Integrand size = 30, antiderivative size = 386 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\left (20 b c e-10 a d e-13 a c f+\frac {3 b c^2 f}{d}+\frac {8 a^2 d f}{b}\right ) x \sqrt {c+d x^2}}{15 b \sqrt {a+b x^2}}+\frac {(5 b d e+3 b c f-4 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}-\frac {\sqrt {a} \left (8 a^2 d^2 f+b^2 c (20 d e+3 c f)-a b d (10 d e+13 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (15 b^2 c e-5 a b d e-6 a b c f+4 a^2 d f\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(20*b*c*e-10*a*d*e-13*a*c*f+3*b*c^2*f/d+8*a^2*d*f/b)*x*(d*x^2+c)^(1/2 )/b/(b*x^2+a)^(1/2)+1/15*(-4*a*d*f+3*b*c*f+5*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x ^2+c)^(1/2)/b^2+1/5*f*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/b-1/15*a^(1/2)*(8* a^2*d^2*f+b^2*c*(3*c*f+20*d*e)-a*b*d*(13*c*f+10*d*e))*(d*x^2+c)^(1/2)*Elli pticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(5/2)/d/(b* x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/15*a^(1/2)*(4*a^2*d*f-6*a*b *c*f-5*a*b*d*e+15*b^2*c*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(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 = 3.78 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.71 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d f-b \left (5 d e+6 c f+3 d f x^2\right )\right )-i c \left (8 a^2 d^2 f+b^2 c (20 d e+3 c f)-a b d (10 d e+13 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) (-5 b d e-3 b c f+4 a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((c + d*x^2)^(3/2)*(e + f*x^2))/Sqrt[a + b*x^2],x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a*d*f - b*(5*d*e + 6*c*f + 3*d *f*x^2))) - I*c*(8*a^2*d^2*f + b^2*c*(20*d*e + 3*c*f) - a*b*d*(10*d*e + 13 *c*f))*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*(-(b*c) + a*d)*(-5*b*d*e - 3*b*c*f + 4*a*d*f)*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.49 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {403, 403, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \frac {\sqrt {d x^2+c} \left ((5 b d e+3 b c f-4 a d f) x^2+c (5 b e-a f)\right )}{\sqrt {b x^2+a}}dx}{5 b}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {\left (c (20 d e+3 c f) b^2-a d (10 d e+13 c f) b+8 a^2 d^2 f\right ) x^2+c \left (4 d f a^2-5 b d e a-6 b c f a+15 b^2 c e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+3 b c f+5 b d e)}{3 b}}{5 b}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\left (8 a^2 d^2 f-a b d (13 c f+10 d e)+b^2 c (3 c f+20 d e)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+c \left (4 a^2 d f-6 a b c f-5 a b d e+15 b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+3 b c f+5 b d e)}{3 b}}{5 b}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\left (8 a^2 d^2 f-a b d (13 c f+10 d e)+b^2 c (3 c f+20 d e)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d f-6 a b c f-5 a b d e+15 b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+3 b c f+5 b d e)}{3 b}}{5 b}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\left (8 a^2 d^2 f-a b d (13 c f+10 d e)+b^2 c (3 c f+20 d e)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d f-6 a b c f-5 a b d e+15 b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+3 b c f+5 b d e)}{3 b}}{5 b}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d f-6 a b c f-5 a b d e+15 b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^2 d^2 f-a b d (13 c f+10 d e)+b^2 c (3 c f+20 d e)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f+3 b c f+5 b d e)}{3 b}}{5 b}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 b}\) |
Input:
Int[((c + d*x^2)^(3/2)*(e + f*x^2))/Sqrt[a + b*x^2],x]
Output:
(f*x*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(5*b) + (((5*b*d*e + 3*b*c*f - 4*a *d*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b) + ((8*a^2*d^2*f + b^2*c*(20 *d*e + 3*c*f) - a*b*d*(10*d*e + 13*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)*(15*b^2*c*e - 5*a*b*d*e - 6*a*b*c*f + 4*a^2*d*f)*Sqr t[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*S qrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*b))/(5*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[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 = 10.26 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {d f \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b}+\frac {\left (2 c d f +d^{2} e -\frac {d f \left (4 a d +4 b c \right )}{5 b}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (c^{2} e -\frac {\left (2 c d f +d^{2} e -\frac {d f \left (4 a d +4 b c \right )}{5 b}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (c^{2} f +2 c e d -\frac {3 a c d f}{5 b}-\frac {\left (2 c d f +d^{2} e -\frac {d f \left (4 a d +4 b c \right )}{5 b}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(448\) |
| risch | \(-\frac {x \left (-3 b d f \,x^{2}+4 a d f -6 b c f -5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2}}+\frac {\left (-\frac {\left (8 f \,d^{2} a^{2}-13 f d c b a -10 a b \,d^{2} e +3 f \,c^{2} b^{2}+20 d \,b^{2} c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {15 b^{2} c^{2} e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {6 a b \,c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {4 a^{2} c d f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {5 a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(628\) |
| default | \(-\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} f \,x^{7}+\sqrt {-\frac {b}{a}}\, a b \,d^{3} f \,x^{5}-9 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} f \,x^{5}-5 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} e \,x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} f \,x^{3}-5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} f \,x^{3}-5 \sqrt {-\frac {b}{a}}\, a b \,d^{3} e \,x^{3}-6 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d f \,x^{3}-5 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} 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^{2} f -7 \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} d f -5 \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^{2} 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^{3} f +5 \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} d 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^{2} f +13 \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} d f +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 c \,d^{2} 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^{3} f -20 \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} d e +4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} f x -6 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d f x -5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} e x \right )}{15 b^{2} d \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}}\) | \(869\) |
Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(1/2),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)*(1/5/b*d*f*x^3 *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(2*c*d*f+d^2*e-1/5/b*d*f*(4*a*d+4 *b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(c^2*e-1/3*(2*c*d*f+d^2*e -1/5/b*d*f*(4*a*d+4*b*c))/b/d*a*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)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))-(c^2*f+2*c*e*d-3/5*a/b*c*d*f-1/3*(2*c*d*f+d^2*e-1/5 /b*d*f*(4*a*d+4*b*c))/b/d*(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 = 0.10 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=-\frac {\sqrt {b d} {\left (10 \, {\left (2 \, b^{2} c^{2} d - a b c d^{2}\right )} e + {\left (3 \, b^{2} c^{3} - 13 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (5 \, {\left (4 \, b^{2} c^{2} d - a b d^{3} - {\left (2 \, a b - 3 \, b^{2}\right )} c d^{2}\right )} e + {\left (3 \, b^{2} c^{3} - 13 \, a b c^{2} d + 4 \, a^{2} d^{3} + 2 \, {\left (4 \, a^{2} - 3 \, a b\right )} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} f x^{4} + {\left (5 \, b^{2} d^{3} e + 2 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} f\right )} x^{2} + 10 \, {\left (2 \, b^{2} c d^{2} - a b d^{3}\right )} e + {\left (3 \, b^{2} c^{2} d - 13 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{3} d^{2} x} \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-1/15*(sqrt(b*d)*(10*(2*b^2*c^2*d - a*b*c*d^2)*e + (3*b^2*c^3 - 13*a*b*c^2 *d + 8*a^2*c*d^2)*f)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b* c)) - sqrt(b*d)*(5*(4*b^2*c^2*d - a*b*d^3 - (2*a*b - 3*b^2)*c*d^2)*e + (3* b^2*c^3 - 13*a*b*c^2*d + 4*a^2*d^3 + 2*(4*a^2 - 3*a*b)*c*d^2)*f)*x*sqrt(-c /d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b^2*d^3*f*x^4 + (5*b^ 2*d^3*e + 2*(3*b^2*c*d^2 - 2*a*b*d^3)*f)*x^2 + 10*(2*b^2*c*d^2 - a*b*d^3)* e + (3*b^2*c^2*d - 13*a*b*c*d^2 + 8*a^2*d^3)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3*d^2*x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}{\sqrt {a + b x^{2}}}\, dx \] Input:
integrate((d*x**2+c)**(3/2)*(f*x**2+e)/(b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x**2)**(3/2)*(e + f*x**2)/sqrt(a + b*x**2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)/sqrt(b*x^2 + a), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((c + d*x^2)^(3/2)*(e + f*x^2))/(a + b*x^2)^(1/2),x)
Output:
int(((c + d*x^2)^(3/2)*(e + f*x^2))/(a + b*x^2)^(1/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d f x +6 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c f x +5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d e x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d f \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} f -13 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d f -10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} f +20 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c d e +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d f -6 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} f -5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d e +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} e}{15 b^{2}} \] Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x + 6*sqrt(c + d*x**2)*sqrt( a + b*x**2)*b*c*f*x + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 3*sqrt (c + d*x**2)*sqrt(a + b*x**2)*b*d*f*x**3 + 8*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*d**2*f - 13 *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*c*d*f - 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*d**2*e + 3*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**2 *c**2*f + 20*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**2*c*d*e + 4*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*c*d*f - 6*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*c**2*f - 5*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*c*d*e + 15*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)*b**2*c**2*e)/(15*b**2)