Integrand size = 35, antiderivative size = 410 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\frac {\left (5 b B c-3 a c C-\frac {2 b c^2 C}{d}+15 A b d-10 a B d+\frac {8 a^2 C d}{b}\right ) x \sqrt {c+d x^2}}{15 b d \sqrt {a+b x^2}}+\frac {(b c C+5 b B d-4 a C d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d}+\frac {C x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\sqrt {a} \left (8 a^2 C d^2-a b d (3 c C+10 B d)-b^2 \left (2 c^2 C-5 B c d-15 A d^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (15 A b^2 d+4 a^2 C d-a b (c C+5 B d)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(5*b*B*c-3*a*c*C-2*b*c^2*C/d+15*A*b*d-10*a*B*d+8*a^2*C*d/b)*x*(d*x^2+ c)^(1/2)/b/d/(b*x^2+a)^(1/2)+1/15*(5*B*b*d-4*C*a*d+C*b*c)*x*(b*x^2+a)^(1/2 )*(d*x^2+c)^(1/2)/b^2/d+1/5*C*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b-1/15*a ^(1/2)*(8*a^2*C*d^2-a*b*d*(10*B*d+3*C*c)-b^2*(-15*A*d^2-5*B*c*d+2*C*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^(5/2)/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/15*a^ (1/2)*(15*A*b^2*d+4*a^2*C*d-a*b*(5*B*d+C*c))*(d*x^2+c)^(1/2)*InverseJacobi AM(arctan(b^(1/2)*x/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)
Result contains complex when optimal does not.
Time = 3.23 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a C d-b \left (c C+5 B d+3 C d x^2\right )\right )-i c \left (8 a^2 C d^2-a b d (3 c C+10 B d)+b^2 \left (-2 c^2 C+5 B c d+15 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) (2 b c C-5 b B d+4 a C d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[c + d*x^2]*(A + B*x^2 + C*x^4))/Sqrt[a + b*x^2],x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a*C*d - b*(c*C + 5*B*d + 3*C*d *x^2))) - I*c*(8*a^2*C*d^2 - a*b*d*(3*c*C + 10*B*d) + b^2*(-2*c^2*C + 5*B* c*d + 15*A*d^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSi nh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(-(b*c) + a*d)*(2*b*c*C - 5*b*B*d + 4* a*C*d)*Sqrt[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^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])
Time = 1.40 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \sqrt {c+d x^2}}{\sqrt {a+b x^2}}+\frac {B x^2 \sqrt {c+d x^2}}{\sqrt {a+b x^2}}+\frac {C x^4 \sqrt {c+d x^2}}{\sqrt {a+b x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {C \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 b}+\frac {B \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 b}+\frac {C (b c-4 a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b^2 d}+\frac {A d \sqrt {b x^2+a} x}{b \sqrt {d x^2+c}}+\frac {B (b c-2 a d) \sqrt {b x^2+a} x}{3 b^2 \sqrt {d x^2+c}}-\frac {C \left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) \sqrt {b x^2+a} x}{15 b^3 d \sqrt {d x^2+c}}-\frac {B \sqrt {c} (b c-2 a d) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} C \left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {A \sqrt {c} \sqrt {d} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} C (b c-4 a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {B c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[(Sqrt[c + d*x^2]*(A + B*x^2 + C*x^4))/Sqrt[a + b*x^2],x]
Output:
(A*d*x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) + (B*(b*c - 2*a*d)*x*Sqrt[a + b*x^2])/(3*b^2*Sqrt[c + d*x^2]) - (C*(2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*x *Sqrt[a + b*x^2])/(15*b^3*d*Sqrt[c + d*x^2]) + (B*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b) + (C*(b*c - 4*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15 *b^2*d) + (C*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b) - (A*Sqrt[c]*Sqrt[ d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] )/(b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (B*Sqrt[c]*( b*c - 2*a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b *c)/(a*d)])/(3*b^2*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*C*(2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*Sqrt[a + b*x^2]*E llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^3*d^(3/2)*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (A*c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d ]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (B*c^(3/2)*Sqrt [a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b* Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)* C*(b*c - 4*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[ c + d*x^2])
Time = 7.36 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.04
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {C \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b}+\frac {\left (B d +C c -\frac {C \left (4 a d +4 b c \right )}{5 b}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (A c -\frac {\left (B d +C c -\frac {C \left (4 a d +4 b c \right )}{5 b}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (A d +B c -\frac {3 a c C}{5 b}-\frac {\left (B d +C c -\frac {C \left (4 a d +4 b c \right )}{5 b}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(425\) |
| risch | \(\frac {x \left (3 C b d \,x^{2}+5 B b d -4 C a d +C b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2} d}+\frac {\left (-\frac {\left (15 A \,b^{2} d^{2}-10 B a b \,d^{2}+5 b^{2} B c d +8 a^{2} C \,d^{2}-3 C a b c d -2 C \,b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {15 A \,b^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {C b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {4 a^{2} c C d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {5 B a b c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 d \,b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(641\) |
| default | \(\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (3 C \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+5 B \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{5}-C \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+4 C \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+5 B \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{3}+5 B \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{3}-4 C \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+C \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+15 A \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 \,d^{2}+5 B \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}-5 B \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 -10 B \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}+5 B \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 -4 C \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}+2 C \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 +2 C \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}+8 C \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}-3 C \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 -2 C \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}+5 B \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x -4 C \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +C \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 b^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) d^{2} \sqrt {-\frac {b}{a}}}\) | \(906\) |
Input:
int((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOS E)
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*C/b*x^3*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(B*d+C*c-1/5*C/b*(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)+(A*c-1/3*(B*d+C*c-1/5*C/b*(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 ))-(A*d+B*c-3/5*a/b*c*C-1/3*(B*d+C*c-1/5*C/b*(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))-El lipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\frac {{\left (2 \, C b^{2} c^{3} + {\left (3 \, C a b - 5 \, B b^{2}\right )} c^{2} d - {\left (8 \, C a^{2} - 10 \, B a b + 15 \, A b^{2}\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, C b^{2} c^{3} + {\left (3 \, C a b - 5 \, B b^{2}\right )} c^{2} d - {\left (8 \, C a^{2} - {\left (10 \, B + C\right )} a b + 15 \, A b^{2}\right )} c d^{2} - {\left (4 \, C a^{2} - 5 \, B a b + 15 \, A b^{2}\right )} d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, C b^{2} d^{3} x^{4} - 2 \, C b^{2} c^{2} d - {\left (3 \, C a b - 5 \, B b^{2}\right )} c d^{2} + {\left (8 \, C a^{2} - 10 \, B a b + 15 \, A b^{2}\right )} d^{3} + {\left (C b^{2} c d^{2} - {\left (4 \, C a b - 5 \, B b^{2}\right )} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{3} d^{3} x} \] Input:
integrate((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="fr icas")
Output:
1/15*((2*C*b^2*c^3 + (3*C*a*b - 5*B*b^2)*c^2*d - (8*C*a^2 - 10*B*a*b + 15* A*b^2)*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/ (b*c)) - (2*C*b^2*c^3 + (3*C*a*b - 5*B*b^2)*c^2*d - (8*C*a^2 - (10*B + C)* a*b + 15*A*b^2)*c*d^2 - (4*C*a^2 - 5*B*a*b + 15*A*b^2)*d^3)*sqrt(b*d)*x*sq rt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (3*C*b^2*d^3*x^4 - 2*C*b^2*c^2*d - (3*C*a*b - 5*B*b^2)*c*d^2 + (8*C*a^2 - 10*B*a*b + 15*A*b^2 )*d^3 + (C*b^2*c*d^2 - (4*C*a*b - 5*B*b^2)*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt( d*x^2 + c))/(b^3*d^3*x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {c + d x^{2}} \left (A + B x^{2} + C x^{4}\right )}{\sqrt {a + b x^{2}}}\, dx \] Input:
integrate((d*x**2+c)**(1/2)*(C*x**4+B*x**2+A)/(b*x**2+a)**(1/2),x)
Output:
Integral(sqrt(c + d*x**2)*(A + B*x**2 + C*x**4)/sqrt(a + b*x**2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="ma xima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(d*x^2 + c)/sqrt(b*x^2 + a), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="gi ac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(d*x^2 + c)/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {d\,x^2+c}\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + b*x^2)^(1/2),x)
Output:
int(((c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + b*x^2)^(1/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\sqrt {a+b x^2}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a c d x +5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d x +\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b \,c^{2} x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c d \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c \,d^{2}+5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} d^{2}-3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} d +5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c d -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{3}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c^{2} d +10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c d -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{3}}{15 b^{2} d} \] Input:
int((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*x + 5*sqrt(c + d*x**2)*sqrt( a + b*x**2)*b**2*d*x + sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2*x + 3*sqrt (c + d*x**2)*sqrt(a + b*x**2)*b*c*d*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*c*d**2 + 5* 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*d**2 - 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)*a*b*c**2*d + 5*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*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**2*c**3 + 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**2*d + 10*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*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*c**3)/(15*b**2*d)