\(\int \frac {\sqrt {a+b x^2} (A+B x^2+C x^4+D x^6)}{(c+d x^2)^{7/2}} \, dx\) [29]

Optimal result

Integrand size = 40, antiderivative size = 567 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {\left (c^2 C d-B c d^2+A d^3-6 c^3 D\right ) x \sqrt {a+b x^2}}{5 c d^3 \left (c+d x^2\right )^{5/2}}+\frac {D x^5 \sqrt {a+b x^2}}{d \left (c+d x^2\right )^{5/2}}-\frac {\left (b c \left (7 c^2 C d-2 B c d^2-3 A d^3-42 c^3 D\right )-a d \left (6 c^2 C d-B c d^2-4 A d^3-41 c^3 D\right )\right ) x \sqrt {a+b x^2}}{15 c^2 d^3 (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {\left (a b c d \left (13 c^2 C d+2 B c d^2+13 A d^3-88 c^3 D\right )-b^2 c^2 \left (8 c^2 C d+2 B c d^2+3 A d^3-48 c^3 D\right )-a^2 d^2 \left (3 c^2 C d+2 B c d^2+8 A d^3-38 c^3 D\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 c^{5/2} d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (15 a^2 c^2 d^2 D+a b d \left (6 c^2 C d-B c d^2-4 A d^3-41 c^3 D\right )-b^2 c \left (4 c^2 C d+B c d^2-6 A d^3-24 c^3 D\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 c^{3/2} d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/5*(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)^(5/2 
)+D*x^5*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(5/2)-1/15*(b*c*(-3*A*d^3-2*B*c*d^2+7* 
C*c^2*d-42*D*c^3)-a*d*(-4*A*d^3-B*c*d^2+6*C*c^2*d-41*D*c^3))*x*(b*x^2+a)^( 
1/2)/c^2/d^3/(-a*d+b*c)/(d*x^2+c)^(3/2)-1/15*(a*b*c*d*(13*A*d^3+2*B*c*d^2+ 
13*C*c^2*d-88*D*c^3)-b^2*c^2*(3*A*d^3+2*B*c*d^2+8*C*c^2*d-48*D*c^3)-a^2*d^ 
2*(8*A*d^3+2*B*c*d^2+3*C*c^2*d-38*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^(5/2)/d^(7/2)/(-a*d+b*c 
)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/15*(15*a^2*c^2*d^2*D 
+a*b*d*(-4*A*d^3-B*c*d^2+6*C*c^2*d-41*D*c^3)-b^2*c*(-6*A*d^3+B*c*d^2+4*C*c 
^2*d-24*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))/c^(3/2)/d^(7/2)/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^ 
(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.76 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 c^2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )+c (b c-a d) \left (-a d \left (-6 c^2 C d+B c d^2+4 A d^3+11 c^3 D\right )+b c \left (-7 c^2 C d+2 B c d^2+3 A d^3+12 c^3 D\right )\right ) \left (c+d x^2\right )+\left (b^2 c^2 \left (8 c^2 C d+2 B c d^2+3 A d^3-33 c^3 D\right )+a^2 d^2 \left (3 c^2 C d+2 B c d^2+8 A d^3-23 c^3 D\right )+a b c d \left (-13 c^2 C d-2 B c d^2-13 A d^3+58 c^3 D\right )\right ) \left (c+d x^2\right )^2\right )-i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (b \left (a b c d \left (13 c^2 C d+2 B c d^2+13 A d^3-88 c^3 D\right )-a^2 d^2 \left (3 c^2 C d+2 B c d^2+8 A d^3-38 c^3 D\right )+b^2 c^2 \left (-8 c^2 C d-2 B c d^2-3 A d^3+48 c^3 D\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (15 a^2 c^2 d^2 D+a b d \left (9 c^2 C d+B c d^2+4 A d^3-64 c^3 D\right )+b^2 c \left (-8 c^2 C d-2 B c d^2-3 A d^3+48 c^3 D\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 \sqrt {\frac {b}{a}} c^3 d^4 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/(c + d*x^2)^(7/2), 
x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(3*c^2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 
 - c^3*D) + c*(b*c - a*d)*(-(a*d*(-6*c^2*C*d + B*c*d^2 + 4*A*d^3 + 11*c^3* 
D)) + b*c*(-7*c^2*C*d + 2*B*c*d^2 + 3*A*d^3 + 12*c^3*D))*(c + d*x^2) + (b^ 
2*c^2*(8*c^2*C*d + 2*B*c*d^2 + 3*A*d^3 - 33*c^3*D) + a^2*d^2*(3*c^2*C*d + 
2*B*c*d^2 + 8*A*d^3 - 23*c^3*D) + a*b*c*d*(-13*c^2*C*d - 2*B*c*d^2 - 13*A* 
d^3 + 58*c^3*D))*(c + d*x^2)^2) - I*c*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)^2*Sq 
rt[1 + (d*x^2)/c]*(b*(a*b*c*d*(13*c^2*C*d + 2*B*c*d^2 + 13*A*d^3 - 88*c^3* 
D) - a^2*d^2*(3*c^2*C*d + 2*B*c*d^2 + 8*A*d^3 - 38*c^3*D) + b^2*c^2*(-8*c^ 
2*C*d - 2*B*c*d^2 - 3*A*d^3 + 48*c^3*D))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], 
 (a*d)/(b*c)] - (b*c - a*d)*(15*a^2*c^2*d^2*D + a*b*d*(9*c^2*C*d + B*c*d^2 
 + 4*A*d^3 - 64*c^3*D) + b^2*c*(-8*c^2*C*d - 2*B*c*d^2 - 3*A*d^3 + 48*c^3* 
D))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*Sqrt[b/a]*c^3*d^4 
*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^(5/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1387\) vs. \(2(567)=1134\).

Time = 2.33 (sec) , antiderivative size = 1387, normalized size of antiderivative = 2.45, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {A \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}+\frac {B x^2 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}+\frac {C x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}+\frac {D x^6 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {D \sqrt {b x^2+a} x^5}{5 d \left (d x^2+c\right )^{5/2}}-\frac {(6 b c-5 a d) D \sqrt {b x^2+a} x^3}{15 d^2 (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {C \sqrt {b x^2+a} x^3}{5 d \left (d x^2+c\right )^{5/2}}-\frac {\left (24 b^2 c^2-41 a b d c+15 a^2 d^2\right ) D \sqrt {b x^2+a} x}{15 d^3 (b c-a d)^2 \sqrt {d x^2+c}}+\frac {2 \left (24 b^2 c^2-44 a b d c+19 a^2 d^2\right ) D \sqrt {b x^2+a} x}{15 d^3 (b c-a d)^2 \sqrt {d x^2+c}}+\frac {B (2 b c-a d) \sqrt {b x^2+a} x}{15 c d (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {A (3 b c-4 a d) \sqrt {b x^2+a} x}{15 c^2 (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {C (4 b c-3 a d) \sqrt {b x^2+a} x}{15 d^2 (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {A \sqrt {b x^2+a} x}{5 c \left (d x^2+c\right )^{5/2}}-\frac {B \sqrt {b x^2+a} x}{5 d \left (d x^2+c\right )^{5/2}}+\frac {2 B \left (b^2 c^2-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 )}{15 c^{3/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {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 \sqrt {c} d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {A \left (3 b^2 c^2-13 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 c^{5/2} \sqrt {d} (b c-a d)^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} \left (24 b^2 c^2-44 a b d c+19 a^2 d^2\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 )}{15 d^{7/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b B (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 \sqrt {c} d^{3/2} (b c-a d)^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 (24 b^2 c^2-41 a b d c+15 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 )}{15 d^{7/2} (b c-a d)^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 \sqrt {c} C (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} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 A b (3 b c-2 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 c^{3/2} \sqrt {d} (b c-a d)^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))/(c + d*x^2)^(7/2),x]
 

Output:

(A*x*Sqrt[a + b*x^2])/(5*c*(c + d*x^2)^(5/2)) - (B*x*Sqrt[a + b*x^2])/(5*d 
*(c + d*x^2)^(5/2)) - (C*x^3*Sqrt[a + b*x^2])/(5*d*(c + d*x^2)^(5/2)) - (D 
*x^5*Sqrt[a + b*x^2])/(5*d*(c + d*x^2)^(5/2)) + (A*(3*b*c - 4*a*d)*x*Sqrt[ 
a + b*x^2])/(15*c^2*(b*c - a*d)*(c + d*x^2)^(3/2)) - (C*(4*b*c - 3*a*d)*x* 
Sqrt[a + b*x^2])/(15*d^2*(b*c - a*d)*(c + d*x^2)^(3/2)) + (B*(2*b*c - a*d) 
*x*Sqrt[a + b*x^2])/(15*c*d*(b*c - a*d)*(c + d*x^2)^(3/2)) - ((6*b*c - 5*a 
*d)*D*x^3*Sqrt[a + b*x^2])/(15*d^2*(b*c - a*d)*(c + d*x^2)^(3/2)) - ((24*b 
^2*c^2 - 41*a*b*c*d + 15*a^2*d^2)*D*x*Sqrt[a + b*x^2])/(15*d^3*(b*c - a*d) 
^2*Sqrt[c + d*x^2]) + (2*(24*b^2*c^2 - 44*a*b*c*d + 19*a^2*d^2)*D*x*Sqrt[a 
 + b*x^2])/(15*d^3*(b*c - a*d)^2*Sqrt[c + d*x^2]) + (2*B*(b^2*c^2 - a*b*c* 
d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b 
*c)/(a*d)])/(15*c^(3/2)*d^(3/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + 
 d*x^2))]*Sqrt[c + d*x^2]) + (C*(8*b^2*c^2 - 13*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)])/(15*Sq 
rt[c]*d^(5/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + 
 d*x^2]) + (A*(3*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*Sqrt[a + b*x^2]*Ellipti 
cE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*c^(5/2)*Sqrt[d]*(b*c 
 - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*Sqrt 
[c]*(24*b^2*c^2 - 44*a*b*c*d + 19*a^2*d^2)*D*Sqrt[a + b*x^2]*EllipticE[Arc 
Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(7/2)*(b*c - a*d)^2*S...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1161\) vs. \(2(534)=1068\).

Time = 9.44 (sec) , antiderivative size = 1162, normalized size of antiderivative = 2.05

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1162\)
default	\(\text {Expression too large to display}\)	\(5928\)

Input:

int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(7/2),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)*(1/5*(A*d^3-B* 
c*d^2+C*c^2*d-D*c^3)/c/d^6*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d) 
^3+1/15*(4*A*a*d^4-3*A*b*c*d^3+B*a*c*d^3-2*B*b*c^2*d^2-6*C*a*c^2*d^2+7*C*b 
*c^3*d+11*D*a*c^3*d-12*D*b*c^4)/c^2/d^5/(a*d-b*c)*x*(b*d*x^4+a*d*x^2+b*c*x 
^2+a*c)^(1/2)/(x^2+c/d)^2+1/15*(b*d*x^2+a*d)/c^3/d^4/(a*d-b*c)^2*x*(8*A*a^ 
2*d^5-13*A*a*b*c*d^4+3*A*b^2*c^2*d^3+2*B*a^2*c*d^4-2*B*a*b*c^2*d^3+2*B*b^2 
*c^3*d^2+3*C*a^2*c^2*d^3-13*C*a*b*c^3*d^2+8*C*b^2*c^4*d-23*D*a^2*c^3*d^2+5 
8*D*a*b*c^4*d-33*D*b^2*c^5)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+((C*b*d+D*a*d- 
3*D*b*c)/d^4+1/15*b*(4*A*a*d^4-3*A*b*c*d^3+B*a*c*d^3-2*B*b*c^2*d^2-6*C*a*c 
^2*d^2+7*C*b*c^3*d+11*D*a*c^3*d-12*D*b*c^4)/c^2/d^4/(a*d-b*c)+1/15/d^4/(a* 
d-b*c)*(8*A*a^2*d^5-13*A*a*b*c*d^4+3*A*b^2*c^2*d^3+2*B*a^2*c*d^4-2*B*a*b*c 
^2*d^3+2*B*b^2*c^3*d^2+3*C*a^2*c^2*d^3-13*C*a*b*c^3*d^2+8*C*b^2*c^4*d-23*D 
*a^2*c^3*d^2+58*D*a*b*c^4*d-33*D*b^2*c^5)/c^3-1/15*a/d^3/c^3/(a*d-b*c)^2*( 
8*A*a^2*d^5-13*A*a*b*c*d^4+3*A*b^2*c^2*d^3+2*B*a^2*c*d^4-2*B*a*b*c^2*d^3+2 
*B*b^2*c^3*d^2+3*C*a^2*c^2*d^3-13*C*a*b*c^3*d^2+8*C*b^2*c^4*d-23*D*a^2*c^3 
*d^2+58*D*a*b*c^4*d-33*D*b^2*c^5))/(-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))-(D*b/d^3-1/15/d^3*b*(8*A*a^2*d^5-13*A*a*b*c*d^4+3*A 
*b^2*c^2*d^3+2*B*a^2*c*d^4-2*B*a*b*c^2*d^3+2*B*b^2*c^3*d^2+3*C*a^2*c^2*d^3 
-13*C*a*b*c^3*d^2+8*C*b^2*c^4*d-23*D*a^2*c^3*d^2+58*D*a*b*c^4*d-33*D*b^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1905 vs. \(2 (533) = 1066\).

Time = 0.16 (sec) , antiderivative size = 1905, normalized size of antiderivative = 3.36 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(7/2),x, algorit 
hm="fricas")
 

Output:

-1/15*(((48*D*b^3*c^6*d^3 - 8*A*a^2*b*c*d^8 - 8*(11*D*a*b^2 + C*b^3)*c^5*d 
^4 + (38*D*a^2*b + 13*C*a*b^2 - 2*B*b^3)*c^4*d^5 - (3*C*a^2*b - 2*B*a*b^2 
+ 3*A*b^3)*c^3*d^6 - (2*B*a^2*b - 13*A*a*b^2)*c^2*d^7)*x^7 + 3*(48*D*b^3*c 
^7*d^2 - 8*A*a^2*b*c^2*d^7 - 8*(11*D*a*b^2 + C*b^3)*c^6*d^3 + (38*D*a^2*b 
+ 13*C*a*b^2 - 2*B*b^3)*c^5*d^4 - (3*C*a^2*b - 2*B*a*b^2 + 3*A*b^3)*c^4*d^ 
5 - (2*B*a^2*b - 13*A*a*b^2)*c^3*d^6)*x^5 + 3*(48*D*b^3*c^8*d - 8*A*a^2*b* 
c^3*d^6 - 8*(11*D*a*b^2 + C*b^3)*c^7*d^2 + (38*D*a^2*b + 13*C*a*b^2 - 2*B* 
b^3)*c^6*d^3 - (3*C*a^2*b - 2*B*a*b^2 + 3*A*b^3)*c^5*d^4 - (2*B*a^2*b - 13 
*A*a*b^2)*c^4*d^5)*x^3 + (48*D*b^3*c^9 - 8*A*a^2*b*c^4*d^5 - 8*(11*D*a*b^2 
 + C*b^3)*c^8*d + (38*D*a^2*b + 13*C*a*b^2 - 2*B*b^3)*c^7*d^2 - (3*C*a^2*b 
 - 2*B*a*b^2 + 3*A*b^3)*c^6*d^3 - (2*B*a^2*b - 13*A*a*b^2)*c^5*d^4)*x)*sqr 
t(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((48*D*b^3 
*c^6*d^3 - 4*A*a^2*b*d^9 - 8*(11*D*a*b^2 + C*b^3)*c^5*d^4 + (38*D*a^2*b + 
(13*C + 24*D)*a*b^2 - 2*B*b^3)*c^4*d^5 - ((3*C + 41*D)*a^2*b - 2*(B - 2*C) 
*a*b^2 + 3*A*b^3)*c^3*d^6 + (15*D*a^3 - 2*(B - 3*C)*a^2*b + (13*A - B)*a*b 
^2)*c^2*d^7 - ((8*A + B)*a^2*b - 6*A*a*b^2)*c*d^8)*x^7 + 3*(48*D*b^3*c^7*d 
^2 - 4*A*a^2*b*c*d^8 - 8*(11*D*a*b^2 + C*b^3)*c^6*d^3 + (38*D*a^2*b + (13* 
C + 24*D)*a*b^2 - 2*B*b^3)*c^5*d^4 - ((3*C + 41*D)*a^2*b - 2*(B - 2*C)*a*b 
^2 + 3*A*b^3)*c^4*d^5 + (15*D*a^3 - 2*(B - 3*C)*a^2*b + (13*A - B)*a*b^2)* 
c^3*d^6 - ((8*A + B)*a^2*b - 6*A*a*b^2)*c^2*d^7)*x^5 + 3*(48*D*b^3*c^8*...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (A + B x^{2} + C x^{4} + D x^{6}\right )}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(D*x**6+C*x**4+B*x**2+A)/(d*x**2+c)**(7/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(a + b*x**2)*(A + B*x**2 + C*x**4 + D*x**6)/(c + d*x**2)**(7/ 
2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(7/2),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)^(7/2), x 
)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (D x^{6} + C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(7/2),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)^(7/2), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(7/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/(c + d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/(d*x^2+c)^(7/2),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*x - 4*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a**2*d**2*x**3 - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2 
*d*x + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c**2*x + 22*sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*a*b*c*d*x**3 + 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b 
*d**2*x**5 - 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*x**3 - 2*sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*x**5 - 16*int((sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*x**4)/(2*a**2*c**4*d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d 
**3*x**4 + 8*a**2*c*d**4*x**6 + 2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d 
*x**2 + 2*a*b*c**3*d**2*x**4 + 8*a*b*c**2*d**3*x**6 + 7*a*b*c*d**4*x**8 + 
2*a*b*d**5*x**10 - b**2*c**5*x**2 - 4*b**2*c**4*d*x**4 - 6*b**2*c**3*d**2* 
x**6 - 4*b**2*c**2*d**3*x**8 - b**2*c*d**4*x**10),x)*a**4*c**3*d**4 - 48*i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a**2*c**4*d + 8*a**2*c**3*d 
**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a**2*c*d**4*x**6 + 2*a**2*d**5*x**8 
- a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*b*c**3*d**2*x**4 + 8*a*b*c**2*d**3*x* 
*6 + 7*a*b*c*d**4*x**8 + 2*a*b*d**5*x**10 - b**2*c**5*x**2 - 4*b**2*c**4*d 
*x**4 - 6*b**2*c**3*d**2*x**6 - 4*b**2*c**2*d**3*x**8 - b**2*c*d**4*x**10) 
,x)*a**4*c**2*d**5*x**2 - 48*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/ 
(2*a**2*c**4*d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a**2*c 
*d**4*x**6 + 2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*b*c**3* 
d**2*x**4 + 8*a*b*c**2*d**3*x**6 + 7*a*b*c*d**4*x**8 + 2*a*b*d**5*x**10...