Integrand size = 40, antiderivative size = 600 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=-\frac {\left (48 a^3 d^3 D-8 a^2 b d^2 (7 C d+2 c D)+a b^2 d \left (21 c C d+70 B d^2-9 c^2 D\right )+b^3 \left (14 c^2 C d-35 B c d^2-105 A d^3-8 c^3 D\right )\right ) x \sqrt {c+d x^2}}{105 b^3 d^3 \sqrt {a+b x^2}}+\frac {\left (24 a^2 d^2 D-a b d (28 C d+5 c D)+b^2 \left (7 c C d+35 B d^2-4 c^2 D\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b^3 d^2}+\frac {(7 b C d+b c D-6 a d D) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b^2 d}+\frac {D x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b}+\frac {\sqrt {a} \left (48 a^3 d^3 D-8 a^2 b d^2 (7 C d+2 c D)+a b^2 d \left (21 c C d+70 B d^2-9 c^2 D\right )+b^3 \left (14 c^2 C d-35 B c d^2-105 A d^3-8 c^3 D\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{7/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (105 A b^3 d^2-24 a^3 d^2 D+a^2 b d (28 C d+5 c D)-a b^2 \left (7 c C d+35 B d^2-4 c^2 D\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{7/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/105*(48*a^3*d^3*D-8*a^2*b*d^2*(7*C*d+2*D*c)+a*b^2*d*(70*B*d^2+21*C*c*d- 9*D*c^2)+b^3*(-105*A*d^3-35*B*c*d^2+14*C*c^2*d-8*D*c^3))*x*(d*x^2+c)^(1/2) /b^3/d^3/(b*x^2+a)^(1/2)+1/105*(24*a^2*d^2*D-a*b*d*(28*C*d+5*D*c)+b^2*(35* B*d^2+7*C*c*d-4*D*c^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^3/d^2+1/35*(7* C*b*d-6*D*a*d+D*b*c)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d+1/7*D*x^5*( b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b+1/105*a^(1/2)*(48*a^3*d^3*D-8*a^2*b*d^2*( 7*C*d+2*D*c)+a*b^2*d*(70*B*d^2+21*C*c*d-9*D*c^2)+b^3*(-105*A*d^3-35*B*c*d^ 2+14*C*c^2*d-8*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^(7/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/( b*x^2+a))^(1/2)+1/105*a^(1/2)*(105*A*b^3*d^2-24*a^3*d^2*D+a^2*b*d*(28*C*d+ 5*D*c)-a*b^2*(35*B*d^2+7*C*c*d-4*D*c^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(a rctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(7/2)/d^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 = 4.66 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (24 a^2 d^2 D-a b d \left (28 C d+5 c D+18 d D x^2\right )+b^2 \left (-4 c^2 D+c d \left (7 C+3 D x^2\right )+d^2 \left (35 B+21 C x^2+15 D x^4\right )\right )\right )+i c \left (48 a^3 d^3 D-8 a^2 b d^2 (7 C d+2 c D)+a b^2 d \left (21 c C d+70 B d^2-9 c^2 D\right )-b^3 \left (-14 c^2 C d+35 B c d^2+105 A d^3+8 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (24 a^2 d^2 D+a b d (-28 C d+13 c D)+b^2 \left (-14 c C d+35 B d^2+8 c^2 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 b^3 \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[c + d*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[a + b*x^2],x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(24*a^2*d^2*D - a*b*d*(28*C*d + 5*c *D + 18*d*D*x^2) + b^2*(-4*c^2*D + c*d*(7*C + 3*D*x^2) + d^2*(35*B + 21*C* x^2 + 15*D*x^4))) + I*c*(48*a^3*d^3*D - 8*a^2*b*d^2*(7*C*d + 2*c*D) + a*b^ 2*d*(21*c*C*d + 70*B*d^2 - 9*c^2*D) - b^3*(-14*c^2*C*d + 35*B*c*d^2 + 105* A*d^3 + 8*c^3*D))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcS inh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*(24*a^2*d^2*D + a*b*d* (-28*C*d + 13*c*D) + b^2*(-14*c*C*d + 35*B*d^2 + 8*c^2*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*b^3*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(1259\) vs. \(2(600)=1200\).
Time = 2.01 (sec) , antiderivative size = 1259, normalized size of antiderivative = 2.10, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \sqrt {c+d x^2}}{\sqrt {a+b x^2}}+\frac {B x^2 \sqrt {c+d x^2}}{\sqrt {a+b x^2}}+\frac {C x^4 \sqrt {c+d x^2}}{\sqrt {a+b x^2}}+\frac {D x^6 \sqrt {c+d x^2}}{\sqrt {a+b x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {D \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 b}+\frac {C \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 b}+\frac {(b c-6 a d) D \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b^2 d}+\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 {\left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right ) D \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 b^3 d^2}+\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 {\left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 a^3 d^3\right ) D \sqrt {b x^2+a} x}{105 b^4 d^2 \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 {\sqrt {c} \left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 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^4 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 {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 {c^{3/2} \left (4 b^2 c^2+5 a b d c-24 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^3 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 {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 + D*x^6))/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]) + ((8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48*a^3*d^3)*D*x*Sqrt[a + b*x^2])/(105*b^4*d^2*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) - ((4*b^2*c^2 + 5*a*b*c*d - 24*a^2*d^2)*D*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*b^3*d^2) + (C*x^3*Sq rt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b) + ((b*c - 6*a*d)*D*x^3*Sqrt[a + b*x^2 ]*Sqrt[c + d*x^2])/(35*b^2*d) + (D*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7 *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]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d )])/(15*b^3*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48*a^3*d^3)*D*Sq rt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(10 5*b^4*d^(5/2)*Sqrt[(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...
Time = 6.96 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.00
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {D x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 b}+\frac {\left (C d +D c -\frac {D \left (6 a d +6 b c \right )}{7 b}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (B d +C c -\frac {5 D a c}{7 b}-\frac {\left (C d +D c -\frac {D \left (6 a d +6 b c \right )}{7 b}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (A c -\frac {\left (B d +C c -\frac {5 D a c}{7 b}-\frac {\left (C d +D c -\frac {D \left (6 a d +6 b c \right )}{7 b}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (A d +B c -\frac {3 \left (C d +D c -\frac {D \left (6 a d +6 b c \right )}{7 b}\right ) a c}{5 b d}-\frac {\left (B d +C c -\frac {5 D a c}{7 b}-\frac {\left (C d +D c -\frac {D \left (6 a d +6 b c \right )}{7 b}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(602\) |
| default | \(\text {Expression too large to display}\) | \(1687\) |
Input:
int((d*x^2+c)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(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/b*x^5*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(C*d+D*c-1/7*D/b*(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*d+C*c-5/7*D/b*a*c-1/5*(C*d +D*c-1/7*D/b*(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*c-1/3*(B*d+C*c-5/7*D/b*a*c-1/5*(C*d+D*c-1/7*D/b*(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*d+B*c-3/5*(C*d+D*c-1/7*D/b*(6*a*d+6*b*c))/b/d*a* c-1/3*(B*d+C*c-5/7*D/b*a*c-1/5*(C*d+D*c-1/7*D/b*(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 = 0.10 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=-\frac {{\left (8 \, D b^{3} c^{4} + {\left (9 \, D a b^{2} - 14 \, C b^{3}\right )} c^{3} d + {\left (16 \, D a^{2} b - 21 \, C a b^{2} + 35 \, B b^{3}\right )} c^{2} d^{2} - {\left (48 \, D a^{3} - 56 \, C a^{2} b + 70 \, B a b^{2} - 105 \, A b^{3}\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, D b^{3} c^{4} + {\left (9 \, D a b^{2} - 14 \, C b^{3}\right )} c^{3} d + {\left (16 \, D a^{2} b - {\left (21 \, C - 4 \, D\right )} a b^{2} + 35 \, B b^{3}\right )} c^{2} d^{2} - {\left (48 \, D a^{3} - {\left (56 \, C + 5 \, D\right )} a^{2} b + 7 \, {\left (10 \, B + C\right )} a b^{2} - 105 \, A b^{3}\right )} c d^{3} - {\left (24 \, D a^{3} - 28 \, C a^{2} b + 35 \, B a b^{2} - 105 \, A b^{3}\right )} d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, D b^{3} d^{4} x^{6} + 8 \, D b^{3} c^{3} d + {\left (9 \, D a b^{2} - 14 \, C b^{3}\right )} c^{2} d^{2} + {\left (16 \, D a^{2} b - 21 \, C a b^{2} + 35 \, B b^{3}\right )} c d^{3} - {\left (48 \, D a^{3} - 56 \, C a^{2} b + 70 \, B a b^{2} - 105 \, A b^{3}\right )} d^{4} + 3 \, {\left (D b^{3} c d^{3} - {\left (6 \, D a b^{2} - 7 \, C b^{3}\right )} d^{4}\right )} x^{4} - {\left (4 \, D b^{3} c^{2} d^{2} + {\left (5 \, D a b^{2} - 7 \, C b^{3}\right )} c d^{3} - {\left (24 \, D a^{2} b - 28 \, C a b^{2} + 35 \, B b^{3}\right )} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{4} d^{4} x} \] Input:
integrate((d*x^2+c)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorit hm="fricas")
Output:
-1/105*((8*D*b^3*c^4 + (9*D*a*b^2 - 14*C*b^3)*c^3*d + (16*D*a^2*b - 21*C*a *b^2 + 35*B*b^3)*c^2*d^2 - (48*D*a^3 - 56*C*a^2*b + 70*B*a*b^2 - 105*A*b^3 )*c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c) ) - (8*D*b^3*c^4 + (9*D*a*b^2 - 14*C*b^3)*c^3*d + (16*D*a^2*b - (21*C - 4* D)*a*b^2 + 35*B*b^3)*c^2*d^2 - (48*D*a^3 - (56*C + 5*D)*a^2*b + 7*(10*B + C)*a*b^2 - 105*A*b^3)*c*d^3 - (24*D*a^3 - 28*C*a^2*b + 35*B*a*b^2 - 105*A* b^3)*d^4)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c )) - (15*D*b^3*d^4*x^6 + 8*D*b^3*c^3*d + (9*D*a*b^2 - 14*C*b^3)*c^2*d^2 + (16*D*a^2*b - 21*C*a*b^2 + 35*B*b^3)*c*d^3 - (48*D*a^3 - 56*C*a^2*b + 70*B *a*b^2 - 105*A*b^3)*d^4 + 3*(D*b^3*c*d^3 - (6*D*a*b^2 - 7*C*b^3)*d^4)*x^4 - (4*D*b^3*c^2*d^2 + (5*D*a*b^2 - 7*C*b^3)*c*d^3 - (24*D*a^2*b - 28*C*a*b^ 2 + 35*B*b^3)*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^4*d^4*x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {c + d x^{2}} \left (A + B x^{2} + C x^{4} + D x^{6}\right )}{\sqrt {a + b x^{2}}}\, dx \] Input:
integrate((d*x**2+c)**(1/2)*(D*x**6+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 + D*x**6)/sqrt(a + b*x**2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorit hm="maxima")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(d*x^2 + c)/sqrt(b*x^2 + a), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorit hm="giac")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(d*x^2 + c)/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {d\,x^2+c}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(1/2),x)
Output:
int(((c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(1/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {24 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x -33 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x -18 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}+35 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} d x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +24 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}+15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}-48 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} d^{3}+72 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+35 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{3} d^{2}-12 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{2} d +35 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{4} c d -6 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c^{3}-24 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} c \,d^{2}+33 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b \,c^{2} d +70 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{3} c d -3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{3}}{105 b^{3} d} \] Input:
int((d*x^2+c)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x)
Output:
(24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*x - 33*sqrt(c + d*x**2)*sq rt(a + b*x**2)*a*b*c*d*x - 18*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 + 3*sqrt(c + d*x**2)*s qrt(a + b*x**2)*b**2*c**2*x + 24*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 - 48*int((sqr t(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 + 72*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 + 35*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**3*d* *2 - 12*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 + 35*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 - 6*int((sqr t(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 - 24*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 + 33*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 + 70*in t((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 - 3*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*b**3*d)