+b e-\frac {-e b^2+c d b+2 a c e}{\sqrt {b^2-4 a c}}\right )}{2 c x^2+b+\sqrt {b^2-4 a c}}+\frac {\left (-c d+b e+\frac {-e b^2+c d b+2 a c e}{\sqrt {b^2-4 a c}}\right ) \sqrt {e x^2+d}}{2 c x^2+b-\sqrt {b^2-4 a c}}\right )dx}{c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{c}-\frac {-\frac {\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 c \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{2 c \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{2 c}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{2 c}}{c}\) |
(e*((x*Sqrt[d + e*x^2])/2 + (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sq rt[e])))/c - (-1/2*(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d - b*e - ( b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt [b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c*Sq rt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2 ])])/(2*c*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (Sqrt[e]*(c*d - b*e - (b*c*d - b^ 2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2 *c) - (Sqrt[e]*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*A rcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c))/c
3.4.72.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[e*(f^2/c) Int[(f*x)^(m - 2)*(d + e*x^2)^ (q - 1), x], x] - Simp[f^2/c Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)*(Simp[ a*e - (c*d - b*e)*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d , e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 )^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Time = 0.75 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(\frac {e x \sqrt {e \,x^{2}+d}}{2 c}-\frac {\frac {\sqrt {e}\, \left (2 b e -3 c d \right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{c}+\frac {a \sqrt {2}\, \left (\frac {\left (2 a c d \,e^{2}-b^{2} d \,e^{2}+2 d^{2} e b c -2 c^{2} d^{3}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}-2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, c d e \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (-2 a c d \,e^{2}+b^{2} d \,e^{2}-2 d^{2} e b c +2 c^{2} d^{3}+\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}-2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, c d e \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{c \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}}{2 c}\) | \(394\) |
| default | \(-\frac {a \left (\left (e^{\frac {3}{2}} c d -\frac {b \,e^{\frac {5}{2}}}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\left (a c -\frac {b^{2}}{2}\right ) e^{\frac {5}{2}}+d c \left (-c d \sqrt {e}+e^{\frac {3}{2}} b \right )\right )\right ) \sqrt {2}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (a \sqrt {2}\, \left (\left (-e^{\frac {3}{2}} c d +\frac {b \,e^{\frac {5}{2}}}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\left (a c -\frac {b^{2}}{2}\right ) e^{\frac {5}{2}}+d c \left (-c d \sqrt {e}+e^{\frac {3}{2}} b \right )\right )\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (-2 b \,e^{2}+3 d c e \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\sqrt {e \,x^{2}+d}\, e^{\frac {3}{2}} c x \right )}{2}\right )}{\sqrt {e}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, c^{2}}\) | \(464\) |
| pseudoelliptic | \(-\frac {a \left (\left (e^{\frac {3}{2}} c d -\frac {b \,e^{\frac {5}{2}}}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\left (a c -\frac {b^{2}}{2}\right ) e^{\frac {5}{2}}+d c \left (-c d \sqrt {e}+e^{\frac {3}{2}} b \right )\right )\right ) \sqrt {2}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (a \sqrt {2}\, \left (\left (-e^{\frac {3}{2}} c d +\frac {b \,e^{\frac {5}{2}}}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\left (a c -\frac {b^{2}}{2}\right ) e^{\frac {5}{2}}+d c \left (-c d \sqrt {e}+e^{\frac {3}{2}} b \right )\right )\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (-2 b \,e^{2}+3 d c e \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\sqrt {e \,x^{2}+d}\, e^{\frac {3}{2}} c x \right )}{2}\right )}{\sqrt {e}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, c^{2}}\) | \(464\) |
1/2*e*x*(e*x^2+d)^(1/2)/c-1/2/c*(e^(1/2)*(2*b*e-3*c*d)/c*ln(x*e^(1/2)+(e*x ^2+d)^(1/2))+1/c*a*2^(1/2)/(-d^2*(4*a*c-b^2))^(1/2)*((2*a*c*d*e^2-b^2*d*e^ 2+2*d^2*e*b*c-2*c^2*d^3+(-d^2*(4*a*c-b^2))^(1/2)*b*e^2-2*(-d^2*(4*a*c-b^2) )^(1/2)*c*d*e)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)*arctan(a/x* (e*x^2+d)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2))-( -2*a*c*d*e^2+b^2*d*e^2-2*d^2*e*b*c+2*c^2*d^3+(-d^2*(4*a*c-b^2))^(1/2)*b*e^ 2-2*(-d^2*(4*a*c-b^2))^(1/2)*c*d*e)/((2*a*e-b*d+(-d^2*(4*a*c-b^2))^(1/2))* a)^(1/2)*arctanh(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((2*a*e-b*d+(-d^2*(4*a*c-b^2) )^(1/2))*a)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 5955 vs. \(2 (415) = 830\).
Time = 54.07 (sec) , antiderivative size = 11917, normalized size of antiderivative = 24.27 \[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
\[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\int \frac {x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c x^{4} + b x^{2} + a} \,d x } \]
Exception generated. \[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^2 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx=\int \frac {x^2\,{\left (e\,x^2+d\right )}^{3/2}}{c\,x^4+b\,x^2+a} \,d x \]