Integrand size = 25, antiderivative size = 408 \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx=\frac {2 (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) x \left (c+d x^2\right )}{35 b d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {x \left (10 a b c d+(b c+a d)^2+3 b d (b c+a d) x^2\right ) \sqrt {a c+(b c+a d) x^2+b d x^4}}{35 b d}+\frac {1}{7} x \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}-\frac {2 \sqrt {a} (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) \left (c+d x^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{35 b^{3/2} d^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {a^{3/2} \left (20 a b c d-(b c+a d)^2\right ) \left (c+d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{35 b^{3/2} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:
2/35*(a*d+b*c)*(8*a*b*c*d-(a*d+b*c)^2)*x*(d*x^2+c)/b/d^2/(a*c+(a*d+b*c)*x^ 2+b*d*x^4)^(1/2)+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^(1/2)*(a*d+b*c)*(8*a*b*c*d-(a*d+b*c)^2)*(d*x^2+c)*EllipticE(b^(1/2)* x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d^2/(a*(d*x^2+c)/c/ (b*x^2+a))^(1/2)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+1/35*a^(3/2)*(20*a*b*c* d-(a*d+b*c)^2)*(d*x^2+c)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/ b/c)^(1/2))/b^(3/2)/d/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(a*c+(a*d+b*c)*x^2+b *d*x^4)^(1/2)
Result contains complex when optimal does not.
Time = 6.11 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.73 \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (17 c+8 d x^2\right )+b^2 \left (c^2+8 c d x^2+5 d^2 x^4\right )\right )+2 i c \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (2 b^3 c^3-11 a b^2 c^2 d+8 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{35 b \sqrt {\frac {b}{a}} d^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] 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]*EllipticE[ 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]*EllipticF[ 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 = 1.33 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.60, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1404, 1490, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {3}{7} \int \left ((b c+a d) x^2+2 a c\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}dx+\frac {1}{7} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {3}{7} \left (\frac {\int \frac {2 (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) x^2+a c \left (20 a b c d-(b c+a d)^2\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (3 b d x^2 (a d+b c)+(a d+b c)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {3}{7} \left (\frac {\sqrt {a} \sqrt {c} \left (\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )}{\sqrt {b} \sqrt {d}}+\sqrt {a} \sqrt {c} \left (20 a b c d-(a d+b c)^2\right )\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {2 \sqrt {a} \sqrt {c} (a d+b c) \left (8 a b c d-(a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (3 b d x^2 (a d+b c)+(a d+b c)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} \left (\frac {\sqrt {a} \sqrt {c} \left (\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )}{\sqrt {b} \sqrt {d}}+\sqrt {a} \sqrt {c} \left (20 a b c d-(a d+b c)^2\right )\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (3 b d x^2 (a d+b c)+(a d+b c)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {3}{7} \left (\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )}{\sqrt {b} \sqrt {d}}+\sqrt {a} \sqrt {c} \left (20 a b c d-(a d+b c)^2\right )\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (3 b d x^2 (a d+b c)+(a d+b c)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {3}{7} \left (\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )}{\sqrt {b} \sqrt {d}}+\sqrt {a} \sqrt {c} \left (20 a b c d-(a d+b c)^2\right )\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}\right )}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (3 b d x^2 (a d+b c)+(a d+b c)^2+10 a b c d\right )}{15 b d}\right )+\frac {1}{7} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}\) |
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) + ((-2*(b*c + a*d)*(8*a*b*c*d - (b*c + a*d)^2)*(-((x*Sqrt[a*c + (b *c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)) + (a^(1 /4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d )*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticE[2*Ar cTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqr t[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b *d*x^4])))/(Sqrt[b]*Sqrt[d]) + (a^(1/4)*c^(1/4)*((2*(b*c + a*d)*(8*a*b*c*d - (b*c + a*d)^2))/(Sqrt[b]*Sqrt[d]) + Sqrt[a]*Sqrt[c]*(20*a*b*c*d - (b*c + a*d)^2))*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d) *x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticF[2*Arc Tan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt [b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]))/(15*b*d)))/7
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 + (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[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
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[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Time = 2.84 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\frac {x \left (5 b^{2} d^{2} x^{4}+8 a \,d^{2} b \,x^{2}+8 b^{2} c d \,x^{2}+a^{2} d^{2}+17 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{35 b d \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}-\frac {-\frac {\left (2 a^{3} d^{3}-10 a^{2} b c \,d^{2}-10 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {a \,b^{2} c^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {18 a^{2} c^{2} b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}}{35 b d}\) | \(544\) |
| default | \(\frac {b d \,x^{5} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{7}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 b d}+\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c^{2}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (2 a^{2} c d +2 a b \,c^{2}-\frac {3 \left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\) | \(624\) |
| elliptic | \(\frac {b d \,x^{5} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{7}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 b d}+\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c^{2}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (2 a^{2} c d +2 a b \,c^{2}-\frac {3 \left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\) | \(624\) |
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)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1 /2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*( -b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b* c)/c/b)^(1/2)))+a*b^2*c^3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2) /(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c )/c/b)^(1/2))+a^3*c*d^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/ c/b)^(1/2))-18*a^2*c^2*b*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2 )/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b* c)/c/b)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.79 \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx=\frac {2 \, {\left (b^{3} c^{4} - 5 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{3} c^{4} - 10 \, a b^{2} c^{3} d + a^{3} d^{4} - {\left (10 \, a^{2} b - a b^{2}\right )} c^{2} d^{2} + 2 \, {\left (a^{3} - 9 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (5 \, b^{3} d^{4} x^{6} - 2 \, b^{3} c^{3} d + 10 \, a b^{2} c^{2} d^{2} + 10 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4} + 8 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} x^{4} + {\left (b^{3} c^{2} d^{2} + 17 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}}{35 \, b^{2} d^{3} x} \] 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 - 10 *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. \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2} \, dx=\int {\left (b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c\right )}^{3/2} \,d x \] 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)*sqrt( 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)*sqrt( 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*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*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)*s qrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*c*d**2 + 1 8*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)