Integrand size = 40, antiderivative size = 604 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\frac {\left (8 a^3 d^3 D-a^2 b d^2 (14 C d-9 c D)-a b^2 d \left (21 c C d-35 B d^2-16 c^2 D\right )+b^3 \left (56 c^2 C d-70 B c d^2+105 A d^3-48 c^3 D\right )\right ) x \sqrt {c+d x^2}}{105 b^2 d^4 \sqrt {a+b x^2}}-\frac {\left (4 a^2 d^2 D-a b d (7 C d-5 c D)+b^2 \left (28 c C d-35 B d^2-24 c^2 D\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b^2 d^3}+\frac {(7 b C d-6 b c D+a d D) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b d^2}+\frac {D x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\sqrt {a} \left (8 a^3 d^3 D-a^2 b d^2 (14 C d-9 c D)-a b^2 d \left (21 c C d-35 B d^2-16 c^2 D\right )+b^3 \left (56 c^2 C d-70 B c d^2+105 A d^3-48 c^3 D\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{5/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^2 c d^2 D-a b c d (7 C d-5 c D)+b^2 \left (28 c^2 C d-35 B c d^2+105 A d^3-24 c^3 D\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{5/2} c d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/105*(8*a^3*d^3*D-a^2*b*d^2*(14*C*d-9*D*c)-a*b^2*d*(-35*B*d^2+21*C*c*d-16 *D*c^2)+b^3*(105*A*d^3-70*B*c*d^2+56*C*c^2*d-48*D*c^3))*x*(d*x^2+c)^(1/2)/ b^2/d^4/(b*x^2+a)^(1/2)-1/105*(4*a^2*d^2*D-a*b*d*(7*C*d-5*D*c)+b^2*(-35*B* d^2+28*C*c*d-24*D*c^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^3+1/35*(7* C*b*d+D*a*d-6*D*b*c)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d^2+1/7*D*x^5*( b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d-1/105*a^(1/2)*(8*a^3*d^3*D-a^2*b*d^2*(14* C*d-9*D*c)-a*b^2*d*(-35*B*d^2+21*C*c*d-16*D*c^2)+b^3*(105*A*d^3-70*B*c*d^2 +56*C*c^2*d-48*D*c^3))*(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^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/( b*x^2+a))^(1/2)+1/105*a^(3/2)*(4*a^2*c*d^2*D-a*b*c*d*(7*C*d-5*D*c)+b^2*(10 5*A*d^3-35*B*c*d^2+28*C*c^2*d-24*D*c^3))*(d*x^2+c)^(1/2)*InverseJacobiAM(a rctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/c/d^3/(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 = 4.83 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 d^2 D+a b d \left (-7 C d+5 c D-3 d D x^2\right )-b^2 \left (24 c^2 D-2 c d \left (14 C+9 D x^2\right )+d^2 \left (35 B+21 C x^2+15 D x^4\right )\right )\right )-i c \left (8 a^3 d^3 D+a^2 b d^2 (-14 C d+9 c D)+a b^2 d \left (-21 c C d+35 B d^2+16 c^2 D\right )+b^3 \left (56 c^2 C d-70 B c d^2+105 A d^3-48 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) \left (4 a^2 c d^2 D+a b c d (-7 C d+8 c D)+b^2 \left (-56 c^2 C d+70 B c d^2-105 A d^3+48 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[c + d*x^2],x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a^2*d^2*D + a*b*d*(-7*C*d + 5* c*D - 3*d*D*x^2) - b^2*(24*c^2*D - 2*c*d*(14*C + 9*D*x^2) + d^2*(35*B + 21 *C*x^2 + 15*D*x^4)))) - I*c*(8*a^3*d^3*D + a^2*b*d^2*(-14*C*d + 9*c*D) + a *b^2*d*(-21*c*C*d + 35*B*d^2 + 16*c^2*D) + b^3*(56*c^2*C*d - 70*B*c*d^2 + 105*A*d^3 - 48*c^3*D))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I *ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(-(b*c) + a*d)*(4*a^2*c*d^2*D + a* b*c*d*(-7*C*d + 8*c*D) + b^2*(-56*c^2*C*d + 70*B*c*d^2 - 105*A*d^3 + 48*c^ 3*D))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a ]*x], (a*d)/(b*c)])/(105*a^2*(b/a)^(5/2)*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])
Leaf count is larger than twice the leaf count of optimal. \(1254\) vs. \(2(604)=1208\).
Time = 1.99 (sec) , antiderivative size = 1254, normalized size of antiderivative = 2.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \sqrt {a+b x^2}}{\sqrt {c+d x^2}}+\frac {B x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2}}+\frac {C x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2}}+\frac {D x^6 \sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {D \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}-\frac {(6 b c-a d) D \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b d^2}+\frac {C \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}-\frac {C (4 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b d^2}-\frac {\left (\frac {4 d a^2}{b}+5 c a-\frac {24 b c^2}{d}\right ) D \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 b d^2}+\frac {B \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 d}+\frac {A \sqrt {b x^2+a} x}{\sqrt {d x^2+c}}-\frac {B (2 b c-a d) \sqrt {b x^2+a} x}{3 b d \sqrt {d x^2+c}}+\frac {C \left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) \sqrt {b x^2+a} x}{15 b^2 d^2 \sqrt {d x^2+c}}-\frac {\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) D \sqrt {b x^2+a} x}{105 b^3 d^3 \sqrt {d x^2+c}}+\frac {B \sqrt {c} (2 b c-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 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 {\sqrt {c} C \left (8 b^2 c^2-3 a b d c-2 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^2 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} \left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) D \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b^3 d^{7/2} \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 {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \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 (4 b c-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 d^{5/2} \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} \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right ) 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 )}{105 b^2 d^{7/2} \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 {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\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 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[c + d*x^2],x]
Output:
(A*x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2] - (B*(2*b*c - a*d)*x*Sqrt[a + b*x^2] )/(3*b*d*Sqrt[c + d*x^2]) + (C*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*x*Sqrt[ a + b*x^2])/(15*b^2*d^2*Sqrt[c + d*x^2]) - ((48*b^3*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*D*x*Sqrt[a + b*x^2])/(105*b^3*d^3*Sqrt[c + d*x ^2]) + (B*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (C*(4*b*c - a*d)*x*Sq rt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b*d^2) - ((5*a*c - (24*b*c^2)/d + (4*a^ 2*d)/b)*D*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*b*d^2) + (C*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - ((6*b*c - a*d)*D*x^3*Sqrt[a + b*x^2]*Sqrt [c + d*x^2])/(35*b*d^2) + (D*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) - (A*Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[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]) + (B*Sqrt[c]*(2*b*c - a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq rt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2) )]*Sqrt[c + d*x^2]) - (Sqrt[c]*C*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*Sqrt[ a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^ 2*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c ]*(48*b^3*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*D*Sqrt[a + b*x ^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b^3*d^(7 /2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (B*c^(3/2)*Sq rt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/...
Time = 6.98 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.00
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {D x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 d}+\frac {\left (C b +D a -\frac {D \left (6 a d +6 b c \right )}{7 d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (B b +C a -\frac {5 D a c}{7 d}-\frac {\left (C b +D a -\frac {D \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (A a -\frac {\left (B b +C a -\frac {5 D a c}{7 d}-\frac {\left (C b +D a -\frac {D \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (A b +B a -\frac {3 \left (C b +D a -\frac {D \left (6 a d +6 b c \right )}{7 d}\right ) a c}{5 b d}-\frac {\left (B b +C a -\frac {5 D a c}{7 d}-\frac {\left (C b +D a -\frac {D \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(602\) |
| default | \(\text {Expression too large to display}\) | \(1803\) |
Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x,method=_RETURN VERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/7*D/d*x^5*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(C*b+D*a-1/7*D/d*(6*a*d+6*b*c))/b/d *x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(B*b+C*a-5/7*D/d*a*c-1/5*(C*b +D*a-1/7*D/d*(6*a*d+6*b*c))/b/d*(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*a-1/3*(B*b+C*a-5/7*D/d*a*c-1/5*(C*b+D*a-1/7*D/d*(6*a*d+6 *b*c))/b/d*(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*b+B*a-3/5*(C*b+D*a-1/7*D/d*(6*a*d+6*b*c))/b/d*a* c-1/3*(B*b+C*a-5/7*D/d*a*c-1/5*(C*b+D*a-1/7*D/d*(6*a*d+6*b*c))/b/d*(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 = 0.09 (sec) , antiderivative size = 563, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\frac {{\left (48 \, D b^{3} c^{5} - 8 \, {\left (2 \, D a b^{2} + 7 \, C b^{3}\right )} c^{4} d - {\left (9 \, D a^{2} b - 21 \, C a b^{2} - 70 \, B b^{3}\right )} c^{3} d^{2} - {\left (8 \, D a^{3} - 14 \, C a^{2} b + 35 \, B a b^{2} + 105 \, A b^{3}\right )} c^{2} d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (48 \, D b^{3} c^{5} - 105 \, A a b^{2} d^{5} - 8 \, {\left (2 \, D a b^{2} + 7 \, C b^{3}\right )} c^{4} d - {\left (9 \, D a^{2} b - 3 \, {\left (7 \, C + 8 \, D\right )} a b^{2} - 70 \, B b^{3}\right )} c^{3} d^{2} - {\left (8 \, D a^{3} - {\left (14 \, C - 5 \, D\right )} a^{2} b + 7 \, {\left (5 \, B + 4 \, C\right )} a b^{2} + 105 \, A b^{3}\right )} c^{2} d^{3} - {\left (4 \, D a^{3} - 7 \, C a^{2} b - 35 \, B a b^{2}\right )} c d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (15 \, D b^{3} c d^{4} x^{6} - 48 \, D b^{3} c^{4} d + 8 \, {\left (2 \, D a b^{2} + 7 \, C b^{3}\right )} c^{3} d^{2} + {\left (9 \, D a^{2} b - 21 \, C a b^{2} - 70 \, B b^{3}\right )} c^{2} d^{3} + {\left (8 \, D a^{3} - 14 \, C a^{2} b + 35 \, B a b^{2} + 105 \, A b^{3}\right )} c d^{4} - 3 \, {\left (6 \, D b^{3} c^{2} d^{3} - {\left (D a b^{2} + 7 \, C b^{3}\right )} c d^{4}\right )} x^{4} + {\left (24 \, D b^{3} c^{3} d^{2} - {\left (5 \, D a b^{2} + 28 \, C b^{3}\right )} c^{2} d^{3} - {\left (4 \, D a^{2} b - 7 \, C a b^{2} - 35 \, B b^{3}\right )} c d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{3} c d^{5} x} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x, algorit hm="fricas")
Output:
1/105*((48*D*b^3*c^5 - 8*(2*D*a*b^2 + 7*C*b^3)*c^4*d - (9*D*a^2*b - 21*C*a *b^2 - 70*B*b^3)*c^3*d^2 - (8*D*a^3 - 14*C*a^2*b + 35*B*a*b^2 + 105*A*b^3) *c^2*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c )) - (48*D*b^3*c^5 - 105*A*a*b^2*d^5 - 8*(2*D*a*b^2 + 7*C*b^3)*c^4*d - (9* D*a^2*b - 3*(7*C + 8*D)*a*b^2 - 70*B*b^3)*c^3*d^2 - (8*D*a^3 - (14*C - 5*D )*a^2*b + 7*(5*B + 4*C)*a*b^2 + 105*A*b^3)*c^2*d^3 - (4*D*a^3 - 7*C*a^2*b - 35*B*a*b^2)*c*d^4)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x ), a*d/(b*c)) + (15*D*b^3*c*d^4*x^6 - 48*D*b^3*c^4*d + 8*(2*D*a*b^2 + 7*C* b^3)*c^3*d^2 + (9*D*a^2*b - 21*C*a*b^2 - 70*B*b^3)*c^2*d^3 + (8*D*a^3 - 14 *C*a^2*b + 35*B*a*b^2 + 105*A*b^3)*c*d^4 - 3*(6*D*b^3*c^2*d^3 - (D*a*b^2 + 7*C*b^3)*c*d^4)*x^4 + (24*D*b^3*c^3*d^2 - (5*D*a*b^2 + 28*C*b^3)*c^2*d^3 - (4*D*a^2*b - 7*C*a*b^2 - 35*B*b^3)*c*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^ 2 + c))/(b^3*c*d^5*x)
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (A + B x^{2} + C x^{4} + D x^{6}\right )}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(D*x**6+C*x**4+B*x**2+A)/(d*x**2+c)**(1/2),x)
Output:
Integral(sqrt(a + b*x**2)*(A + B*x**2 + C*x**4 + D*x**6)/sqrt(c + d*x**2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x, algorit hm="maxima")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x, algorit hm="giac")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x +2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}+35 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} d x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}+15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}+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^{3} d^{3}-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^{2} b c \,d^{2}+140 \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^{3} 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} c^{2} d -70 \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^{4} c d +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 ) b^{3} 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^{3} c \,d^{2}+105 \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^{2} d^{2}-2 \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 -35 \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^{3} c d +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 \,b^{2} c^{3}}{105 b^{2} d^{2}} \] Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*x + 2*sqrt(c + d*x**2)*s qrt(a + b*x**2)*a*b*c*d*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*x **3 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d*x - 4*sqrt(c + d*x**2)*s qrt(a + b*x**2)*b**2*c**2*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d *x**3 + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*x**5 + 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**3*d**3 - 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**2*b*c*d**2 + 140*int((sqrt(c + d*x**2)*sq rt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**3*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*c**2*d - 70*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**4*c*d + 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 )*b**3*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**3*c*d**2 + 105*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**2*d**2 - 2*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 - 35*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**3*c*d + 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*b**2*c**3)/(105*...