\(\int \frac {A+B x^2+C x^4}{(a c+(b c-a d) x^2-b d x^4)^{5/2}} \, dx\) [14]

Optimal result

Integrand size = 40, antiderivative size = 713 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\frac {x \left (A \left (b^2 c^2+a^2 d^2\right )-a c (b B c-a (2 c C+B d))-\left (A b^2 c d+a^2 c C d-a b \left (c^2 C+2 B c d+A d^2\right )\right ) x^2\right )}{3 a c (b c+a d)^2 \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}}+\frac {x \left (3 a c (b c-a d) \left (A b^2 c d+a^2 c C d-a b \left (c^2 C+2 B c d+A d^2\right )\right )+\left (b^2 c^2+a^2 d^2\right ) \left (2 A \left (b^2 c^2+3 a b c d+a^2 d^2\right )+a c (b B c-a (2 c C+B d))\right )-b d \left (2 A \left (b^3 c^3+5 a b^2 c^2 d-5 a^2 b c d^2-a^3 d^3\right )+a c \left (b^2 B c^2+a^2 d (8 c C+B d)-2 a b c (4 c C+7 B d)\right )\right ) x^2\right )}{3 a^2 c^2 (b c+a d)^4 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {\sqrt {d} \left (2 A \left (b^3 c^3+5 a b^2 c^2 d-5 a^2 b c d^2-a^3 d^3\right )+a c \left (b^2 B c^2+a^2 d (8 c C+B d)-2 a b c (4 c C+7 B d)\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{3 a c^{3/2} (b c+a d)^4 \sqrt {a c+(b c-a d) x^2-b d x^4}}-\frac {\left (A b^2 c^2 d+a^2 d \left (5 c^2 C+B c d-2 A d^2\right )-a b c \left (3 c^2 C+7 B c d+9 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} (b c+a d)^3 \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

1/3*x*(A*(a^2*d^2+b^2*c^2)-a*c*(B*b*c-a*(B*d+2*C*c))-(A*b^2*c*d+a^2*c*C*d- 
a*b*(A*d^2+2*B*c*d+C*c^2))*x^2)/a/c/(a*d+b*c)^2/(a*c+(-a*d+b*c)*x^2-b*d*x^ 
4)^(3/2)+1/3*x*(3*a*c*(-a*d+b*c)*(A*b^2*c*d+a^2*c*C*d-a*b*(A*d^2+2*B*c*d+C 
*c^2))+(a^2*d^2+b^2*c^2)*(2*A*(a^2*d^2+3*a*b*c*d+b^2*c^2)+a*c*(B*b*c-a*(B* 
d+2*C*c)))-b*d*(2*A*(-a^3*d^3-5*a^2*b*c*d^2+5*a*b^2*c^2*d+b^3*c^3)+a*c*(b^ 
2*B*c^2+a^2*d*(B*d+8*C*c)-2*a*b*c*(7*B*d+4*C*c)))*x^2)/a^2/c^2/(a*d+b*c)^4 
/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)+1/3*d^(1/2)*(2*A*(-a^3*d^3-5*a^2*b*c*d 
^2+5*a*b^2*c^2*d+b^3*c^3)+a*c*(b^2*B*c^2+a^2*d*(B*d+8*C*c)-2*a*b*c*(7*B*d+ 
4*C*c)))*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2),( 
-b*c/a/d)^(1/2))/a/c^(3/2)/(a*d+b*c)^4/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)- 
1/3*(A*b^2*c^2*d+a^2*d*(-2*A*d^2+B*c*d+5*C*c^2)-a*b*c*(9*A*d^2+7*B*c*d+3*C 
*c^2))*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(-b 
*c/a/d)^(1/2))/a/c^(3/2)/d^(1/2)/(a*d+b*c)^3/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^ 
(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.43 (sec) , antiderivative size = 520, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (a^2 c d (b c+a d) \left (c^2 C+B c d+A d^2\right ) \left (a+b x^2\right )^2+a^2 d \left (a d \left (-4 c^2 C-B c d+2 A d^2\right )+b c \left (4 c^2 C+7 B c d+10 A d^2\right )\right ) \left (a+b x^2\right )^2 \left (c-d x^2\right )+a b c^2 \left (A b^2+a (-b B+a C)\right ) (b c+a d) \left (c-d x^2\right )^2+b c^2 \left (2 A b^2 (b c+5 a d)+a \left (b^2 B c+4 a^2 C d-a b (4 c C+7 B d)\right )\right ) \left (a+b x^2\right ) \left (c-d x^2\right )^2\right )+i c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (-c+d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \left (-b \left (2 A \left (b^3 c^3+5 a b^2 c^2 d-5 a^2 b c d^2-a^3 d^3\right )+a c \left (b^2 B c^2+a^2 d (8 c C+B d)-2 a b c (4 c C+7 B d)\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+(b c+a d) \left (A b \left (2 b^2 c^2+9 a b c d-a^2 d^2\right )+a c \left (b^2 B c+3 a^2 C d-a b (5 c C+7 B d)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c+a d)^4 \left (\left (a+b x^2\right ) \left (c-d x^2\right )\right )^{3/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(5/2),x]
 

Output:

(Sqrt[b/a]*x*(a^2*c*d*(b*c + a*d)*(c^2*C + B*c*d + A*d^2)*(a + b*x^2)^2 + 
a^2*d*(a*d*(-4*c^2*C - B*c*d + 2*A*d^2) + b*c*(4*c^2*C + 7*B*c*d + 10*A*d^ 
2))*(a + b*x^2)^2*(c - d*x^2) + a*b*c^2*(A*b^2 + a*(-(b*B) + a*C))*(b*c + 
a*d)*(c - d*x^2)^2 + b*c^2*(2*A*b^2*(b*c + 5*a*d) + a*(b^2*B*c + 4*a^2*C*d 
 - a*b*(4*c*C + 7*B*d)))*(a + b*x^2)*(c - d*x^2)^2) + I*c*(a + b*x^2)*Sqrt 
[1 + (b*x^2)/a]*(-c + d*x^2)*Sqrt[1 - (d*x^2)/c]*(-(b*(2*A*(b^3*c^3 + 5*a* 
b^2*c^2*d - 5*a^2*b*c*d^2 - a^3*d^3) + a*c*(b^2*B*c^2 + a^2*d*(8*c*C + B*d 
) - 2*a*b*c*(4*c*C + 7*B*d)))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b 
*c))]) + (b*c + a*d)*(A*b*(2*b^2*c^2 + 9*a*b*c*d - a^2*d^2) + a*c*(b^2*B*c 
 + 3*a^2*C*d - a*b*(5*c*C + 7*B*d)))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], -(( 
a*d)/(b*c))]))/(3*a^2*Sqrt[b/a]*c^2*(b*c + a*d)^4*((a + b*x^2)*(c - d*x^2) 
)^(3/2))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 671, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2206, 25, 1492, 25, 1514, 399, 321, 327}

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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}-\frac {\int -\frac {-3 \left (c C d a^2-b \left (C c^2+2 B d c+A d^2\right ) a+A b^2 c d\right ) x^2+a c (b B c-2 a C c-a B d)+2 A \left (b^2 c^2+3 a b d c+a^2 d^2\right )}{\left (-b d x^4+(b c-a d) x^2+a c\right )^{3/2}}dx}{3 a c (a d+b c)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-3 \left (c C d a^2-b \left (C c^2+2 B d c+A d^2\right ) a+A b^2 c d\right ) x^2+2 A \left (b^2 c^2+3 a b d c+a^2 d^2\right )+a c (b B c-a (2 c C+B d))}{\left (-b d x^4+(b c-a d) x^2+a c\right )^{3/2}}dx}{3 a c (a d+b c)^2}+\frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {x \left (3 a c (b c-a d) \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )+\left (a^2 d^2+b^2 c^2\right ) \left (2 A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c (b B c-a (B d+2 c C))\right )-b d x^2 \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right )\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {\int -\frac {b d \left (2 A \left (b^3 c^3+5 a b^2 d c^2-5 a^2 b d^2 c-a^3 d^3\right )+a c \left (d (8 c C+B d) a^2-2 b c (4 c C+7 B d) a+b^2 B c^2\right )\right ) x^2+a c \left (3 c C d^2 a^3-b d \left (10 C c^2+8 B d c-A d^2\right ) a^2+b^2 c \left (3 C c^2+8 B d c+18 A d^2\right ) a+A b^3 c^2 d\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}}{3 a c (a d+b c)^2}+\frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b d \left (2 A \left (b^3 c^3+5 a b^2 d c^2-5 a^2 b d^2 c-a^3 d^3\right )+a c \left (d (8 c C+B d) a^2-2 b c (4 c C+7 B d) a+b^2 B c^2\right )\right ) x^2+a c \left (3 c C d^2 a^3-b d \left (10 C c^2+8 B d c-A d^2\right ) a^2+b^2 c \left (3 C c^2+8 B d c+18 A d^2\right ) a+A b^3 c^2 d\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}+\frac {x \left (3 a c (b c-a d) \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )+\left (a^2 d^2+b^2 c^2\right ) \left (2 A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c (b B c-a (B d+2 c C))\right )-b d x^2 \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right )\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {b d \left (2 A \left (b^3 c^3+5 a b^2 d c^2-5 a^2 b d^2 c-a^3 d^3\right )+a c \left (d (8 c C+B d) a^2-2 b c (4 c C+7 B d) a+b^2 B c^2\right )\right ) x^2+a c \left (3 c C d^2 a^3-b d \left (10 C c^2+8 B d c-A d^2\right ) a^2+b^2 c \left (3 C c^2+8 B d c+18 A d^2\right ) a+A b^3 c^2 d\right )}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (3 a c (b c-a d) \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )+\left (a^2 d^2+b^2 c^2\right ) \left (2 A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c (b B c-a (B d+2 c C))\right )-b d x^2 \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right )\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (a d \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx-a (a d+b c) \left (a^2 d \left (-2 A d^2+B c d+5 c^2 C\right )-a b c \left (9 A d^2+7 B c d+3 c^2 C\right )+A b^2 c^2 d\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (3 a c (b c-a d) \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )+\left (a^2 d^2+b^2 c^2\right ) \left (2 A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c (b B c-a (B d+2 c C))\right )-b d x^2 \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right )\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (a d \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx-\frac {a \sqrt {c} (a d+b c) \left (a^2 d \left (-2 A d^2+B c d+5 c^2 C\right )-a b c \left (9 A d^2+7 B c d+3 c^2 C\right )+A b^2 c^2 d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d}}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (3 a c (b c-a d) \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )+\left (a^2 d^2+b^2 c^2\right ) \left (2 A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c (b B c-a (B d+2 c C))\right )-b d x^2 \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right )\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x \left (-\left (x^2 \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )\right )+A \left (a^2 d^2+b^2 c^2\right )-a c (b B c-a (B d+2 c C))\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}+\frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (a \sqrt {c} \sqrt {d} \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )-\frac {a \sqrt {c} (a d+b c) \left (a^2 d \left (-2 A d^2+B c d+5 c^2 C\right )-a b c \left (9 A d^2+7 B c d+3 c^2 C\right )+A b^2 c^2 d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d}}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (3 a c (b c-a d) \left (a^2 c C d-a b \left (A d^2+2 B c d+c^2 C\right )+A b^2 c d\right )+\left (a^2 d^2+b^2 c^2\right ) \left (2 A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c (b B c-a (B d+2 c C))\right )-b d x^2 \left (a c \left (a^2 d (B d+8 c C)-2 a b c (7 B d+4 c C)+b^2 B c^2\right )+2 A \left (-a^3 d^3-5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )\right )\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}\)

Input:

Int[(A + B*x^2 + C*x^4)/(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(5/2),x]
 

Output:

(x*(A*(b^2*c^2 + a^2*d^2) - a*c*(b*B*c - a*(2*c*C + B*d)) - (A*b^2*c*d + a 
^2*c*C*d - a*b*(c^2*C + 2*B*c*d + A*d^2))*x^2))/(3*a*c*(b*c + a*d)^2*(a*c 
+ (b*c - a*d)*x^2 - b*d*x^4)^(3/2)) + ((x*(3*a*c*(b*c - a*d)*(A*b^2*c*d + 
a^2*c*C*d - a*b*(c^2*C + 2*B*c*d + A*d^2)) + (b^2*c^2 + a^2*d^2)*(2*A*(b^2 
*c^2 + 3*a*b*c*d + a^2*d^2) + a*c*(b*B*c - a*(2*c*C + B*d))) - b*d*(2*A*(b 
^3*c^3 + 5*a*b^2*c^2*d - 5*a^2*b*c*d^2 - a^3*d^3) + a*c*(b^2*B*c^2 + a^2*d 
*(8*c*C + B*d) - 2*a*b*c*(4*c*C + 7*B*d)))*x^2))/(a*c*(b*c + a*d)^2*Sqrt[a 
*c + (b*c - a*d)*x^2 - b*d*x^4]) + (Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c 
]*(a*Sqrt[c]*Sqrt[d]*(2*A*(b^3*c^3 + 5*a*b^2*c^2*d - 5*a^2*b*c*d^2 - a^3*d 
^3) + a*c*(b^2*B*c^2 + a^2*d*(8*c*C + B*d) - 2*a*b*c*(4*c*C + 7*B*d)))*Ell 
ipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))] - (a*Sqrt[c]*(b*c + a* 
d)*(A*b^2*c^2*d + a^2*d*(5*c^2*C + B*c*d - 2*A*d^2) - a*b*c*(3*c^2*C + 7*B 
*c*d + 9*A*d^2))*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/S 
qrt[d]))/(a*c*(b*c + a*d)^2*Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4]))/(3*a*c 
*(b*c + a*d)^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

rule 1514

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt 
[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4])   Int[(d + e*x^2)/(Sqrt[1 
+ 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, 
c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 3.67 (sec) , antiderivative size = 1147, normalized size of antiderivative = 1.61

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1147\)
default	\(\text {Expression too large to display}\)	\(2104\)

Input:

int((C*x^4+B*x^2+A)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x,method=_RETURNVER 
BOSE)
 

Output:

(1/3/b^2/d^2*(A*a*b*d^2-A*b^2*c*d+2*B*a*b*c*d-C*a^2*c*d+C*a*b*c^2)/a/c/(a^ 
2*d^2+2*a*b*c*d+b^2*c^2)*x^3+1/3/b^2/d^2*(A*a^2*d^2+A*b^2*c^2+B*a^2*c*d-B* 
a*b*c^2+2*C*a^2*c^2)/a/c/(a^2*d^2+2*a*b*c*d+b^2*c^2)*x)*(-b*d*x^4-a*d*x^2+ 
b*c*x^2+a*c)^(1/2)/(x^4+(a*d-b*c)/b/d*x^2-a*c/b/d)^2+2*b*d*(1/6*(2*A*a^3*d 
^3+10*A*a^2*b*c*d^2-10*A*a*b^2*c^2*d-2*A*b^3*c^3-B*a^3*c*d^2+14*B*a^2*b*c^ 
2*d-B*a*b^2*c^3-8*C*a^3*c^2*d+8*C*a^2*b*c^3)/a^2/c^2/(a^2*d^2+2*a*b*c*d+b^ 
2*c^2)^2*x^3+1/6*(2*A*a^4*d^4+9*A*a^3*b*c*d^3-2*A*a^2*b^2*c^2*d^2+9*A*a*b^ 
3*c^3*d+2*A*b^4*c^4-B*a^4*c*d^3+7*B*a^3*b*c^2*d^2-7*B*a^2*b^2*c^3*d+B*a*b^ 
3*c^4-5*C*a^4*c^2*d^2+6*C*a^3*b*c^3*d-5*C*a^2*b^2*c^4)/a^2/c^2/(a^2*d^2+2* 
a*b*c*d+b^2*c^2)^2/b/d*x)/(-(x^4+(a*d-b*c)/b/d*x^2-a*c/b/d)*b*d)^(1/2)+(1/ 
3/(a^2*d^2+2*a*b*c*d+b^2*c^2)*(2*A*a^2*d^2+6*A*a*b*c*d+2*A*b^2*c^2-B*a^2*c 
*d+B*a*b*c^2-2*C*a^2*c^2)/a^2/c^2-1/3*(2*A*a^4*d^4+9*A*a^3*b*c*d^3-2*A*a^2 
*b^2*c^2*d^2+9*A*a*b^3*c^3*d+2*A*b^4*c^4-B*a^4*c*d^3+7*B*a^3*b*c^2*d^2-7*B 
*a^2*b^2*c^3*d+B*a*b^3*c^4-5*C*a^4*c^2*d^2+6*C*a^3*b*c^3*d-5*C*a^2*b^2*c^4 
)/a^2/c^2/(a^2*d^2+2*a*b*c*d+b^2*c^2)^2)/(d/c)^(1/2)*(1-d*x^2/c)^(1/2)*(1+ 
b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(d/c)^(1/2 
),(-1-(-a*d+b*c)/a/d)^(1/2))+1/3*d*(2*A*a^3*d^3+10*A*a^2*b*c*d^2-10*A*a*b^ 
2*c^2*d-2*A*b^3*c^3-B*a^3*c*d^2+14*B*a^2*b*c^2*d-B*a*b^2*c^3-8*C*a^3*c^2*d 
+8*C*a^2*b*c^3)/(a^2*d^2+2*a*b*c*d+b^2*c^2)^2/a/c^2/(d/c)^(1/2)*(1-d*x^2/c 
)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*(Ellipti...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2641 vs. \(2 (674) = 1348\).

Time = 0.26 (sec) , antiderivative size = 2641, normalized size of antiderivative = 3.70 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^4+B*x^2+A)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x, algorithm= 
"fricas")
 

Output:

-1/3*((2*A*a^5*c^2*d^5 + (2*A*a^3*b^2*d^7 + (8*C*a^2*b^3 - B*a*b^4 - 2*A*b 
^5)*c^3*d^4 - 2*(4*C*a^3*b^2 - 7*B*a^2*b^3 + 5*A*a*b^4)*c^2*d^5 - (B*a^3*b 
^2 - 10*A*a^2*b^3)*c*d^6)*x^8 + (8*C*a^4*b - B*a^3*b^2 - 2*A*a^2*b^3)*c^5* 
d^2 - 2*(4*C*a^5 - 7*B*a^4*b + 5*A*a^3*b^2)*c^4*d^3 - (B*a^5 - 10*A*a^4*b) 
*c^3*d^4 + 2*(2*A*a^4*b*d^7 - (8*C*a^2*b^3 - B*a*b^4 - 2*A*b^5)*c^4*d^3 + 
(16*C*a^3*b^2 - 15*B*a^2*b^3 + 8*A*a*b^4)*c^3*d^4 - (8*C*a^4*b - 15*B*a^3* 
b^2 + 20*A*a^2*b^3)*c^2*d^5 - (B*a^4*b - 8*A*a^3*b^2)*c*d^6)*x^6 + (2*A*a^ 
5*d^7 + (8*C*a^2*b^3 - B*a*b^4 - 2*A*b^5)*c^5*d^2 - 2*(20*C*a^3*b^2 - 9*B* 
a^2*b^3 + A*a*b^4)*c^4*d^3 + 2*(20*C*a^4*b - 29*B*a^3*b^2 + 24*A*a^2*b^3)* 
c^3*d^4 - 2*(4*C*a^5 - 9*B*a^4*b + 24*A*a^3*b^2)*c^2*d^5 - (B*a^5 - 2*A*a^ 
4*b)*c*d^6)*x^4 - 2*(2*A*a^5*c*d^6 - (8*C*a^3*b^2 - B*a^2*b^3 - 2*A*a*b^4) 
*c^5*d^2 + (16*C*a^4*b - 15*B*a^3*b^2 + 8*A*a^2*b^3)*c^4*d^3 - (8*C*a^5 - 
15*B*a^4*b + 20*A*a^3*b^2)*c^3*d^4 - (B*a^5 - 8*A*a^4*b)*c^2*d^5)*x^2)*sqr 
t(a*c)*sqrt(d/c)*elliptic_e(arcsin(x*sqrt(d/c)), -b*c/(a*d)) - (3*C*a^3*b^ 
2*c^7 + 2*A*a^5*c^2*d^5 + (3*C*a*b^4*c^5*d^2 + 2*A*a^3*b^2*d^7 - (10*C*a^2 
*b^3 - 8*B*a*b^4 - A*b^5)*c^4*d^3 + (3*C*a^3*b^2 - 8*(B - C)*a^2*b^3 + (18 
*A - B)*a*b^4 - 2*A*b^5)*c^3*d^4 - (8*C*a^3*b^2 - (A + 14*B)*a^2*b^3 + 10* 
A*a*b^4)*c^2*d^5 - (B*a^3*b^2 - 10*A*a^2*b^3)*c*d^6)*x^8 - (10*C*a^4*b - 8 
*B*a^3*b^2 - A*a^2*b^3)*c^6*d + (3*C*a^5 - 8*(B - C)*a^4*b + (18*A - B)*a^ 
3*b^2 - 2*A*a^2*b^3)*c^5*d^2 - (8*C*a^5 - (A + 14*B)*a^4*b + 10*A*a^3*b...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((C*x**4+B*x**2+A)/(a*c+(-a*d+b*c)*x**2-b*d*x**4)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x, algorithm= 
"maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(-b*d*x^4 + (b*c - a*d)*x^2 + a*c)^(5/2), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x, algorithm= 
"giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(-b*d*x^4 + (b*c - a*d)*x^2 + a*c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c\right )}^{5/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/(a*c - x^2*(a*d - b*c) - b*d*x^4)^(5/2),x)
 

Output:

int((A + B*x^2 + C*x^4)/(a*c - x^2*(a*d - b*c) - b*d*x^4)^(5/2), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int((C*x^4+B*x^2+A)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x)
 

Output:

(sqrt(c - d*x**2)*sqrt(a + b*x**2)*b*x + 2*int((sqrt(c - d*x**2)*sqrt(a + 
b*x**2)*x**4)/(a**4*c**3*d - 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 - 
a**4*d**4*x**6 - a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 - 12*a**3*b*c**2*d**2* 
x**4 + 10*a**3*b*c*d**3*x**6 - 3*a**3*b*d**4*x**8 - 3*a**2*b**2*c**4*x**2 
+ 12*a**2*b**2*c**3*d*x**4 - 18*a**2*b**2*c**2*d**2*x**6 + 12*a**2*b**2*c* 
d**3*x**8 - 3*a**2*b**2*d**4*x**10 - 3*a*b**3*c**4*x**4 + 10*a*b**3*c**3*d 
*x**6 - 12*a*b**3*c**2*d**2*x**8 + 6*a*b**3*c*d**3*x**10 - a*b**3*d**4*x** 
12 - b**4*c**4*x**6 + 3*b**4*c**3*d*x**8 - 3*b**4*c**2*d**2*x**10 + b**4*c 
*d**3*x**12),x)*a**4*c**3*d**2 - 4*int((sqrt(c - d*x**2)*sqrt(a + b*x**2)* 
x**4)/(a**4*c**3*d - 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 - a**4*d** 
4*x**6 - a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 - 12*a**3*b*c**2*d**2*x**4 + 1 
0*a**3*b*c*d**3*x**6 - 3*a**3*b*d**4*x**8 - 3*a**2*b**2*c**4*x**2 + 12*a** 
2*b**2*c**3*d*x**4 - 18*a**2*b**2*c**2*d**2*x**6 + 12*a**2*b**2*c*d**3*x** 
8 - 3*a**2*b**2*d**4*x**10 - 3*a*b**3*c**4*x**4 + 10*a*b**3*c**3*d*x**6 - 
12*a*b**3*c**2*d**2*x**8 + 6*a*b**3*c*d**3*x**10 - a*b**3*d**4*x**12 - b** 
4*c**4*x**6 + 3*b**4*c**3*d*x**8 - 3*b**4*c**2*d**2*x**10 + b**4*c*d**3*x* 
*12),x)*a**4*c**2*d**3*x**2 + 2*int((sqrt(c - d*x**2)*sqrt(a + b*x**2)*x** 
4)/(a**4*c**3*d - 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 - a**4*d**4*x 
**6 - a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 - 12*a**3*b*c**2*d**2*x**4 + 10*a 
**3*b*c*d**3*x**6 - 3*a**3*b*d**4*x**8 - 3*a**2*b**2*c**4*x**2 + 12*a**...