Integrand size = 40, antiderivative size = 841 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\frac {\left (8 a^4 d^4 D-a^3 b d^3 (18 C d-11 c D)-9 a^2 b^2 d^2 \left (4 c C d-7 B d^2-3 c^2 D\right )+a b^3 d \left (216 c^2 C d-273 B c d^2+420 A d^3-184 c^3 D\right )-2 b^4 c \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) x \sqrt {c+d x^2}}{315 b^2 d^5 \sqrt {a+b x^2}}-\frac {\left (4 a^3 d^3 D-3 a^2 b d^2 (3 C d-2 c D)+3 a b^2 d \left (33 c C d-42 B d^2-28 c^2 D\right )-b^3 \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^2 d^4}+\frac {\left (3 a^2 d^2 D+a b d (72 C d-61 c D)-b^2 \left (54 c C d-63 B d^2-48 c^2 D\right )\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^3}+\frac {(9 b C d-8 b c D+10 a d D) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b D x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}}{9 d}-\frac {\sqrt {a} \left (8 a^4 d^4 D-a^3 b d^3 (18 C d-11 c D)-9 a^2 b^2 d^2 \left (4 c C d-7 B d^2-3 c^2 D\right )+a b^3 d \left (216 c^2 C d-273 B c d^2+420 A d^3-184 c^3 D\right )-2 b^4 c \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{5/2} d^5 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^3 c d^3 D-3 a^2 b c d^2 (3 C d-2 c D)-b^3 c \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )+3 a b^2 d \left (33 c^2 C d-42 B c d^2+105 A d^3-28 c^3 D\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{5/2} c d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/315*(8*a^4*d^4*D-a^3*b*d^3*(18*C*d-11*D*c)-9*a^2*b^2*d^2*(-7*B*d^2+4*C*c *d-3*D*c^2)+a*b^3*d*(420*A*d^3-273*B*c*d^2+216*C*c^2*d-184*D*c^3)-2*b^4*c* (105*A*d^3-84*B*c*d^2+72*C*c^2*d-64*D*c^3))*x*(d*x^2+c)^(1/2)/b^2/d^5/(b*x ^2+a)^(1/2)-1/315*(4*a^3*d^3*D-3*a^2*b*d^2*(3*C*d-2*D*c)+3*a*b^2*d*(-42*B* d^2+33*C*c*d-28*D*c^2)-b^3*(105*A*d^3-84*B*c*d^2+72*C*c^2*d-64*D*c^3))*x*( b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^4+1/315*(3*a^2*d^2*D+a*b*d*(72*C*d-61 *D*c)-b^2*(-63*B*d^2+54*C*c*d-48*D*c^2))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/ 2)/b/d^3+1/63*(9*C*b*d+10*D*a*d-8*D*b*c)*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/ 2)/d^2+1/9*b*D*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d-1/315*a^(1/2)*(8*a^4* d^4*D-a^3*b*d^3*(18*C*d-11*D*c)-9*a^2*b^2*d^2*(-7*B*d^2+4*C*c*d-3*D*c^2)+a *b^3*d*(420*A*d^3-273*B*c*d^2+216*C*c^2*d-184*D*c^3)-2*b^4*c*(105*A*d^3-84 *B*c*d^2+72*C*c^2*d-64*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^5/(b*x^2+a)^(1/2)/(a*(d*x^ 2+c)/c/(b*x^2+a))^(1/2)+1/315*a^(3/2)*(4*a^3*c*d^3*D-3*a^2*b*c*d^2*(3*C*d- 2*D*c)-b^3*c*(105*A*d^3-84*B*c*d^2+72*C*c^2*d-64*D*c^3)+3*a*b^2*d*(105*A*d ^3-42*B*c*d^2+33*C*c^2*d-28*D*c^3))*(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)/c/d^4/(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 = 9.82 (sec) , antiderivative size = 568, normalized size of antiderivative = 0.68 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-4 a^3 d^3 D+3 a^2 b d^2 \left (3 C d-2 c D+d D x^2\right )+a b^2 d \left (84 c^2 D-c d \left (99 C+61 D x^2\right )+2 d^2 \left (63 B+36 C x^2+25 D x^4\right )\right )+b^3 \left (-64 c^3 D+24 c^2 d \left (3 C+2 D x^2\right )-2 c d^2 \left (42 B+27 C x^2+20 D x^4\right )+d^3 \left (105 A+63 B x^2+45 C x^4+35 D x^6\right )\right )\right )-i c \left (8 a^4 d^4 D+a^3 b d^3 (-18 C d+11 c D)+9 a^2 b^2 d^2 \left (-4 c C d+7 B d^2+3 c^2 D\right )+a b^3 d \left (216 c^2 C d-273 B c d^2+420 A d^3-184 c^3 D\right )+2 b^4 c \left (-72 c^2 C d+84 B c d^2-105 A d^3+64 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) \left (4 a^3 c d^3 D+9 a^2 b c d^2 (-C d+c D)+3 a b^2 d \left (-48 c^2 C d+63 B c d^2-105 A d^3+40 c^3 D\right )-2 b^3 c \left (-72 c^2 C d+84 B c d^2-105 A d^3+64 c^3 D\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 a^2 \left (\frac {b}{a}\right )^{5/2} d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[c + d*x^2], x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*a^3*d^3*D + 3*a^2*b*d^2*(3*C*d - 2*c*D + d*D*x^2) + a*b^2*d*(84*c^2*D - c*d*(99*C + 61*D*x^2) + 2*d^2*(63 *B + 36*C*x^2 + 25*D*x^4)) + b^3*(-64*c^3*D + 24*c^2*d*(3*C + 2*D*x^2) - 2 *c*d^2*(42*B + 27*C*x^2 + 20*D*x^4) + d^3*(105*A + 63*B*x^2 + 45*C*x^4 + 3 5*D*x^6))) - I*c*(8*a^4*d^4*D + a^3*b*d^3*(-18*C*d + 11*c*D) + 9*a^2*b^2*d ^2*(-4*c*C*d + 7*B*d^2 + 3*c^2*D) + a*b^3*d*(216*c^2*C*d - 273*B*c*d^2 + 4 20*A*d^3 - 184*c^3*D) + 2*b^4*c*(-72*c^2*C*d + 84*B*c*d^2 - 105*A*d^3 + 64 *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^3*c*d^3*D + 9*a^2*b*c*d^2*(- (C*d) + c*D) + 3*a*b^2*d*(-48*c^2*C*d + 63*B*c*d^2 - 105*A*d^3 + 40*c^3*D) - 2*b^3*c*(-72*c^2*C*d + 84*B*c*d^2 - 105*A*d^3 + 64*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)] )/(315*a^2*(b/a)^(5/2)*d^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 2.84 (sec) , antiderivative size = 1598, normalized size of antiderivative = 1.90, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}+\frac {B x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}+\frac {C x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}+\frac {D x^6 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b D \sqrt {b x^2+a} \sqrt {d x^2+c} x^7}{9 d}-\frac {2 (4 b c-5 a d) D \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{63 d^2}+\frac {b C \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}-\frac {2 C (3 b c-4 a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 d^2}+\frac {\left (48 b^2 c^2-61 a b d c+3 a^2 d^2\right ) D \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{315 b d^3}+\frac {b B \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}-\frac {2 B (2 b c-3 a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 d^2}+\frac {C \left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{35 b d^3}-\frac {2 \left (32 b^3 c^3-42 a b^2 d c^2+3 a^2 b d^2 c+2 a^3 d^3\right ) D \sqrt {b x^2+a} \sqrt {d x^2+c} x}{315 b^2 d^4}+\frac {A b \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 d}-\frac {2 A (b c-2 a d) \sqrt {b x^2+a} x}{3 d \sqrt {d x^2+c}}-\frac {B \left (-\frac {3 d a^2}{b}+13 c a-\frac {8 b c^2}{d}\right ) \sqrt {b x^2+a} x}{15 d \sqrt {d x^2+c}}-\frac {2 C (2 b c-a d) \left (4 b^2 c^2-4 a b d c-a^2 d^2\right ) \sqrt {b x^2+a} x}{35 b^2 d^3 \sqrt {d x^2+c}}+\frac {\left (128 b^4 c^4-184 a b^3 d c^3+27 a^2 b^2 d^2 c^2+11 a^3 b d^3 c+8 a^4 d^4\right ) D \sqrt {b x^2+a} x}{315 b^3 d^4 \sqrt {d x^2+c}}+\frac {2 A \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 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 {2 \sqrt {c} C (2 b c-a d) \left (4 b^2 c^2-4 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 )}{35 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 {B \sqrt {c} \left (8 b^2 c^2-13 a b d c+3 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 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 (128 b^4 c^4-184 a b^3 d c^3+27 a^2 b^2 d^2 c^2+11 a^3 b d^3 c+8 a^4 d^4\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 )}{315 b^3 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 \sqrt {c} (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^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 B c^{3/2} (2 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 )}{15 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} C \left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) \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^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 c^{3/2} \left (32 b^3 c^3-42 a b^2 d c^2+3 a^2 b d^2 c+2 a^3 d^3\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 )}{315 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}}\) |
Input:
Int[((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[c + d*x^2],x]
Output:
(-2*A*(b*c - 2*a*d)*x*Sqrt[a + b*x^2])/(3*d*Sqrt[c + d*x^2]) - (B*(13*a*c - (8*b*c^2)/d - (3*a^2*d)/b)*x*Sqrt[a + b*x^2])/(15*d*Sqrt[c + d*x^2]) - ( 2*C*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*x*Sqrt[a + b*x^2])/(35 *b^2*d^3*Sqrt[c + d*x^2]) + ((128*b^4*c^4 - 184*a*b^3*c^3*d + 27*a^2*b^2*c ^2*d^2 + 11*a^3*b*c*d^3 + 8*a^4*d^4)*D*x*Sqrt[a + b*x^2])/(315*b^3*d^4*Sqr t[c + d*x^2]) + (A*b*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (2*B*(2*b* c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (C*(8*b^2*c^2 - 1 1*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b*d^3) - (2*(3 2*b^3*c^3 - 42*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3)*D*x*Sqrt[a + b*x^2 ]*Sqrt[c + d*x^2])/(315*b^2*d^4) + (b*B*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2 ])/(5*d) - (2*C*(3*b*c - 4*a*d)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*d ^2) + ((48*b^2*c^2 - 61*a*b*c*d + 3*a^2*d^2)*D*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*b*d^3) + (b*C*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) - (2*(4*b*c - 5*a*d)*D*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*d^2) + (b*D* x^7*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(9*d) + (2*A*Sqrt[c]*(b*c - 2*a*d)*Sq rt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3* d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c ]*C*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*Sqrt[a + b*x^2]*Ellipt icE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^2*d^(7/2)*Sqrt[(c *(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (B*Sqrt[c]*(8*b^2*c^2...
Time = 6.69 (sec) , antiderivative size = 1020, normalized size of antiderivative = 1.21
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1020\) |
| default | \(\text {Expression too large to display}\) | \(2764\) |
Input:
int((b*x^2+a)^(3/2)*(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)*(1/9*D*b/d*x^7 *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8*a*d+8 *b*c))/b/d*x^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(B*b^2+2*C*a*b+D*a^ 2-7/9*D*b/d*a*c-1/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8*a*d+8*b*c))/b/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^2*A+2*a*b*B+a^2*C- 5/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8*a*d+8*b*c))/b/d*a*c-1/5*(B*b^2+2*C*a*b+D*a ^2-7/9*D*b/d*a*c-1/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8*a*d+8*b*c))/b/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^2*A- 1/3*(b^2*A+2*a*b*B+a^2*C-5/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8*a*d+8*b*c))/b/d*a *c-1/5*(B*b^2+2*C*a*b+D*a^2-7/9*D*b/d*a*c-1/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8* a*d+8*b*c))/b/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)*Elli pticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(2*a*b*A+a^2*B-3/5*(B*b^2+2 *C*a*b+D*a^2-7/9*D*b/d*a*c-1/7*(b^2*C+2*a*b*D-1/9*D*b/d*(8*a*d+8*b*c))/b/d *(6*a*d+6*b*c))/b/d*a*c-1/3*(b^2*A+2*a*b*B+a^2*C-5/7*(b^2*C+2*a*b*D-1/9*D* b/d*(8*a*d+8*b*c))/b/d*a*c-1/5*(B*b^2+2*C*a*b+D*a^2-7/9*D*b/d*a*c-1/7*(b^2 *C+2*a*b*D-1/9*D*b/d*(8*a*d+8*b*c))/b/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 = 0.10 (sec) , antiderivative size = 822, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x, algorit hm="fricas")
Output:
-1/315*((128*D*b^4*c^6 - 8*(23*D*a*b^3 + 18*C*b^4)*c^5*d + 3*(9*D*a^2*b^2 + 72*C*a*b^3 + 56*B*b^4)*c^4*d^2 + (11*D*a^3*b - 36*C*a^2*b^2 - 273*B*a*b^ 3 - 210*A*b^4)*c^3*d^3 + (8*D*a^4 - 18*C*a^3*b + 63*B*a^2*b^2 + 420*A*a*b^ 3)*c^2*d^4)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b *c)) - (128*D*b^4*c^6 + 315*A*a^2*b^2*d^6 - 8*(23*D*a*b^3 + 18*C*b^4)*c^5* d + (27*D*a^2*b^2 + 8*(27*C + 8*D)*a*b^3 + 168*B*b^4)*c^4*d^2 + (11*D*a^3* b - 12*(3*C + 7*D)*a^2*b^2 - 3*(91*B + 24*C)*a*b^3 - 210*A*b^4)*c^3*d^3 + (8*D*a^4 - 6*(3*C - D)*a^3*b + 9*(7*B + 11*C)*a^2*b^2 + 84*(5*A + B)*a*b^3 )*c^2*d^4 + (4*D*a^4 - 9*C*a^3*b - 126*B*a^2*b^2 - 105*A*a*b^3)*c*d^5)*sqr t(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (35*D*b^ 4*c*d^5*x^8 + 128*D*b^4*c^5*d - 8*(23*D*a*b^3 + 18*C*b^4)*c^4*d^2 + 3*(9*D *a^2*b^2 + 72*C*a*b^3 + 56*B*b^4)*c^3*d^3 + (11*D*a^3*b - 36*C*a^2*b^2 - 2 73*B*a*b^3 - 210*A*b^4)*c^2*d^4 + (8*D*a^4 - 18*C*a^3*b + 63*B*a^2*b^2 + 4 20*A*a*b^3)*c*d^5 - 5*(8*D*b^4*c^2*d^4 - (10*D*a*b^3 + 9*C*b^4)*c*d^5)*x^6 + (48*D*b^4*c^3*d^3 - (61*D*a*b^3 + 54*C*b^4)*c^2*d^4 + 3*(D*a^2*b^2 + 24 *C*a*b^3 + 21*B*b^4)*c*d^5)*x^4 - (64*D*b^4*c^4*d^2 - 12*(7*D*a*b^3 + 6*C* b^4)*c^3*d^3 + 3*(2*D*a^2*b^2 + 33*C*a*b^3 + 28*B*b^4)*c^2*d^4 + (4*D*a^3* b - 9*C*a^2*b^2 - 126*B*a*b^3 - 105*A*b^4)*c*d^5)*x^2)*sqrt(b*x^2 + a)*sqr t(d*x^2 + c))/(b^3*c*d^6*x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (A + B x^{2} + C x^{4} + D x^{6}\right )}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(D*x**6+C*x**4+B*x**2+A)/(d*x**2+c)**(1/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(A + B*x**2 + C*x**4 + D*x**6)/sqrt(c + d*x** 2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(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)*(b*x^2 + a)^(3/2)/sqrt(d*x^2 + c), x )
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(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)*(b*x^2 + a)^(3/2)/sqrt(d*x^2 + c), x )
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(1/2),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*x + 3*sqrt(c + d*x**2)*s qrt(a + b*x**2)*a**2*b*c*d**2*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2 *b*d**3*x**3 + 231*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*d**2*x - 15*sq rt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*x + 11*sqrt(c + d*x**2)*sqrt (a + b*x**2)*a*b**2*c*d**2*x**3 + 50*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b **2*d**3*x**5 - 84*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c*d*x + 63*sqrt( c + d*x**2)*sqrt(a + b*x**2)*b**4*d**2*x**3 + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**2*d*x**3 + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*x**5 + 35*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*b**3*d**3*x**7 + 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**4*d**4 - 7*int((sq rt(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*b*c*d**3 + 483*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**3*d**3 - 9*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**2*c**2*d**2 - 483*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**4*c*d**2 + 32*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 *c**3*d + 168*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**5*c**2*d - 16*int((sqrt(c + d*x**2)*sqrt(...