\(\int \frac {A+B x^2+C x^4}{(a c+(b c+a d) x^2+b d x^4)^{5/2}} \, dx\) [9]

Optimal result

Integrand size = 38, antiderivative size = 593 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=-\frac {x \left (a c (b B c-2 a c C+a B d)-A \left (b^2 c^2+a^2 d^2\right )-\left (A b^2 c d+a^2 c C d+a b \left (c^2 C-2 B c d+A d^2\right )\right ) x^2\right )}{3 a c (b c-a d)^2 \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}}+\frac {\left (A b^2 c^2 d+a^2 d \left (5 c^2 C-B c d-2 A d^2\right )+a b c \left (3 c^2 C-7 B c d+9 A d^2\right )\right ) x}{3 a c^2 (b c-a d)^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {\sqrt {b} \left (2 A \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right )+a c \left (b^2 B c^2-2 a b c (4 c C-7 B d)-a^2 d (8 c C-B d)\right )\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{5/2} c (b c-a d)^4 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {\left (A b^3 c^2 d-3 a^3 c C d^2-a^2 b d \left (10 c^2 C-8 B c d-A d^2\right )-a b^2 c \left (3 c^2 C-8 B c d+18 A d^2\right )\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} c (b c-a d)^4 \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-1/3*x*(a*c*(B*a*d+B*b*c-2*C*a*c)-A*(a^2*d^2+b^2*c^2)-(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*(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))*x/a/c^2/(-a*d+b*c)^3/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+1/ 
3*b^(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^ 
2-2*a*b*c*(-7*B*d+4*C*c)-a^2*d*(-B*d+8*C*c)))*(b*x^2+a)*(a*(d*x^2+c)/c/(b* 
x^2+a))^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1 
/2))/a^(5/2)/c/(-a*d+b*c)^4/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)-1/3*(A*b^3*c 
^2*d-3*a^3*c*C*d^2-a^2*b*d*(-A*d^2-8*B*c*d+10*C*c^2)-a*b^2*c*(18*A*d^2-8*B 
*c*d+3*C*c^2))*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*InverseJacobiAM(a 
rctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(3/2)/b^(1/2)/c/(-a*d+b*c)^4 
/(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 = 13.30 (sec) , antiderivative size = 517, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\frac {i \left (i \sqrt {\frac {b}{a}} x \left (a^2 c d (b c-a d) \left (c^2 C-B c d+A d^2\right ) \left (a+b x^2\right )^2+a^2 d \left (-a d \left (-4 c^2 C+B c d+2 A d^2\right )+b c \left (4 c^2 C-7 B c d+10 A d^2\right )\right ) \left (a+b x^2\right )^2 \left (c+d x^2\right )+a b c^2 \left (A b^2+a (-b B+a C)\right ) (-b c+a d) \left (c+d x^2\right )^2-b c^2 \left (2 A b^2 (b c-5 a d)+a \left (b^2 B c-4 a b c C+7 a b B d-4 a^2 C d\right )\right ) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )-c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (-b \left (2 A \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right )+a c \left (b^2 B c^2+a^2 d (-8 c C+B d)+2 a b c (-4 c C+7 B d)\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) \left (a c \left (b^2 B c-5 a b c C+7 a b B d-3 a^2 C d\right )+A b \left (2 b^2 c^2-9 a b c d-a^2 d^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c-a d)^4 \left (\left (a+b x^2\right ) \left (c+d x^2\right )\right )^{3/2}} \] 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:

((I/3)*(I*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*b*c*C + 7*a*b*B*d - 4*a^2*C*d))*(a + b*x^2)*(c + d*x^2)^2) - 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*c*(b^2*B*c - 5*a*b*c*C + 7*a*b*B*d - 
3*a^2*C*d) + A*b*(2*b^2*c^2 - 9*a*b*c*d - a^2*d^2))*EllipticF[I*ArcSinh[Sq 
rt[b/a]*x], (a*d)/(b*c)])))/(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 = 1.50 (sec) , antiderivative size = 1094, normalized size of antiderivative = 1.84, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2206, 25, 1492, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2206

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1492

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {x \left (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+\left (b^2 c^2+a^2 d^2\right ) \left (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 )\right )-3 a c (b c+a d) \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 )\right )}{a c (b c-a d)^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {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} \sqrt {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 )\right ) \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\sqrt {b} \sqrt {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 ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {x \sqrt {b d x^4+(b c+a d) x^2+a c}}{\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}-\frac {x \left (-\left (\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\right )+a c (b B c-2 a C c+a B d)-A \left (b^2 c^2+a^2 d^2\right )\right )}{3 a c (b c-a d)^2 \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}\)

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:

-1/3*(x*(a*c*(b*B*c - 2*a*c*C + a*B*d) - A*(b^2*c^2 + a^2*d^2) - (A*b^2*c* 
d + a^2*c*C*d + a*b*(c^2*C - 2*B*c*d + A*d^2))*x^2))/(a*c*(b*c - a*d)^2*(a 
*c + (b*c + a*d)*x^2 + b*d*x^4)^(3/2)) + ((x*((b^2*c^2 + a^2*d^2)*(a*c*(b* 
B*c - 2*a*c*C + a*B*d) + 2*A*(b^2*c^2 - 3*a*b*c*d + a^2*d^2)) - 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*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 - 2* 
a*b*c*(4*c*C - 7*B*d) - a^2*d*(8*c*C - B*d)))*x^2))/(a*c*(b*c - a*d)^2*Sqr 
t[a*c + (b*c + a*d)*x^2 + b*d*x^4]) - (-(Sqrt[b]*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 - 2*a*b*c*(4*c*C 
- 7*B*d) - a^2*d*(8*c*C - B*d)))*(-((x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^ 
4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)) + (a^(1/4)*c^(1/4)*(Sqrt[a]*S 
qrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt 
[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*d^(1/4)* 
x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])) 
/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]))) + (a^(1/4)* 
c^(1/4)*(Sqrt[a]*Sqrt[c]*(A*b^3*c^2*d - 3*a^3*c*C*d^2 - a^2*b*d*(10*c^2*C 
- 8*B*c*d - A*d^2) - a*b^2*c*(3*c^2*C - 8*B*c*d + 18*A*d^2)) + Sqrt[b]*Sqr 
t[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 - 2*a*b*c*(4*c*C - 7*B*d) - a^2*d*(8*c*C - B*d))))*(Sqrt[a]*Sqrt[c] + 
 Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*S...
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

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 1509

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

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[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 = 1125, normalized size of antiderivative = 1.90

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1125\)
default	\(\text {Expression too large to display}\)	\(2051\)

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=_RETURNVERB 
OSE)
 

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)/(-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))+1/3*b*(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^2/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)*(Elliptic...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2688 vs. \(2 (572) = 1144\).

Time = 0.30 (sec) , antiderivative size = 2688, normalized size of antiderivative = 4.53 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\text {Too large to display} \] 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*b^2*c^2*d^3 + (2*A*a^3*b^4*d^5 - (8*C*a^2*b^5 - B*a*b^6 - 2 
*A*b^7)*c^3*d^2 - 2*(4*C*a^3*b^4 - 7*B*a^2*b^5 + 5*A*a*b^6)*c^2*d^3 + (B*a 
^3*b^4 - 10*A*a^2*b^5)*c*d^4)*x^8 + 2*(2*A*a^4*b^3*d^5 - (8*C*a^2*b^5 - B* 
a*b^6 - 2*A*b^7)*c^4*d - (16*C*a^3*b^4 - 15*B*a^2*b^5 + 8*A*a*b^6)*c^3*d^2 
 - (8*C*a^4*b^3 - 15*B*a^3*b^4 + 20*A*a^2*b^5)*c^2*d^3 + (B*a^4*b^3 - 8*A* 
a^3*b^4)*c*d^4)*x^6 - (8*C*a^4*b^3 - B*a^3*b^4 - 2*A*a^2*b^5)*c^5 - 2*(4*C 
*a^5*b^2 - 7*B*a^4*b^3 + 5*A*a^3*b^4)*c^4*d + (B*a^5*b^2 - 10*A*a^4*b^3)*c 
^3*d^2 + (2*A*a^5*b^2*d^5 - (8*C*a^2*b^5 - B*a*b^6 - 2*A*b^7)*c^5 - 2*(20* 
C*a^3*b^4 - 9*B*a^2*b^5 + A*a*b^6)*c^4*d - 2*(20*C*a^4*b^3 - 29*B*a^3*b^4 
+ 24*A*a^2*b^5)*c^3*d^2 - 2*(4*C*a^5*b^2 - 9*B*a^4*b^3 + 24*A*a^3*b^4)*c^2 
*d^3 + (B*a^5*b^2 - 2*A*a^4*b^3)*c*d^4)*x^4 + 2*(2*A*a^5*b^2*c*d^4 - (8*C* 
a^3*b^4 - B*a^2*b^5 - 2*A*a*b^6)*c^5 - (16*C*a^4*b^3 - 15*B*a^3*b^4 + 8*A* 
a^2*b^5)*c^4*d - (8*C*a^5*b^2 - 15*B*a^4*b^3 + 20*A*a^3*b^4)*c^3*d^2 + (B* 
a^5*b^2 - 8*A*a^4*b^3)*c^2*d^3)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsi 
n(x*sqrt(-b/a)), a*d/(b*c)) + (((3*C*a^3*b^4 + 8*C*a^2*b^5 - B*a*b^6 - 2*A 
*b^7)*c^3*d^2 + (10*C*a^4*b^3 - 8*(B - C)*a^3*b^4 - (A + 14*B)*a^2*b^5 + 1 
0*A*a*b^6)*c^2*d^3 + (3*C*a^5*b^2 - 8*B*a^4*b^3 + (18*A - B)*a^3*b^4 + 10* 
A*a^2*b^5)*c*d^4 - (A*a^4*b^3 + 2*A*a^3*b^4)*d^5)*x^8 + 2*((3*C*a^3*b^4 + 
8*C*a^2*b^5 - B*a*b^6 - 2*A*b^7)*c^4*d + (13*C*a^4*b^3 - 8*(B - 2*C)*a^3*b 
^4 - (A + 15*B)*a^2*b^5 + 8*A*a*b^6)*c^3*d^2 + (13*C*a^5*b^2 - 8*(2*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. \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c\right )}^{5/2}} \,d x \] 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 
+ 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**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 + 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*...