Integrand size = 26, antiderivative size = 428 \[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=-\frac {e x \left (a+b x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{d}+\frac {4 b e x \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{3 d^2}-\frac {b (8 b c-7 a d) e x \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{3 d^3 \left (a+b x^2\right )}+\frac {\sqrt {a} \sqrt {b} (8 b c-7 a d) e \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 d^3 \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (4 b c-3 a d) e \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} c d^2 \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-e*x*(b*x^2+a)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/d+4/3*b*e*x*(d*x^2+c )*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/d^2-1/3*b*(-7*a*d+8*b*c)*e*x*(d*x ^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/d^3/(b*x^2+a)+1/3*a^(1/2)*b^( 1/2)*(-7*a*d+8*b*c)*e*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)*Ell ipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/d^3/(b*x^2+a )/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*a^(3/2)*(-3*a*d+4*b*c)*e*(d*x^2+c)*( b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^( 1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/d^2/(b*x^2+a)/(a*(d*x^2+c)/c/(b*x^2+a)) ^(1/2)
Result contains complex when optimal does not.
Time = 5.07 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.55 \[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=-\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 a d-b \left (4 c+d x^2\right )\right )+i b c (-8 b c+7 a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \left (8 b^2 c^2-11 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 \sqrt {\frac {b}{a}} d^3 \left (a+b x^2\right )} \] Input:
Integrate[x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]
Output:
-1/3*(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a + b*x^2)*(3*a* d - b*(4*c + d*x^2)) + I*b*c*(-8*b*c + 7*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(8*b^2*c^2 - 11*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic F[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*d^3*(a + b*x^2))
Time = 0.88 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2058, 369, 403, 25, 406, 320, 388, 313}
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.
| \(\displaystyle \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {x^2 \left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2}}dx}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\int \frac {\sqrt {b x^2+a} \left (4 b x^2+a\right )}{\sqrt {d x^2+c}}dx}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\frac {\int -\frac {b (8 b c-7 a d) x^2+a (4 b c-3 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {4 b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\frac {4 b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\int \frac {b (8 b c-7 a d) x^2+a (4 b c-3 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\frac {4 b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {a (4 b c-3 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+b (8 b c-7 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\frac {4 b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {b (8 b c-7 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (4 b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\frac {4 b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {b (8 b c-7 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (4 b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\frac {4 b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (4 b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b (8 b c-7 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{d}-\frac {x \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
Input:
Int[x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]
Output:
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-((x*(a + b*x^2)^(3/ 2))/(d*Sqrt[c + d*x^2])) + ((4*b*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (b*(8*b*c - 7*a*d)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*S qrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b *Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c ]*(4*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d *x^2]))/(3*d))/d))/Sqrt[a + b*x^2]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 13.29 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(\frac {{\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} d^{2} x^{5}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{3}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{3}-3 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{3}+3 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{3}+3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} d^{2}-11 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d +8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2}+7 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d -8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b c d x -3 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x +3 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c d x \right )}{3 \left (b \,x^{2}+a \right )^{2} d^{3} \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(734\) |
| risch | \(\frac {b x \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{3 d^{2}}+\frac {\left (\frac {-\frac {2 b d \left (4 a d -5 b c \right ) a c e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}+\frac {3 a^{2} d^{2} \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {3 b^{2} c^{2} \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {7 a b c d \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}}{d}-\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (b d \,x^{2} e +a d e \right ) x}{c \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2} e +a d e \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 b d a e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}\right )}{d}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{3 d^{2} \left (b \,x^{2}+a \right )}\) | \(934\) |
Input:
int(x^2*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*(e*(b*x^2+a)/(d*x^2+c))^(3/2)*(d*x^2+c)*(((d*x^2+c)*(b*x^2+a))^(1/2)*( -b/a)^(1/2)*b^2*d^2*x^5+((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*a*b*d^2*x ^3+((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*b^2*c*d*x^3-3*(b*d*x^4+a*d*x^2 +b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*b*d^2*x^3+3*(b*d*x^4+a*d*x^2+b*c*x^2+a* c)^(1/2)*(-b/a)^(1/2)*b^2*c*d*x^3+3*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a) /a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^ 2*d^2-11*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/ 2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c*d+8*((d*x^2+c)*(b*x^2+a ))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2), (a*d/b/c)^(1/2))*b^2*c^2+7*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2) *((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c*d-8*( (d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti cE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^2+((d*x^2+c)*(b*x^2+a))^(1/2)*(-b /a)^(1/2)*a*b*c*d*x-3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a^2 *d^2*x+3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*b*c*d*x)/(b*x^ 2+a)^2/d^3/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.54 \[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\frac {{\left (8 \, b^{2} c^{3} - 7 \, a b c^{2} d\right )} \sqrt {\frac {b e}{d}} e x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{2} c^{3} - 7 \, a b c^{2} d + 4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} \sqrt {\frac {b e}{d}} e x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (b^{2} c d^{2} e x^{4} - 4 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} e x^{2} - {\left (8 \, b^{2} c^{3} - 7 \, a b c^{2} d\right )} e\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{3 \, b c d^{3} x} \] Input:
integrate(x^2*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")
Output:
1/3*((8*b^2*c^3 - 7*a*b*c^2*d)*sqrt(b*e/d)*e*x*sqrt(-c/d)*elliptic_e(arcsi n(sqrt(-c/d)/x), a*d/(b*c)) - (8*b^2*c^3 - 7*a*b*c^2*d + 4*a*b*c*d^2 - 3*a ^2*d^3)*sqrt(b*e/d)*e*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b *c)) + (b^2*c*d^2*e*x^4 - 4*(b^2*c^2*d - a*b*c*d^2)*e*x^2 - (8*b^2*c^3 - 7 *a*b*c^2*d)*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b*c*d^3*x)
Timed out. \[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate(x**2*(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)
Output:
Timed out
\[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\int { \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="maxima")
Output:
integrate(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*x^2, x)
\[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\int { \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")
Output:
integrate(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*x^2, x)
Timed out. \[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\int x^2\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2} \,d x \] Input:
int(x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x)
Output:
int(x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x)
\[ \int x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int(x^2*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x)
Output:
(sqrt(e)*e*(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d*x - 3*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a*b*c*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c* x**3 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x* *2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*c*d **2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x** 2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*d**3 *x**2 + 11*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d* x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b**2*c **2*d + 11*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d* x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b**2*c *d**2*x**2 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a* c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b**3 *c**3 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x **2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b**3*c**2 *d*x**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*c**2*d - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d** 2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*c*d**2*x**2 + 3 *int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x **4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*c**3 + 3*int(...