Integrand size = 40, antiderivative size = 624 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {\left (6 a^3 d^3 D-3 a^2 b d^2 (7 C d-10 c D)+a b^2 d \left (189 c C d-140 B d^2-228 c^2 D\right )-b^3 \left (168 c^2 C d-140 B c d^2+105 A d^3-192 c^3 D\right )\right ) x \sqrt {a+b x^2}}{105 b^2 d^4 \sqrt {c+d x^2}}+\frac {\left (3 a^2 d^2 D+3 a b d (14 C d-17 c D)-b^2 \left (42 c C d-35 B d^2-48 c^2 D\right )\right ) x^3 \sqrt {a+b x^2}}{105 b d^3 \sqrt {c+d x^2}}+\frac {(7 b C d-8 b c D+8 a d D) x^5 \sqrt {a+b x^2}}{35 d^2 \sqrt {c+d x^2}}+\frac {b D x^7 \sqrt {a+b x^2}}{7 d \sqrt {c+d x^2}}+\frac {\left (6 a^3 c d^3 D-3 a^2 b c d^2 (7 C d-11 c D)+a b^2 d \left (336 c^2 C d-245 B c d^2+105 A d^3-408 c^3 D\right )-2 b^3 c \left (168 c^2 C d-140 B c d^2+105 A d^3-192 c^3 D\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b^2 \sqrt {c} d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \left (3 a^2 c d^2 D+3 a b d \left (49 c C d-35 B d^2-60 c^2 D\right )-b^2 \left (168 c^2 C d-140 B c d^2+105 A d^3-192 c^3 D\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 b d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-1/105*(6*a^3*d^3*D-3*a^2*b*d^2*(7*C*d-10*D*c)+a*b^2*d*(-140*B*d^2+189*C*c *d-228*D*c^2)-b^3*(105*A*d^3-140*B*c*d^2+168*C*c^2*d-192*D*c^3))*x*(b*x^2+ a)^(1/2)/b^2/d^4/(d*x^2+c)^(1/2)+1/105*(3*a^2*d^2*D+3*a*b*d*(14*C*d-17*D*c )-b^2*(-35*B*d^2+42*C*c*d-48*D*c^2))*x^3*(b*x^2+a)^(1/2)/b/d^3/(d*x^2+c)^( 1/2)+1/35*(7*C*b*d+8*D*a*d-8*D*b*c)*x^5*(b*x^2+a)^(1/2)/d^2/(d*x^2+c)^(1/2 )+1/7*b*D*x^7*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(1/2)+1/105*(6*a^3*c*d^3*D-3*a^2 *b*c*d^2*(7*C*d-11*D*c)+a*b^2*d*(105*A*d^3-245*B*c*d^2+336*C*c^2*d-408*D*c ^3)-2*b^3*c*(105*A*d^3-140*B*c*d^2+168*C*c^2*d-192*D*c^3))*(b*x^2+a)^(1/2) *EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/b^2/c^(1 /2)/d^(9/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/105*c^(1/2)* (3*a^2*c*d^2*D+3*a*b*d*(-35*B*d^2+49*C*c*d-60*D*c^2)-b^2*(105*A*d^3-140*B* c*d^2+168*C*c^2*d-192*D*c^3))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/ 2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/b/d^(9/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) /(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 7.97 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 a^2 c d^2 D \left (c+d x^2\right )+3 a b d \left (35 A d^3-60 c^3 D+c^2 d \left (49 C-17 D x^2\right )+c d^2 \left (-35 B+14 C x^2+8 D x^4\right )\right )+b^2 c \left (192 c^3 D-24 c^2 d \left (7 C-2 D x^2\right )+d^3 \left (-105 A+35 B x^2+21 C x^4+15 D x^6\right )+2 c d^2 \left (70 B-3 \left (7 C x^2+4 D x^4\right )\right )\right )\right )+i c \left (6 a^3 c d^3 D+3 a^2 b c d^2 (-7 C d+11 c D)+a b^2 d \left (336 c^2 C d-245 B c d^2+105 A d^3-408 c^3 D\right )+2 b^3 c \left (-168 c^2 C d+140 B c d^2-105 A d^3+192 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (3 a^2 c d^2 D+3 a b d \left (-56 c C d+35 B d^2+72 c^2 D\right )+b^2 \left (336 c^2 C d-280 B c d^2+210 A d^3-384 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 b \sqrt {\frac {b}{a}} c d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + D*x^6))/(c + d*x^2)^(3/2 ),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(3*a^2*c*d^2*D*(c + d*x^2) + 3*a*b*d*(35*A*d^3 - 60*c^3*D + c^2*d*(49*C - 17*D*x^2) + c*d^2*(-35*B + 14*C*x^2 + 8*D*x^4)) + b^2*c*(192*c^3*D - 24*c^2*d*(7*C - 2*D*x^2) + d^3*(-105*A + 35*B*x^2 + 21*C*x^4 + 15*D*x^6) + 2*c*d^2*(70*B - 3*(7*C*x^2 + 4*D*x^4)))) + I*c*(6*a ^3*c*d^3*D + 3*a^2*b*c*d^2*(-7*C*d + 11*c*D) + a*b^2*d*(336*c^2*C*d - 245* B*c*d^2 + 105*A*d^3 - 408*c^3*D) + 2*b^3*c*(-168*c^2*C*d + 140*B*c*d^2 - 1 05*A*d^3 + 192*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*c*(-(b*c) + a*d)*(3*a^2*c*d^2*D + 3*a*b*d*(-56*c*C*d + 35*B*d^2 + 72*c^2*D) + b^2*(336*c^2*C*d - 280*B*c*d^2 + 210*A*d^3 - 384*c^3*D))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti cF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*b*Sqrt[b/a]*c*d^5*Sqrt[a + b *x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(1401\) vs. \(2(624)=1248\).
Time = 2.55 (sec) , antiderivative size = 1401, normalized size of antiderivative = 2.25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}+\frac {B x^2 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}+\frac {C x^4 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}+\frac {D x^6 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 b D \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d^2}-\frac {D \left (b x^2+a\right )^{3/2} x^5}{d \sqrt {d x^2+c}}-\frac {(48 b c-43 a d) D \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 d^3}+\frac {6 b C \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d^2}-\frac {C \left (b x^2+a\right )^{3/2} x^3}{d \sqrt {d x^2+c}}-\frac {C (8 b c-7 a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{5 d^3}+\frac {\left (64 b^2 c^2-60 a b d c+a^2 d^2\right ) D \sqrt {b x^2+a} \sqrt {d x^2+c} x}{35 b d^4}+\frac {4 b B \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 d^2}-\frac {B \left (b x^2+a\right )^{3/2} x}{d \sqrt {d x^2+c}}-\frac {B (8 b c-7 a d) \sqrt {b x^2+a} x}{3 d^2 \sqrt {d x^2+c}}-\frac {A (b c-a d) \sqrt {b x^2+a} x}{c d \sqrt {d x^2+c}}+\frac {A (2 b c-a d) \sqrt {b x^2+a} x}{c d \sqrt {d x^2+c}}-\frac {C \left (-\frac {d a^2}{b}+16 c a-\frac {16 b c^2}{d}\right ) \sqrt {b x^2+a} x}{5 d^2 \sqrt {d x^2+c}}-\frac {\left (128 b^3 c^3-136 a b^2 d c^2+11 a^2 b d^2 c+2 a^3 d^3\right ) D \sqrt {b x^2+a} x}{35 b^2 d^4 \sqrt {d x^2+c}}+\frac {B \sqrt {c} (8 b c-7 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 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 (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 )}{\sqrt {c} 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 (16 b^2 c^2-16 a b d c+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 )}{5 b 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 {\sqrt {c} \left (128 b^3 c^3-136 a b^2 d c^2+11 a^2 b d^2 c+2 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 )}{35 b^2 d^{9/2} \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} C (8 b c-7 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 )}{5 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 {B \sqrt {c} (4 b c-3 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 )}{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 {c^{3/2} \left (64 b^2 c^2-60 a b d c+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 )}{35 b d^{9/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A b \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 )}{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[((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + D*x^6))/(c + d*x^2)^(3/2),x]
Output:
-1/3*(B*(8*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(d^2*Sqrt[c + d*x^2]) - (A*(b*c - a*d)*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) + (A*(2*b*c - a*d)*x*Sqrt [a + b*x^2])/(c*d*Sqrt[c + d*x^2]) - (C*(16*a*c - (16*b*c^2)/d - (a^2*d)/b )*x*Sqrt[a + b*x^2])/(5*d^2*Sqrt[c + d*x^2]) - ((128*b^3*c^3 - 136*a*b^2*c ^2*d + 11*a^2*b*c*d^2 + 2*a^3*d^3)*D*x*Sqrt[a + b*x^2])/(35*b^2*d^4*Sqrt[c + d*x^2]) - (B*x*(a + b*x^2)^(3/2))/(d*Sqrt[c + d*x^2]) - (C*x^3*(a + b*x ^2)^(3/2))/(d*Sqrt[c + d*x^2]) - (D*x^5*(a + b*x^2)^(3/2))/(d*Sqrt[c + d*x ^2]) + (4*b*B*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d^2) - (C*(8*b*c - 7*a *d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d^3) + ((64*b^2*c^2 - 60*a*b*c*d + a^2*d^2)*D*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b*d^4) + (6*b*C*x^3*S qrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d^2) - ((48*b*c - 43*a*d)*D*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*d^3) + (8*b*D*x^5*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(7*d^2) + (B*Sqrt[c]*(8*b*c - 7*a*d)*Sqrt[a + b*x^2]*EllipticE[ArcT an[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(5/2)*Sqrt[(c*(a + b*x^2)) /(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (A*(2*b*c - a*d)*Sqrt[a + b*x^2]*Elli pticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*d^(3/2)*Sqrt [(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*C*(16*b^2*c^ 2 - 16*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqr t[c]], 1 - (b*c)/(a*d)])/(5*b*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2)) ]*Sqrt[c + d*x^2]) + (Sqrt[c]*(128*b^3*c^3 - 136*a*b^2*c^2*d + 11*a^2*b...
Leaf count of result is larger than twice the leaf count of optimal. \(1329\) vs. \(2(583)=1166\).
Time = 9.53 (sec) , antiderivative size = 1330, normalized size of antiderivative = 2.13
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1330\) |
| default | \(\text {Expression too large to display}\) | \(2092\) |
Input:
int((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(3/2),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)*((b*d*x^2+a*d) *(A*a*d^4-A*b*c*d^3-B*a*c*d^3+B*b*c^2*d^2+C*a*c^2*d^2-C*b*c^3*d-D*a*c^3*d+ D*b*c^4)/c/d^5*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+1/7*b/d^2*D*x^5*(b*d*x^4+ a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(b/d^2*(C*b*d+2*D*a*d-D*b*c)-1/7*b/d^2*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*(1/d^3*(B*b^2 *d^2+2*C*a*b*d^2-C*b^2*c*d+D*a^2*d^2-2*D*a*b*c*d+D*b^2*c^2)-5/7*b/d^2*D*a* c-1/5*(b/d^2*(C*b*d+2*D*a*d-D*b*c)-1/7*b/d^2*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)+((2*A*a*b*d^4-A*b^2*c*d^3 +B*a^2*d^4-2*B*a*b*c*d^3+B*b^2*c^2*d^2-C*a^2*c*d^3+2*C*a*b*c^2*d^2-C*b^2*c ^3*d+D*a^2*c^2*d^2-2*D*a*b*c^3*d+D*b^2*c^4)/d^5+(A*a*d^4-A*b*c*d^3-B*a*c*d ^3+B*b*c^2*d^2+C*a*c^2*d^2-C*b*c^3*d-D*a*c^3*d+D*b*c^4)/d^5*(a*d-b*c)/c-a/ d^4*(A*a*d^4-A*b*c*d^3-B*a*c*d^3+B*b*c^2*d^2+C*a*c^2*d^2-C*b*c^3*d-D*a*c^3 *d+D*b*c^4)/c-1/3*(1/d^3*(B*b^2*d^2+2*C*a*b*d^2-C*b^2*c*d+D*a^2*d^2-2*D*a* b*c*d+D*b^2*c^2)-5/7*b/d^2*D*a*c-1/5*(b/d^2*(C*b*d+2*D*a*d-D*b*c)-1/7*b/d^ 2*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))-(1/d^4*(A*b^2*d^3+2*B*a*b*d^3-B*b^2*c*d ^2+C*a^2*d^3-2*C*a*b*c*d^2+C*b^2*c^2*d-D*a^2*c*d^2+2*D*a*b*c^2*d-D*b^2*c^3 )-(A*a*d^4-A*b*c*d^3-B*a*c*d^3+B*b*c^2*d^2+C*a*c^2*d^2-C*b*c^3*d-D*a*c^3*d +D*b*c^4)/d^4*b/c-3/5*(b/d^2*(C*b*d+2*D*a*d-D*b*c)-1/7*b/d^2*D*(6*a*d+6...
Time = 0.11 (sec) , antiderivative size = 997, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(3/2),x, algorit hm="fricas")
Output:
1/105*(((384*D*b^3*c^5*d + 105*A*a*b^2*c*d^5 - 24*(17*D*a*b^2 + 14*C*b^3)* c^4*d^2 + (33*D*a^2*b + 336*C*a*b^2 + 280*B*b^3)*c^3*d^3 + (6*D*a^3 - 21*C *a^2*b - 245*B*a*b^2 - 210*A*b^3)*c^2*d^4)*x^3 + (384*D*b^3*c^6 + 105*A*a* b^2*c^2*d^4 - 24*(17*D*a*b^2 + 14*C*b^3)*c^5*d + (33*D*a^2*b + 336*C*a*b^2 + 280*B*b^3)*c^4*d^2 + (6*D*a^3 - 21*C*a^2*b - 245*B*a*b^2 - 210*A*b^3)*c ^3*d^3)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c) ) - ((384*D*b^3*c^5*d - 24*(17*D*a*b^2 + 14*C*b^3)*c^4*d^2 + (33*D*a^2*b + 48*(7*C + 4*D)*a*b^2 + 280*B*b^3)*c^3*d^3 + (6*D*a^3 - 3*(7*C + 60*D)*a^2 *b - 7*(35*B + 24*C)*a*b^2 - 210*A*b^3)*c^2*d^4 + (3*D*a^3 + 147*C*a^2*b + 35*(3*A + 4*B)*a*b^2)*c*d^5 - 105*(B*a^2*b + A*a*b^2)*d^6)*x^3 + (384*D*b ^3*c^6 - 24*(17*D*a*b^2 + 14*C*b^3)*c^5*d + (33*D*a^2*b + 48*(7*C + 4*D)*a *b^2 + 280*B*b^3)*c^4*d^2 + (6*D*a^3 - 3*(7*C + 60*D)*a^2*b - 7*(35*B + 24 *C)*a*b^2 - 210*A*b^3)*c^3*d^3 + (3*D*a^3 + 147*C*a^2*b + 35*(3*A + 4*B)*a *b^2)*c^2*d^4 - 105*(B*a^2*b + A*a*b^2)*c*d^5)*x)*sqrt(b*d)*sqrt(-c/d)*ell iptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (15*D*b^3*c*d^5*x^8 - 384*D*b^3 *c^5*d - 105*A*a*b^2*c*d^5 + 24*(17*D*a*b^2 + 14*C*b^3)*c^4*d^2 - (33*D*a^ 2*b + 336*C*a*b^2 + 280*B*b^3)*c^3*d^3 - (6*D*a^3 - 21*C*a^2*b - 245*B*a*b ^2 - 210*A*b^3)*c^2*d^4 - 3*(8*D*b^3*c^2*d^4 - (8*D*a*b^2 + 7*C*b^3)*c*d^5 )*x^6 + (48*D*b^3*c^3*d^3 - 3*(17*D*a*b^2 + 14*C*b^3)*c^2*d^4 + (3*D*a^2*b + 42*C*a*b^2 + 35*B*b^3)*c*d^5)*x^4 - (192*D*b^3*c^4*d^2 - 12*(19*D*a*...
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (A + B x^{2} + C x^{4} + D x^{6}\right )}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(D*x**6+C*x**4+B*x**2+A)/(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(A + B*x**2 + C*x**4 + D*x**6)/(c + d*x**2)** (3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(3/2),x, algorit hm="maxima")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*(b*x^2 + a)^(3/2)/(d*x^2 + c)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(3/2),x, algorit hm="giac")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*(b*x^2 + a)^(3/2)/(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(3/2),x)
Output:
int(((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(3/2),x)
Output:
( - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c*d**2*x + 315*sqrt(c + d*x** 2)*sqrt(a + b*x**2)*a**2*b**2*d**2*x + 27*sqrt(c + d*x**2)*sqrt(a + b*x**2 )*a**2*b*c**2*d*x + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c*d**2*x**3 - 105*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d*x - 18*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*a*b**2*c**3*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a* b**2*c**2*d*x**3 + 48*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*x**5 + 70*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c*d*x**3 + 12*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b**3*c**3*x**3 - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b* *3*c**2*d*x**5 + 30*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*x**7 - 3 *int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d **2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*b*c**2*d**3 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a *d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*b*c*d**4*x* *2 - 315*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x* *2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b**3* c*d**3 - 315*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c* d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b **3*d**4*x**2 - 45*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)* a**2*b**2*c**3*d**2 - 45*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(...