Integrand size = 40, antiderivative size = 780 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\left (c^2 C d-B c d^2+A d^3+6 c^3 D\right ) x \sqrt {a+b x^2}}{7 c d^3 \left (c+d x^2\right )^{7/2}}-\frac {D x^5 \sqrt {a+b x^2}}{d \left (c+d x^2\right )^{7/2}}-\frac {\left (b c \left (9 c^2 C d-2 B c d^2-5 A d^3+54 c^3 D\right )-a d \left (8 c^2 C d-B c d^2-6 A d^3+55 c^3 D\right )\right ) x \sqrt {a+b x^2}}{35 c^2 d^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {\left (b^2 c^2 \left (8 c^2 C d+6 B c d^2+15 A d^3+48 c^3 D\right )+a^2 d^2 \left (3 c^2 C d+4 B c d^2+24 A d^3+60 c^3 D\right )-a b c d \left (15 c^2 C d+6 B c d^2+43 A d^3+104 c^3 D\right )\right ) x \sqrt {a+b x^2}}{105 c^3 d^3 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {\left (a^3 d^3 \left (6 c^2 C d+8 B c d^2+48 A d^3+15 c^3 D\right )-b^3 c^3 \left (8 c^2 C d+6 B c d^2+15 A d^3+48 c^3 D\right )-a^2 b c d^2 \left (9 c^2 C d+19 B c d^2+128 A d^3+103 c^3 D\right )+a b^2 c^2 d \left (19 c^2 C d+9 B c d^2+103 A d^3+128 c^3 D\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 c^{7/2} d^{7/2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \left (a^2 d^2 \left (3 c^2 C d+4 B c d^2+24 A d^3-45 c^3 D\right )-b^2 c^2 \left (4 c^2 C d+3 B c d^2-45 A d^3+24 c^3 D\right )+a b c d \left (9 c^2 C d-9 B c d^2-61 A d^3+61 c^3 D\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 c^{5/2} d^{7/2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/7*(A*d^3-B*c*d^2+C*c^2*d+6*D*c^3)*x*(b*x^2+a)^(1/2)/c/d^3/(d*x^2+c)^(7/2 )-D*x^5*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(7/2)-1/35*(b*c*(-5*A*d^3-2*B*c*d^2+9* C*c^2*d+54*D*c^3)-a*d*(-6*A*d^3-B*c*d^2+8*C*c^2*d+55*D*c^3))*x*(b*x^2+a)^( 1/2)/c^2/d^3/(-a*d+b*c)/(d*x^2+c)^(5/2)+1/105*(b^2*c^2*(15*A*d^3+6*B*c*d^2 +8*C*c^2*d+48*D*c^3)+a^2*d^2*(24*A*d^3+4*B*c*d^2+3*C*c^2*d+60*D*c^3)-a*b*c *d*(43*A*d^3+6*B*c*d^2+15*C*c^2*d+104*D*c^3))*x*(b*x^2+a)^(1/2)/c^3/d^3/(- a*d+b*c)^2/(d*x^2+c)^(3/2)-1/105*(a^3*d^3*(48*A*d^3+8*B*c*d^2+6*C*c^2*d+15 *D*c^3)-b^3*c^3*(15*A*d^3+6*B*c*d^2+8*C*c^2*d+48*D*c^3)-a^2*b*c*d^2*(128*A *d^3+19*B*c*d^2+9*C*c^2*d+103*D*c^3)+a*b^2*c^2*d*(103*A*d^3+9*B*c*d^2+19*C *c^2*d+128*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))/c^(7/2)/d^(7/2)/(-a*d+b*c)^3/(c*(b*x^2+a)/a/(d*x ^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/105*b*(a^2*d^2*(24*A*d^3+4*B*c*d^2+3*C*c^2* d-45*D*c^3)-b^2*c^2*(-45*A*d^3+3*B*c*d^2+4*C*c^2*d+24*D*c^3)+a*b*c*d*(-61* A*d^3-9*B*c*d^2+9*C*c^2*d+61*D*c^3))*(b*x^2+a)^(1/2)*InverseJacobiAM(arcta n(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/c^(5/2)/d^(7/2)/(-a*d+b*c)^3/(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 = 7.83 (sec) , antiderivative size = 820, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (15 c^3 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )+3 c^2 (b c-a d)^2 \left (-a d \left (-8 c^2 C d+B c d^2+6 A d^3+15 c^3 D\right )+b c \left (-9 c^2 C d+2 B c d^2+5 A d^3+16 c^3 D\right )\right ) \left (c+d x^2\right )+c (b c-a d) \left (b^2 c^2 \left (8 c^2 C d+6 B c d^2+15 A d^3-57 c^3 D\right )+a^2 d^2 \left (3 c^2 C d+4 B c d^2+24 A d^3-45 c^3 D\right )+a b c d \left (-15 c^2 C d-6 B c d^2-43 A d^3+106 c^3 D\right )\right ) \left (c+d x^2\right )^2+\left (-a^3 d^3 \left (6 c^2 C d+8 B c d^2+48 A d^3+15 c^3 D\right )+b^3 c^3 \left (8 c^2 C d+6 B c d^2+15 A d^3+48 c^3 D\right )+a^2 b c d^2 \left (9 c^2 C d+19 B c d^2+128 A d^3+103 c^3 D\right )-a b^2 c^2 d \left (19 c^2 C d+9 B c d^2+103 A d^3+128 c^3 D\right )\right ) \left (c+d x^2\right )^3\right )+i b c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^3 \sqrt {1+\frac {d x^2}{c}} \left (\left (-a^3 d^3 \left (6 c^2 C d+8 B c d^2+48 A d^3+15 c^3 D\right )+b^3 c^3 \left (8 c^2 C d+6 B c d^2+15 A d^3+48 c^3 D\right )+a^2 b c d^2 \left (9 c^2 C d+19 B c d^2+128 A d^3+103 c^3 D\right )-a b^2 c^2 d \left (19 c^2 C d+9 B c d^2+103 A d^3+128 c^3 D\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (b^2 c^2 \left (8 c^2 C d+6 B c d^2+15 A d^3+48 c^3 D\right )+a^2 d^2 \left (3 c^2 C d+4 B c d^2+24 A d^3+60 c^3 D\right )-a b c d \left (15 c^2 C d+6 B c d^2+43 A d^3+104 c^3 D\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{105 \sqrt {\frac {b}{a}} c^4 d^4 (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/(c + d*x^2)^(9/2), x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(15*c^3*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^ 3 - c^3*D) + 3*c^2*(b*c - a*d)^2*(-(a*d*(-8*c^2*C*d + B*c*d^2 + 6*A*d^3 + 15*c^3*D)) + b*c*(-9*c^2*C*d + 2*B*c*d^2 + 5*A*d^3 + 16*c^3*D))*(c + d*x^2 ) + c*(b*c - a*d)*(b^2*c^2*(8*c^2*C*d + 6*B*c*d^2 + 15*A*d^3 - 57*c^3*D) + a^2*d^2*(3*c^2*C*d + 4*B*c*d^2 + 24*A*d^3 - 45*c^3*D) + a*b*c*d*(-15*c^2* C*d - 6*B*c*d^2 - 43*A*d^3 + 106*c^3*D))*(c + d*x^2)^2 + (-(a^3*d^3*(6*c^2 *C*d + 8*B*c*d^2 + 48*A*d^3 + 15*c^3*D)) + b^3*c^3*(8*c^2*C*d + 6*B*c*d^2 + 15*A*d^3 + 48*c^3*D) + a^2*b*c*d^2*(9*c^2*C*d + 19*B*c*d^2 + 128*A*d^3 + 103*c^3*D) - a*b^2*c^2*d*(19*c^2*C*d + 9*B*c*d^2 + 103*A*d^3 + 128*c^3*D) )*(c + d*x^2)^3) + I*b*c*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)^3*Sqrt[1 + (d*x^2 )/c]*((-(a^3*d^3*(6*c^2*C*d + 8*B*c*d^2 + 48*A*d^3 + 15*c^3*D)) + b^3*c^3* (8*c^2*C*d + 6*B*c*d^2 + 15*A*d^3 + 48*c^3*D) + a^2*b*c*d^2*(9*c^2*C*d + 1 9*B*c*d^2 + 128*A*d^3 + 103*c^3*D) - a*b^2*c^2*d*(19*c^2*C*d + 9*B*c*d^2 + 103*A*d^3 + 128*c^3*D))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*(b^2*c^2*(8*c^2*C*d + 6*B*c*d^2 + 15*A*d^3 + 48*c^3*D) + a^2*d ^2*(3*c^2*C*d + 4*B*c*d^2 + 24*A*d^3 + 60*c^3*D) - a*b*c*d*(15*c^2*C*d + 6 *B*c*d^2 + 43*A*d^3 + 104*c^3*D))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/ (b*c)]))/(105*Sqrt[b/a]*c^4*d^4*(b*c - a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)^ (7/2))
Leaf count is larger than twice the leaf count of optimal. \(1613\) vs. \(2(780)=1560\).
Time = 2.71 (sec) , antiderivative size = 1613, normalized size of antiderivative = 2.07, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \sqrt {a+b x^2}}{\left (c+d x^2\right )^{9/2}}+\frac {B x^2 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{9/2}}+\frac {C x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{9/2}}+\frac {D x^6 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{9/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {D \sqrt {b x^2+a} x^5}{7 d \left (d x^2+c\right )^{7/2}}-\frac {(6 b c-5 a d) D \sqrt {b x^2+a} x^3}{35 d^2 (b c-a d) \left (d x^2+c\right )^{5/2}}-\frac {C \sqrt {b x^2+a} x^3}{7 d \left (d x^2+c\right )^{7/2}}+\frac {2 B \left (3 b^2 c^2-3 a b d c+2 a^2 d^2\right ) \sqrt {b x^2+a} x}{105 c^2 d (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {C \left (8 b^2 c^2-15 a b d c+3 a^2 d^2\right ) \sqrt {b x^2+a} x}{105 c d^2 (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {A \left (15 b^2 c^2-43 a b d c+24 a^2 d^2\right ) \sqrt {b x^2+a} x}{105 c^3 (b c-a d)^2 \left (d x^2+c\right )^{3/2}}-\frac {\left (24 b^2 c^2-43 a b d c+15 a^2 d^2\right ) D \sqrt {b x^2+a} x}{105 d^3 (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {B (2 b c-a d) \sqrt {b x^2+a} x}{35 c d (b c-a d) \left (d x^2+c\right )^{5/2}}+\frac {A (5 b c-6 a d) \sqrt {b x^2+a} x}{35 c^2 (b c-a d) \left (d x^2+c\right )^{5/2}}-\frac {C (4 b c-3 a d) \sqrt {b x^2+a} x}{35 d^2 (b c-a d) \left (d x^2+c\right )^{5/2}}+\frac {A \sqrt {b x^2+a} x}{7 c \left (d x^2+c\right )^{7/2}}-\frac {B \sqrt {b x^2+a} x}{7 d \left (d x^2+c\right )^{7/2}}+\frac {C (b c-2 a d) \left (8 b^2 c^2-3 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 )}{105 c^{3/2} d^{5/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {B (2 b c-a d) \left (3 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 )}{105 c^{5/2} d^{3/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A \left (15 b^3 c^3-103 a b^2 d c^2+128 a^2 b d^2 c-48 a^3 d^3\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 c^{7/2} \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 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 \sqrt {c} d^{7/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b B \left (3 b^2 c^2+9 a b d c-4 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 )}{105 c^{3/2} d^{3/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b C \left (4 b^2 c^2-9 a b d c-3 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 )}{105 \sqrt {c} d^{5/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A b \left (45 b^2 c^2-61 a b d c+24 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 )}{105 c^{5/2} \sqrt {d} (b c-a d)^3 \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 (24 b^2 c^2-61 a b d c+45 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 d^{7/2} (b c-a d)^3 \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))/(c + d*x^2)^(9/2),x]
Output:
(A*x*Sqrt[a + b*x^2])/(7*c*(c + d*x^2)^(7/2)) - (B*x*Sqrt[a + b*x^2])/(7*d *(c + d*x^2)^(7/2)) - (C*x^3*Sqrt[a + b*x^2])/(7*d*(c + d*x^2)^(7/2)) - (D *x^5*Sqrt[a + b*x^2])/(7*d*(c + d*x^2)^(7/2)) + (A*(5*b*c - 6*a*d)*x*Sqrt[ a + b*x^2])/(35*c^2*(b*c - a*d)*(c + d*x^2)^(5/2)) - (C*(4*b*c - 3*a*d)*x* Sqrt[a + b*x^2])/(35*d^2*(b*c - a*d)*(c + d*x^2)^(5/2)) + (B*(2*b*c - a*d) *x*Sqrt[a + b*x^2])/(35*c*d*(b*c - a*d)*(c + d*x^2)^(5/2)) - ((6*b*c - 5*a *d)*D*x^3*Sqrt[a + b*x^2])/(35*d^2*(b*c - a*d)*(c + d*x^2)^(5/2)) + (2*B*( 3*b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(105*c^2*d*(b*c - a* d)^2*(c + d*x^2)^(3/2)) + (C*(8*b^2*c^2 - 15*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(105*c*d^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (A*(15*b^2*c^2 - 43*a*b*c*d + 24*a^2*d^2)*x*Sqrt[a + b*x^2])/(105*c^3*(b*c - a*d)^2*(c + d* x^2)^(3/2)) - ((24*b^2*c^2 - 43*a*b*c*d + 15*a^2*d^2)*D*x*Sqrt[a + b*x^2]) /(105*d^3*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (C*(b*c - 2*a*d)*(8*b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[ c]], 1 - (b*c)/(a*d)])/(105*c^(3/2)*d^(5/2)*(b*c - a*d)^3*Sqrt[(c*(a + b*x ^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (B*(2*b*c - a*d)*(3*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)])/(105*c^(5/2)*d^(3/2)*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2)) /(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (A*(15*b^3*c^3 - 103*a*b^2*c^2*d + 12 8*a^2*b*c*d^2 - 48*a^3*d^3)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x...
Leaf count of result is larger than twice the leaf count of optimal. \(1618\) vs. \(2(741)=1482\).
Time = 10.21 (sec) , antiderivative size = 1619, normalized size of antiderivative = 2.08
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1619\) |
| default | \(\text {Expression too large to display}\) | \(10184\) |
Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(9/2),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*(A*d^3-B* c*d^2+C*c^2*d-D*c^3)/c/d^7*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d) ^4+1/35*(6*A*a*d^4-5*A*b*c*d^3+B*a*c*d^3-2*B*b*c^2*d^2-8*C*a*c^2*d^2+9*C*b *c^3*d+15*D*a*c^3*d-16*D*b*c^4)/d^6/(a*d-b*c)/c^2*x*(b*d*x^4+a*d*x^2+b*c*x ^2+a*c)^(1/2)/(x^2+c/d)^3+1/105*(24*A*a^2*d^5-43*A*a*b*c*d^4+15*A*b^2*c^2* d^3+4*B*a^2*c*d^4-6*B*a*b*c^2*d^3+6*B*b^2*c^3*d^2+3*C*a^2*c^2*d^3-15*C*a*b *c^3*d^2+8*C*b^2*c^4*d-45*D*a^2*c^3*d^2+106*D*a*b*c^4*d-57*D*b^2*c^5)/d^5/ (a*d-b*c)^2/c^3*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^2+1/105*(b *d*x^2+a*d)/c^4/d^4/(a*d-b*c)^3*x*(48*A*a^3*d^6-128*A*a^2*b*c*d^5+103*A*a* b^2*c^2*d^4-15*A*b^3*c^3*d^3+8*B*a^3*c*d^5-19*B*a^2*b*c^2*d^4+9*B*a*b^2*c^ 3*d^3-6*B*b^3*c^4*d^2+6*C*a^3*c^2*d^4-9*C*a^2*b*c^3*d^3+19*C*a*b^2*c^4*d^2 -8*C*b^3*c^5*d+15*D*a^3*c^3*d^3-103*D*a^2*b*c^4*d^2+128*D*a*b^2*c^5*d-48*D *b^3*c^6)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(D*b/d^4+1/105*b*(24*A*a^2*d^5-4 3*A*a*b*c*d^4+15*A*b^2*c^2*d^3+4*B*a^2*c*d^4-6*B*a*b*c^2*d^3+6*B*b^2*c^3*d ^2+3*C*a^2*c^2*d^3-15*C*a*b*c^3*d^2+8*C*b^2*c^4*d-45*D*a^2*c^3*d^2+106*D*a *b*c^4*d-57*D*b^2*c^5)/d^4/(a*d-b*c)^2/c^3+1/105/d^4/(a*d-b*c)^2*(48*A*a^3 *d^6-128*A*a^2*b*c*d^5+103*A*a*b^2*c^2*d^4-15*A*b^3*c^3*d^3+8*B*a^3*c*d^5- 19*B*a^2*b*c^2*d^4+9*B*a*b^2*c^3*d^3-6*B*b^3*c^4*d^2+6*C*a^3*c^2*d^4-9*C*a ^2*b*c^3*d^3+19*C*a*b^2*c^4*d^2-8*C*b^3*c^5*d+15*D*a^3*c^3*d^3-103*D*a^2*b *c^4*d^2+128*D*a*b^2*c^5*d-48*D*b^3*c^6)/c^4-1/105*a/d^3/c^4/(a*d-b*c)^...
Leaf count of result is larger than twice the leaf count of optimal. 3122 vs. \(2 (741) = 1482\).
Time = 0.25 (sec) , antiderivative size = 3122, normalized size of antiderivative = 4.00 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(9/2),x, algorit hm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(1/2)*(D*x**6+C*x**4+B*x**2+A)/(d*x**2+c)**(9/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(9/2),x, algorit hm="maxima")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(b*x^2 + a)/(d*x^2 + c)^(9/2), x )
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(9/2),x, algorit hm="giac")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(b*x^2 + a)/(d*x^2 + c)^(9/2), x )
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(9/2),x)
Output:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(9/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(9/2),x)
Output:
(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*x + 6*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a**2*d**2*x**3 - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d* x - 21*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c**2*x - 44*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a*b*c*d*x**3 - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d **2*x**5 + 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*x**3 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*x**5 + 72*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(3*a**2*c**5*d + 15*a**2*c**4*d**2*x**2 + 30*a**2*c**3*d* *3*x**4 + 30*a**2*c**2*d**4*x**6 + 15*a**2*c*d**5*x**8 + 3*a**2*d**6*x**10 - a*b*c**6 - 2*a*b*c**5*d*x**2 + 5*a*b*c**4*d**2*x**4 + 20*a*b*c**3*d**3* x**6 + 25*a*b*c**2*d**4*x**8 + 14*a*b*c*d**5*x**10 + 3*a*b*d**6*x**12 - b* *2*c**6*x**2 - 5*b**2*c**5*d*x**4 - 10*b**2*c**4*d**2*x**6 - 10*b**2*c**3* d**3*x**8 - 5*b**2*c**2*d**4*x**10 - b**2*c*d**5*x**12),x)*a**4*c**4*d**4 + 288*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(3*a**2*c**5*d + 15*a** 2*c**4*d**2*x**2 + 30*a**2*c**3*d**3*x**4 + 30*a**2*c**2*d**4*x**6 + 15*a* *2*c*d**5*x**8 + 3*a**2*d**6*x**10 - a*b*c**6 - 2*a*b*c**5*d*x**2 + 5*a*b* c**4*d**2*x**4 + 20*a*b*c**3*d**3*x**6 + 25*a*b*c**2*d**4*x**8 + 14*a*b*c* d**5*x**10 + 3*a*b*d**6*x**12 - b**2*c**6*x**2 - 5*b**2*c**5*d*x**4 - 10*b **2*c**4*d**2*x**6 - 10*b**2*c**3*d**3*x**8 - 5*b**2*c**2*d**4*x**10 - b** 2*c*d**5*x**12),x)*a**4*c**3*d**5*x**2 + 432*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(3*a**2*c**5*d + 15*a**2*c**4*d**2*x**2 + 30*a**2*c**3*...