Integrand size = 30, antiderivative size = 295 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {(4 a d f-3 b (d e+c f)) x \sqrt {c+d x^2}}{3 b^2 \sqrt {a+b x^2}}+\frac {f x \left (c+d x^2\right )^{3/2}}{3 b \sqrt {a+b x^2}}+\frac {\left (3 b^2 c e-6 a b d e-7 a b c f+8 a^2 d f\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} (4 a d f-3 b (d e+c f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/3*(4*a*d*f-3*b*(c*f+d*e))*x*(d*x^2+c)^(1/2)/b^2/(b*x^2+a)^(1/2)+1/3*f*x *(d*x^2+c)^(3/2)/b/(b*x^2+a)^(1/2)+1/3*(8*a^2*d*f-7*a*b*c*f-6*a*b*d*e+3*b^ 2*c*e)*(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))/a^(1/2)/b^(5/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1 /2)-1/3*a^(1/2)*(4*a*d*f-3*b*(c*f+d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(ar ctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/(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 = 5.51 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (3 b^2 c e+4 a^2 d f+a b \left (-3 d e-3 c f+d f x^2\right )\right )+i c \left (3 b^2 c e+8 a^2 d f-a b (6 d e+7 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) (-3 b e+4 a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((c + d*x^2)^(3/2)*(e + f*x^2))/(a + b*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*x*(c + d*x^2)*(3*b^2*c*e + 4*a^2*d*f + a*b*(-3*d*e - 3*c*f + d*f*x^2)) + I*c*(3*b^2*c*e + 8*a^2*d*f - a*b*(6*d*e + 7*c*f))*Sqr t[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a* d)/(b*c)] - I*c*(-(b*c) + a*d)*(-3*b*e + 4*a*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b^3*Sqrt [a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.50 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {401, 25, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}-\frac {\int -\frac {\sqrt {d x^2+c} \left (a c f-d (3 b e-4 a f) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {d x^2+c} \left (a c f-d (3 b e-4 a f) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int -\frac {d \left (8 d f a^2-b (6 d e+7 c f) a+3 b^2 c e\right ) x^2+a c (4 a d f-3 b (d e+c f))}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b e-4 a f)}{3 b}}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {d \left (8 d f a^2-b (6 d e+7 c f) a+3 b^2 c e\right ) x^2+a c (4 a d f-3 b (d e+c f))}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b e-4 a f)}{3 b}}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {-\frac {d \left (8 a^2 d f-a b (7 c f+6 d e)+3 b^2 c e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (4 a d f-3 b (c f+d e)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b e-4 a f)}{3 b}}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {-\frac {d \left (8 a^2 d f-a b (7 c f+6 d e)+3 b^2 c e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (4 a d f-3 b (c f+d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}-\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b e-4 a f)}{3 b}}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {-\frac {d \left (8 a^2 d f-a b (7 c f+6 d e)+3 b^2 c e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (4 a d f-3 b (c f+d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}-\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b e-4 a f)}{3 b}}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {-\frac {d \left (8 a^2 d f-a b (7 c f+6 d e)+3 b^2 c e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (4 a d f-3 b (c f+d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}-\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b e-4 a f)}{3 b}}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{a b \sqrt {a+b x^2}}\) |
Input:
Int[((c + d*x^2)^(3/2)*(e + f*x^2))/(a + b*x^2)^(3/2),x]
Output:
((b*e - a*f)*x*(c + d*x^2)^(3/2))/(a*b*Sqrt[a + b*x^2]) + (-1/3*(d*(3*b*e - 4*a*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/b - (d*(3*b^2*c*e + 8*a^2*d*f - a*b*(6*d*e + 7*c*f))*((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)*(4*a*d*f - 3*b*(d*e + c*f))*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*b))/(a*b)
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[(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[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(266)=532\).
Time = 11.56 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.85
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) x}{a \,b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {d f x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2}}+\frac {\left (\frac {f \,d^{2} a^{2}-2 f d c b a -a b \,d^{2} e +f \,c^{2} b^{2}+2 d \,b^{2} c e}{b^{3}}-\frac {\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) \left (a d -b c \right )}{b^{3} a}-\frac {c \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right )}{b^{2} a}-\frac {a c d f}{3 b^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {d \left (a d f -2 b c f -b d e \right )}{b^{2}}-\frac {\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) d}{b^{2} a}-\frac {d f \left (2 a d +2 b c \right )}{3 b^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(545\) |
| default | \(\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, a b \,d^{2} f \,x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} f \,x^{3}-2 \sqrt {-\frac {b}{a}}\, a b c d f \,x^{3}-3 \sqrt {-\frac {b}{a}}\, a b \,d^{2} e \,x^{3}+3 \sqrt {-\frac {b}{a}}\, b^{2} c d e \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d f -4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d f +7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f +6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e +4 \sqrt {-\frac {b}{a}}\, a^{2} c d f x -3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} f x -3 \sqrt {-\frac {b}{a}}\, a b c d e x +3 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} e x \right )}{3 b^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}\, a}\) | \(670\) |
| risch | \(\frac {f x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, d}{3 b^{2}}-\frac {\left (-\frac {\left (5 a d f -4 b c f -3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 \left (a^{3} d^{2} f -2 a^{2} b c d f -a^{2} b \,d^{2} e +b^{2} c^{2} f a +2 a \,b^{2} c d e -b^{3} c^{2} e \right ) \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{b}-\frac {\left (3 f \,d^{2} a^{2}-7 f d c b a -3 a b \,d^{2} e +3 f \,c^{2} b^{2}+6 d \,b^{2} c e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{b \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(686\) |
Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*((b*d*x^2+b*c) *(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/a/b^3*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2) +1/3*d*f/b^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+((a^2*d^2*f-2*a*b*c*d*f -a*b*d^2*e+b^2*c^2*f+2*b^2*c*d*e)/b^3-(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^ 3*(a*d-b*c)/a-1/b^2*c*(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/a-1/3*a/b^2*c*d*f) /(-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))-(-1/b^2*d*( a*d*f-2*b*c*f-b*d*e)-(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2*d/a-1/3*d*f/b^2 *(2*a*d+2*b*c))*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))))
Time = 0.11 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.48 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d\right )} e - {\left (7 \, a b^{2} c^{2} - 8 \, a^{2} b c d\right )} f\right )} x^{3} + {\left (3 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} e - {\left (7 \, a^{2} b c^{2} - 8 \, a^{3} c d\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d - a b^{2} d^{2}\right )} e - {\left (7 \, a b^{2} c^{2} - 4 \, a^{2} b d^{2} - {\left (8 \, a^{2} b - 3 \, a b^{2}\right )} c d\right )} f\right )} x^{3} + {\left (3 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d - a^{2} b d^{2}\right )} e - {\left (7 \, a^{2} b c^{2} - 4 \, a^{3} d^{2} - {\left (8 \, a^{3} - 3 \, a^{2} b\right )} c d\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (a b^{2} d^{2} f x^{4} + {\left (3 \, a b^{2} d^{2} e + 4 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} f\right )} x^{2} - 3 \, {\left (a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e + {\left (7 \, a^{2} b c d - 8 \, a^{3} d^{2}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a b^{4} d x^{3} + a^{2} b^{3} d x\right )}} \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/3*(((3*(b^3*c^2 - 2*a*b^2*c*d)*e - (7*a*b^2*c^2 - 8*a^2*b*c*d)*f)*x^3 + (3*(a*b^2*c^2 - 2*a^2*b*c*d)*e - (7*a^2*b*c^2 - 8*a^3*c*d)*f)*x)*sqrt(b*d) *sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((3*(b^3*c^2 - 2 *a*b^2*c*d - a*b^2*d^2)*e - (7*a*b^2*c^2 - 4*a^2*b*d^2 - (8*a^2*b - 3*a*b^ 2)*c*d)*f)*x^3 + (3*(a*b^2*c^2 - 2*a^2*b*c*d - a^2*b*d^2)*e - (7*a^2*b*c^2 - 4*a^3*d^2 - (8*a^3 - 3*a^2*b)*c*d)*f)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_ f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (a*b^2*d^2*f*x^4 + (3*a*b^2*d^2*e + 4 *(a*b^2*c*d - a^2*b*d^2)*f)*x^2 - 3*(a*b^2*c*d - 2*a^2*b*d^2)*e + (7*a^2*b *c*d - 8*a^3*d^2)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a*b^4*d*x^3 + a^2*b ^3*d*x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(3/2)*(f*x**2+e)/(b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x**2)**(3/2)*(e + f*x**2)/(a + b*x**2)**(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)/(b*x^2 + a)^(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)/(b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(((c + d*x^2)^(3/2)*(e + f*x^2))/(a + b*x^2)^(3/2),x)
Output:
int(((c + d*x^2)^(3/2)*(e + f*x^2))/(a + b*x^2)^(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*f*x + 2*sqrt(c + d*x**2)*sqr t(a + b*x**2)*a*d**2*f*x**3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2*f *x + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*d*e*x - 8*int((sqrt(c + d*x** 2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x **4 + b**2*c*x**4 + b**2*d*x**6),x)*a**3*d**3*f + 11*int((sqrt(c + d*x**2) *sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x** 4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c*d**2*f + 6*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x* *4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**3*e - 8*int((sqrt(c + d*x**2) *sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x** 4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**3*f*x**2 - 3*int((sqrt(c + d*x **2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d *x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c**2*d*f - 6*int((sqrt(c + d* x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b* d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d**2*e + 11*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a* b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d**2*f*x**2 + 6*int((sqr t(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*d**3*e*x**2 - 3*int( (sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*...