Integrand size = 27, antiderivative size = 355 \[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\frac {x \left (10 a b c d-(b c-a d)^2+3 b d (b c-a d) x^2\right ) \sqrt {a c+(b c-a d) x^2-b d x^4}}{35 b d}+\frac {1}{7} x \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}+\frac {2 a \sqrt {c} (b c-a d) \left (8 a b c d+(b c-a d)^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt {a c+(b c-a d) x^2-b d x^4}}-\frac {a \sqrt {c} (b c+a d) \left (b^2 c^2-9 a b c d-2 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:
1/35*x*(10*a*b*c*d-(-a*d+b*c)^2+3*b*d*(-a*d+b*c)*x^2)*(a*c+(-a*d+b*c)*x^2- b*d*x^4)^(1/2)/b/d+1/7*x*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2)+2/35*a*c^(1/2) *(-a*d+b*c)*(8*a*b*c*d+(-a*d+b*c)^2)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*E llipticE(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/b^2/d^(3/2)/(a*c+(-a*d+b*c)*x ^2-b*d*x^4)^(1/2)-1/35*a*c^(1/2)*(a*d+b*c)*(-2*a^2*d^2-9*a*b*c*d+b^2*c^2)* (1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(-b*c/a/d) ^(1/2))/b^2/d^(3/2)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)
Result contains complex when optimal does not.
Time = 5.94 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.86 \[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (-c+d x^2\right ) \left (a^2 d^2+a b d \left (-17 c+8 d x^2\right )+b^2 \left (c^2-8 c d x^2+5 d^2 x^4\right )\right )-2 i c \left (-b^3 c^3-5 a b^2 c^2 d+5 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c \left (-2 b^3 c^3-11 a b^2 c^2 d-8 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{35 b \sqrt {\frac {b}{a}} d^2 \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] Input:
Integrate[(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(3/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(-c + d*x^2)*(a^2*d^2 + a*b*d*(-17*c + 8*d*x^2) + b^2*(c^2 - 8*c*d*x^2 + 5*d^2*x^4)) - (2*I)*c*(-(b^3*c^3) - 5*a*b^2*c^2* d + 5*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Ellip ticE[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))] + I*c*(-2*b^3*c^3 - 11*a*b^2* c^2*d - 8*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*E llipticF[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))])/(35*b*Sqrt[b/a]*d^2*Sqrt [(a + b*x^2)*(c - d*x^2)])
Time = 0.90 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1404, 1490, 25, 1514, 399, 321, 327}
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.
| \(\displaystyle \int \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {3}{7} \int \left ((b c-a d) x^2+2 a c\right ) \sqrt {-b d x^4+(b c-a d) x^2+a c}dx+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {3}{7} \left (\frac {x \left (3 b d x^2 (b c-a d)-(b c-a d)^2+10 a b c d\right ) \sqrt {x^2 (b c-a d)+a c-b d x^4}}{15 b d}-\frac {\int -\frac {2 (b c-a d) \left ((b c-a d)^2+8 a b c d\right ) x^2+a c \left ((b c-a d)^2+20 a b c d\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{15 b d}\right )+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{7} \left (\frac {\int \frac {2 (b c-a d) \left ((b c-a d)^2+8 a b c d\right ) x^2+a c \left ((b c-a d)^2+20 a b c d\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{15 b d}+\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (3 b d x^2 (b c-a d)-(b c-a d)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1514 |
| \(\displaystyle \frac {3}{7} \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {2 (b c-a d) \left ((b c-a d)^2+8 a b c d\right ) x^2+a c \left ((b c-a d)^2+20 a b c d\right )}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (3 b d x^2 (b c-a d)-(b c-a d)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {3}{7} \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {2 a (b c-a d) \left ((b c-a d)^2+8 a b c d\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a (a d+b c) \left (-2 a^2 d^2-9 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{b}\right )}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (3 b d x^2 (b c-a d)-(b c-a d)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {3}{7} \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {2 a (b c-a d) \left ((b c-a d)^2+8 a b c d\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a \sqrt {c} (a d+b c) \left (-2 a^2 d^2-9 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (3 b d x^2 (b c-a d)-(b c-a d)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3}{7} \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {2 a \sqrt {c} (b c-a d) \left ((b c-a d)^2+8 a b c d\right ) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d}}-\frac {a \sqrt {c} (a d+b c) \left (-2 a^2 d^2-9 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (3 b d x^2 (b c-a d)-(b c-a d)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}\) |
Input:
Int[(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(3/2),x]
Output:
(x*(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(3/2))/7 + (3*((x*(10*a*b*c*d - (b*c - a*d)^2 + 3*b*d*(b*c - a*d)*x^2)*Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4])/( 15*b*d) + (Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*((2*a*Sqrt[c]*(b*c - a* d)*(8*a*b*c*d + (b*c - a*d)^2)*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b *c)/(a*d))])/(b*Sqrt[d]) - (a*Sqrt[c]*(b*c + a*d)*(b^2*c^2 - 9*a*b*c*d - 2 *a^2*d^2)*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[ d])))/(15*b*d*Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4])))/7
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b *x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1)) Int[(2*a + b*x^2)*( a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* c, 0] && GtQ[p, 0] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt [1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[(d + e*x^2)/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Time = 3.93 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(-\frac {x \left (5 b^{2} d^{2} x^{4}+8 a \,d^{2} b \,x^{2}-8 b^{2} c d \,x^{2}+a^{2} d^{2}-17 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}{35 b d \sqrt {-\left (b \,x^{2}+a \right ) \left (d \,x^{2}-c \right )}}+\frac {\frac {\left (2 a^{3} d^{3}+10 a^{2} b c \,d^{2}-10 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}+\frac {a \,b^{2} c^{3} \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}+\frac {a^{3} c \,d^{2} \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}+\frac {18 a^{2} c^{2} b d \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}}{35 b d}\) | \(560\) |
| default | \(-\frac {b d \,x^{5} \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{7}-\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{5 b d}-\frac {\left (a^{2} d^{2}-\frac {23 a b c d}{7}+b^{2} c^{2}+\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c^{2}+\frac {\left (a^{2} d^{2}-\frac {23 a b c d}{7}+b^{2} c^{2}+\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-2 a^{2} c d +2 a b \,c^{2}+\frac {3 \left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) a c}{5 b d}+\frac {\left (a^{2} d^{2}-\frac {23 a b c d}{7}+b^{2} c^{2}+\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) \left (-2 a d +2 b c \right )}{3 b d}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\) | \(637\) |
| elliptic | \(-\frac {b d \,x^{5} \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{7}-\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{5 b d}-\frac {\left (a^{2} d^{2}-\frac {23 a b c d}{7}+b^{2} c^{2}+\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c^{2}+\frac {\left (a^{2} d^{2}-\frac {23 a b c d}{7}+b^{2} c^{2}+\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-2 a^{2} c d +2 a b \,c^{2}+\frac {3 \left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) a c}{5 b d}+\frac {\left (a^{2} d^{2}-\frac {23 a b c d}{7}+b^{2} c^{2}+\frac {\left (2 a b \,d^{2}-2 b^{2} c d +\frac {b d \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) \left (-2 a d +2 b c \right )}{3 b d}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\) | \(637\) |
Input:
int((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/35/b/d*x*(5*b^2*d^2*x^4+8*a*b*d^2*x^2-8*b^2*c*d*x^2+a^2*d^2-17*a*b*c*d+ b^2*c^2)*(b*x^2+a)*(-d*x^2+c)/(-(b*x^2+a)*(d*x^2-c))^(1/2)+1/35/b/d*((2*a^ 3*d^3+10*a^2*b*c*d^2-10*a*b^2*c^2*d-2*b^3*c^3)*a/(d/c)^(1/2)*(1-d*x^2/c)^( 1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)/b*(EllipticF(x *(d/c)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))-EllipticE(x*(d/c)^(1/2),(-1-(-a*d+ b*c)/a/d)^(1/2)))+a*b^2*c^3/(d/c)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2 )/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(d/c)^(1/2),(-1-(-a*d+b *c)/a/d)^(1/2))+a^3*c*d^2/(d/c)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/ (-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(d/c)^(1/2),(-1-(-a*d+b*c )/a/d)^(1/2))+18*a^2*c^2*b*d/(d/c)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/ 2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(d/c)^(1/2),(-1-(-a*d+ b*c)/a/d)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.92 \[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=-\frac {2 \, {\left (b^{3} c^{4} + 5 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - {\left (2 \, b^{3} c^{4} + 10 \, a b^{2} c^{3} d + a^{3} d^{4} - {\left (10 \, a^{2} b - a b^{2}\right )} c^{2} d^{2} - 2 \, {\left (a^{3} - 9 \, a^{2} b\right )} c d^{3}\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + {\left (5 \, b^{3} d^{4} x^{6} + 2 \, b^{3} c^{3} d + 10 \, a b^{2} c^{2} d^{2} - 10 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4} - 8 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{4} + {\left (b^{3} c^{2} d^{2} - 17 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c}}{35 \, b^{2} d^{3} x} \] Input:
integrate((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x, algorithm="fricas")
Output:
-1/35*(2*(b^3*c^4 + 5*a*b^2*c^3*d - 5*a^2*b*c^2*d^2 - a^3*c*d^3)*sqrt(-b*d )*x*sqrt(c/d)*elliptic_e(arcsin(sqrt(c/d)/x), -a*d/(b*c)) - (2*b^3*c^4 + 1 0*a*b^2*c^3*d + a^3*d^4 - (10*a^2*b - a*b^2)*c^2*d^2 - 2*(a^3 - 9*a^2*b)*c *d^3)*sqrt(-b*d)*x*sqrt(c/d)*elliptic_f(arcsin(sqrt(c/d)/x), -a*d/(b*c)) + (5*b^3*d^4*x^6 + 2*b^3*c^3*d + 10*a*b^2*c^2*d^2 - 10*a^2*b*c*d^3 - 2*a^3* d^4 - 8*(b^3*c*d^3 - a*b^2*d^4)*x^4 + (b^3*c^2*d^2 - 17*a*b^2*c*d^3 + a^2* b*d^4)*x^2)*sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c))/(b^2*d^3*x)
\[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\int \left (a c - b d x^{4} + x^{2} \left (- a d + b c\right )\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a*c+(-a*d+b*c)*x**2-b*d*x**4)**(3/2),x)
Output:
Integral((a*c - b*d*x**4 + x**2*(-a*d + b*c))**(3/2), x)
\[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\int { {\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x, algorithm="maxima")
Output:
integrate((-b*d*x^4 + (b*c - a*d)*x^2 + a*c)^(3/2), x)
\[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\int { {\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x, algorithm="giac")
Output:
integrate((-b*d*x^4 + (b*c - a*d)*x^2 + a*c)^(3/2), x)
Timed out. \[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\int {\left (-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c\right )}^{3/2} \,d x \] Input:
int((a*c - x^2*(a*d - b*c) - b*d*x^4)^(3/2),x)
Output:
int((a*c - x^2*(a*d - b*c) - b*d*x^4)^(3/2), x)
\[ \int \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2} \, dx=\frac {-\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x +17 \sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x -8 \sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}-\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +8 \sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}-5 \sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}-2 \left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} d^{3}-10 \left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+10 \left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{2} d +2 \left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c^{3}+\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} c \,d^{2}+18 \left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b \,c^{2} d +\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{3}}{35 b d} \] Input:
int((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(3/2),x)
Output:
( - sqrt(c - d*x**2)*sqrt(a + b*x**2)*a**2*d**2*x + 17*sqrt(c - d*x**2)*sq rt(a + b*x**2)*a*b*c*d*x - 8*sqrt(c - d*x**2)*sqrt(a + b*x**2)*a*b*d**2*x* *3 - sqrt(c - d*x**2)*sqrt(a + b*x**2)*b**2*c**2*x + 8*sqrt(c - d*x**2)*sq rt(a + b*x**2)*b**2*c*d*x**3 - 5*sqrt(c - d*x**2)*sqrt(a + b*x**2)*b**2*d* *2*x**5 - 2*int((sqrt(c - d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c - a*d*x**2 + b*c*x**2 - b*d*x**4),x)*a**3*d**3 - 10*int((sqrt(c - d*x**2)*sqrt(a + b*x **2)*x**2)/(a*c - a*d*x**2 + b*c*x**2 - b*d*x**4),x)*a**2*b*c*d**2 + 10*in t((sqrt(c - d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c - a*d*x**2 + b*c*x**2 - b* d*x**4),x)*a*b**2*c**2*d + 2*int((sqrt(c - d*x**2)*sqrt(a + b*x**2)*x**2)/ (a*c - a*d*x**2 + b*c*x**2 - b*d*x**4),x)*b**3*c**3 + int((sqrt(c - d*x**2 )*sqrt(a + b*x**2))/(a*c - a*d*x**2 + b*c*x**2 - b*d*x**4),x)*a**3*c*d**2 + 18*int((sqrt(c - d*x**2)*sqrt(a + b*x**2))/(a*c - a*d*x**2 + b*c*x**2 - b*d*x**4),x)*a**2*b*c**2*d + int((sqrt(c - d*x**2)*sqrt(a + b*x**2))/(a*c - a*d*x**2 + b*c*x**2 - b*d*x**4),x)*a*b**2*c**3)/(35*b*d)