Integrand size = 23, antiderivative size = 404 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=-\frac {2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x \sqrt {c+d x^2}}{35 b d^2 \sqrt {a+b x^2}}+\frac {1}{35} \left (9 a c-\frac {2 b c^2}{d}+\frac {a^2 d}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}+\frac {3 (b c+a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{35 d}+\frac {1}{7} x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}+\frac {2 \sqrt {a} (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{35 b^{3/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{35 b^{3/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-2/35*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2*c^2)*x*(d*x^2+c)^(1/2)/b/d^2/(b*x^2 +a)^(1/2)+1/35*(9*a*c-2*b*c^2/d+a^2*d/b)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2) +3/35*(a*d+b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/d+1/7*x*(b*x^2+a)^(3/2)* (d*x^2+c)^(3/2)+2/35*a^(1/2)*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2*c^2)*(d*x^2+ c)^(1/2)*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/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/35*a^(3/2)*( a^2*d^2-18*a*b*c*d+b^2*c^2)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2) *x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b *x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 3.96 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.75 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (17 c+8 d x^2\right )+b^2 \left (c^2+8 c d x^2+5 d^2 x^4\right )\right )+2 i c \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (2 b^3 c^3-11 a b^2 c^2 d+8 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{35 b \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)^(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]* Sqrt[c + d*x^2])
Time = 0.56 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {318, 403, 27, 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.
| \(\displaystyle \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (2 d (4 b c-a d) x^2+c (7 b c-a d)\right )}{\sqrt {d x^2+c}}dx}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {3 d \sqrt {b x^2+a} \left (\left (b^2 c^2+9 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)\right )}{\sqrt {d x^2+c}}dx}{5 d}+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{5} \int \frac {\sqrt {b x^2+a} \left (\left (b^2 c^2+9 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)\right )}{\sqrt {d x^2+c}}dx+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {\int -\frac {2 (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2-18 a b d c+a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^2 d^2+9 a b c d+b^2 c^2\right )}{3 d}\right )+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^2 d^2+9 a b c d+b^2 c^2\right )}{3 d}-\frac {\int \frac {2 (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2-18 a b d c+a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}\right )+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^2 d^2+9 a b c d+b^2 c^2\right )}{3 d}-\frac {a c \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}\right )+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^2 d^2+9 a b c d+b^2 c^2\right )}{3 d}-\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^2 d^2+9 a b c d+b^2 c^2\right )}{3 d}-\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^2 d^2+9 a b c d+b^2 c^2\right )}{3 d}-\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )+\frac {2}{5} x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (4 b c-a d)}{7 b}+\frac {d x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 b}\) |
Input:
Int[(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2),x]
Output:
(d*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*b) + ((2*(4*b*c - a*d)*x*(a + b *x^2)^(3/2)*Sqrt[c + d*x^2])/5 + (3*(((b^2*c^2 + 9*a*b*c*d - 2*a^2*d^2)*x* Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c *d + a^2*d^2)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[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])) + (c^(3/2)*(b^2* c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/S qrt[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)))/5)/(7*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[(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]
Time = 5.20 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(\frac {x \left (5 b^{2} d^{2} x^{4}+8 x^{2} a b \,d^{2}+8 x^{2} b^{2} c d +a^{2} d^{2}+17 a b c d +b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{35 b d}-\frac {\left (-\frac {\left (2 a^{3} d^{3}-10 a^{2} b c \,d^{2}-10 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b^{2} c^{3} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {18 a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{35 b d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(566\) |
| elliptic | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b d \,x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a^{2} c^{2}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 a^{2} c d +2 b \,c^{2} a -\frac {3 \left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (a^{2} d^{2}+\frac {23 a b c d}{7}+b^{2} c^{2}-\frac {\left (2 a b \,d^{2}+2 b^{2} c d -\frac {b d \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\) | \(684\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (5 \sqrt {-\frac {b}{a}}\, b^{3} d^{4} x^{9}+13 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{4} x^{7}+13 \sqrt {-\frac {b}{a}}\, b^{3} c \,d^{3} x^{7}+9 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{4} x^{5}+38 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{3} x^{5}+9 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a^{3} d^{4} x^{3}+26 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{3} x^{3}+26 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{3} c^{3} d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} c \,d^{3}+8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,c^{2} d^{2}-11 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{3} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{4}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} c \,d^{3}+10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,c^{2} d^{2}+10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{3} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{4}+\sqrt {-\frac {b}{a}}\, a^{3} c \,d^{3} x +17 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d^{2} x +\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} d x \right )}{35 b \,d^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}}\) | \(780\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^(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)^(1/2)*(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^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))+b^ 2*c^3*a/(-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*b*c^2*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)))* ((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.78 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\frac {2 \, {\left (b^{3} c^{4} - 5 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{3} c^{4} - 10 \, a b^{2} c^{3} d + a^{3} d^{4} - {\left (10 \, a^{2} b - a b^{2}\right )} c^{2} d^{2} + 2 \, {\left (a^{3} - 9 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (5 \, b^{3} d^{4} x^{6} - 2 \, b^{3} c^{3} d + 10 \, a b^{2} c^{2} d^{2} + 10 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4} + 8 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} x^{4} + {\left (b^{3} c^{2} d^{2} + 17 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{35 \, b^{2} d^{3} x} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(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*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^3*x)
\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\int \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2), x)
\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2), x)
\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\int {\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2} \,d x \] Input:
int((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2),x)
Output:
int((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2), x)
\[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x +17 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x +8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}+\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}+5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} d^{3}+10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{2} d -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c^{3}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} c \,d^{2}+18 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b \,c^{2} d -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{3}}{35 b d} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^(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)