\(\int \frac {e+f x^2}{(a+b x^2)^{5/4} (c+d x^2)^3} \, dx\) [564]

Optimal result

Integrand size = 28, antiderivative size = 493 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=-\frac {(d e-c f) x}{4 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2}-\frac {(b c (15 d e-7 c f)-2 a d (3 d e+c f)) x}{16 c^2 (b c-a d)^2 \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (32 b^2 c^2 e+a b c (19 d e-43 c f)-2 a^2 d (3 d e+c f)\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{16 \sqrt {a} c^2 (b c-a d)^3 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \left (7 b^2 c^2 (11 d e-3 c f)+4 a^2 d^2 (3 d e+c f)-4 a b c d (11 d e+7 c f)\right ) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{32 c^2 \sqrt {d} (-b c+a d)^{7/2} x}-\frac {\sqrt [4]{a} \left (7 b^2 c^2 (11 d e-3 c f)+4 a^2 d^2 (3 d e+c f)-4 a b c d (11 d e+7 c f)\right ) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{32 c^2 \sqrt {d} (-b c+a d)^{7/2} x} \] Output:

-1/4*(-c*f+d*e)*x/c/(-a*d+b*c)/(b*x^2+a)^(1/4)/(d*x^2+c)^2-1/16*(b*c*(-7*c 
*f+15*d*e)-2*a*d*(c*f+3*d*e))*x/c^2/(-a*d+b*c)^2/(b*x^2+a)^(1/4)/(d*x^2+c) 
+1/16*b^(1/2)*(32*b^2*c^2*e+a*b*c*(-43*c*f+19*d*e)-2*a^2*d*(c*f+3*d*e))*(1 
+b*x^2/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(1 
/2)/c^2/(-a*d+b*c)^3/(b*x^2+a)^(1/4)+1/32*a^(1/4)*(7*b^2*c^2*(-3*c*f+11*d* 
e)+4*a^2*d^2*(c*f+3*d*e)-4*a*b*c*d*(7*c*f+11*d*e))*(-b*x^2/a)^(1/2)*Ellipt 
icPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c^2/d^(1/ 
2)/(a*d-b*c)^(7/2)/x-1/32*a^(1/4)*(7*b^2*c^2*(-3*c*f+11*d*e)+4*a^2*d^2*(c* 
f+3*d*e)-4*a*b*c*d*(7*c*f+11*d*e))*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^( 
1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c^2/d^(1/2)/(a*d-b*c)^(7/2 
)/x
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.12 (sec) , antiderivative size = 1540, normalized size of antiderivative = 3.12 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)^3),x]
 

Output:

(x*(-576*a*b^3*c^5*e*(c + d*x^2)*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), 
-((d*x^2)/c)] - 1728*a^2*b^2*c^4*d*e*(c + d*x^2)*AppellF1[1/2, 1/4, 1, 3/2 
, -((b*x^2)/a), -((d*x^2)/c)] + 900*a^3*b*c^3*d^2*e*(c + d*x^2)*AppellF1[1 
/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 216*a^4*c^2*d^3*e*(c + d*x^ 
2)*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 1152*a^2*b^2*c 
^5*f*(c + d*x^2)*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 
540*a^3*b*c^4*d*f*(c + d*x^2)*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -(( 
d*x^2)/c)] - 72*a^4*c^3*d^2*f*(c + d*x^2)*AppellF1[1/2, 1/4, 1, 3/2, -((b* 
x^2)/a), -((d*x^2)/c)] + 6*c*(4*a*c*d*(-(b*c) + a*d)*(-(d*e) + c*f)*(a + b 
*x^2) - a*d*(2*a*d*(3*d*e + c*f) + b*c*(-19*d*e + 11*c*f))*(a + b*x^2)*(c 
+ d*x^2) + 32*b^2*c^2*(b*e - a*f)*(c + d*x^2)^2)*(6*a*c*AppellF1[1/2, 1/4, 
 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/ 
2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/ 
a), -((d*x^2)/c)])) + 6*a^2*b*d^3*e*x^2*(1 + (b*x^2)/a)^(1/4)*(c + d*x^2)^ 
2*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]*(6*a*c*AppellF1[1 
/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/ 
4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -( 
(b*x^2)/a), -((d*x^2)/c)])) + 43*a*b^2*c^2*d*f*x^2*(1 + (b*x^2)/a)^(1/4)*( 
c + d*x^2)^2*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]*(6*a*c 
*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*Ap...
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {402, 27, 402, 27, 402, 27, 405, 227, 225, 212, 310, 993, 1542}

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.

\(\displaystyle \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {2 \int \frac {-7 d (b e-a f) x^2+b c e+a d e-2 a c f}{2 \sqrt [4]{b x^2+a} \left (d x^2+c\right )^3}dx}{a (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int \frac {-7 d (b e-a f) x^2+b c e+a d e-2 a c f}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )^3}dx}{a (b c-a d)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\int \frac {-2 d (3 d e+c f) a^2+16 b c (d e-c f) a-3 b d (8 b c e+a d e-9 a c f) x^2+8 b^2 c^2 e}{2 \sqrt [4]{b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\int \frac {2 \left (-d (3 d e+c f) a^2+8 b c (d e-c f) a+4 b^2 c^2 e\right )-3 b d (8 b c e+a d e-9 a c f) x^2}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )^2}dx}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {\int \frac {4 d^2 (3 d e+c f) a^3-10 b c d (5 d e+3 c f) a^2+32 b^2 c^2 (3 d e-2 c f) a+b d \left (-2 d (3 d e+c f) a^2+b c (19 d e-43 c f) a+32 b^2 c^2 e\right ) x^2+32 b^3 c^3 e}{2 \sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {\int \frac {b d \left (-2 d (3 d e+c f) a^2+b c (19 d e-43 c f) a+32 b^2 c^2 e\right ) x^2+2 \left (2 d^2 (3 d e+c f) a^3-5 b c d (5 d e+3 c f) a^2+16 b^2 c^2 (3 d e-2 c f) a+16 b^3 c^3 e\right )}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 405

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {a \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+b \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right ) \int \frac {1}{\sqrt [4]{b x^2+a}}dx}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {a \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right ) \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{\sqrt [4]{a+b x^2}}}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 225

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {a \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{\sqrt [4]{a+b x^2}}}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {a \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right ) \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{\sqrt [4]{a+b x^2}}}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {\frac {2 a \sqrt {-\frac {b x^2}{a}} \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c-a d+d \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right ) \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{\sqrt [4]{a+b x^2}}}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {\frac {2 a \sqrt {-\frac {b x^2}{a}} \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \left (\frac {\int \frac {1}{\left (\sqrt {a d-b c}+\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {a d-b c}-\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}\right )}{x}+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right ) \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{\sqrt [4]{a+b x^2}}}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\frac {\frac {2 a \sqrt {-\frac {b x^2}{a}} \left (4 a^2 d^2 (c f+3 d e)-4 a b c d (7 c f+11 d e)+7 b^2 c^2 (11 d e-3 c f)\right ) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}\right )}{x}+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right ) \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{\sqrt [4]{a+b x^2}}}{4 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} \left (-2 a^2 d (c f+3 d e)+a b c (19 d e-43 c f)+32 b^2 c^2 e\right )}{2 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4} (-9 a c f+a d e+8 b c e)}{4 c \left (c+d x^2\right )^2 (b c-a d)}}{a (b c-a d)}\)

Input:

Int[(e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)^3),x]
 

Output:

(2*(b*e - a*f)*x)/(a*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)^2) - (-1/4* 
(d*(8*b*c*e + a*d*e - 9*a*c*f)*x*(a + b*x^2)^(3/4))/(c*(b*c - a*d)*(c + d* 
x^2)^2) + (-1/2*(d*(32*b^2*c^2*e + a*b*c*(19*d*e - 43*c*f) - 2*a^2*d*(3*d* 
e + c*f))*x*(a + b*x^2)^(3/4))/(c*(b*c - a*d)*(c + d*x^2)) + ((b*(32*b^2*c 
^2*e + a*b*c*(19*d*e - 43*c*f) - 2*a^2*d*(3*d*e + c*f))*(1 + (b*x^2)/a)^(1 
/4)*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt[b]*x) 
/Sqrt[a]]/2, 2])/Sqrt[b]))/(a + b*x^2)^(1/4) + (2*a*(7*b^2*c^2*(11*d*e - 3 
*c*f) + 4*a^2*d^2*(3*d*e + c*f) - 4*a*b*c*d*(11*d*e + 7*c*f))*Sqrt[-((b*x^ 
2)/a)]*((a^(1/4)*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSi 
n[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqrt[-(b*c) + a*d]) - (a^(1/ 
4)*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/ 
4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqrt[-(b*c) + a*d])))/x)/(4*c*(b*c - a*d)))/( 
8*c*(b*c - a*d)))/(a*(b*c - a*d))
 

Defintions of rubi rules used

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 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 225

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) 
, x] - Simp[a   Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ 
a, 0] && PosQ[b/a]
 

rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a + b*x^2)^(1/4)   Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] 
&& PosQ[a]
 

rule 310

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[2*(Sqrt[(-b)*(x^2/a)]/x)   Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* 
x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 405

Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 
), x_Symbol] :> Simp[f/d   Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d 
 Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
 

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1542

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

Maple [F]

\[\int \frac {f \,x^{2}+e}{\left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (x^{2} d +c \right )^{3}}d x\]

Input:

int((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c)^3,x)
 

Output:

int((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c)^3,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((f*x**2+e)/(b*x**2+a)**(5/4)/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c)^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/4)*(d*x^2 + c)^3), x)
 

Giac [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{3}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/4)*(d*x^2 + c)^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=\int \frac {f\,x^2+e}{{\left (b\,x^2+a\right )}^{5/4}\,{\left (d\,x^2+c\right )}^3} \,d x \] Input:

int((e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)^3),x)
 

Output:

int((e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3} \, dx=\left (\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,c^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,c^{2} d \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a c \,d^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,d^{3} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b \,c^{3} x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} b \,c^{2} d \,x^{4}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} b c \,d^{2} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b \,d^{3} x^{8}}d x \right ) f +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,c^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,c^{2} d \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a c \,d^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,d^{3} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b \,c^{3} x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} b \,c^{2} d \,x^{4}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} b c \,d^{2} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b \,d^{3} x^{8}}d x \right ) e \] Input:

int((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c)^3,x)
 

Output:

int(x**2/((a + b*x**2)**(1/4)*a*c**3 + 3*(a + b*x**2)**(1/4)*a*c**2*d*x**2 
 + 3*(a + b*x**2)**(1/4)*a*c*d**2*x**4 + (a + b*x**2)**(1/4)*a*d**3*x**6 + 
 (a + b*x**2)**(1/4)*b*c**3*x**2 + 3*(a + b*x**2)**(1/4)*b*c**2*d*x**4 + 3 
*(a + b*x**2)**(1/4)*b*c*d**2*x**6 + (a + b*x**2)**(1/4)*b*d**3*x**8),x)*f 
 + int(1/((a + b*x**2)**(1/4)*a*c**3 + 3*(a + b*x**2)**(1/4)*a*c**2*d*x**2 
 + 3*(a + b*x**2)**(1/4)*a*c*d**2*x**4 + (a + b*x**2)**(1/4)*a*d**3*x**6 + 
 (a + b*x**2)**(1/4)*b*c**3*x**2 + 3*(a + b*x**2)**(1/4)*b*c**2*d*x**4 + 3 
*(a + b*x**2)**(1/4)*b*c*d**2*x**6 + (a + b*x**2)**(1/4)*b*d**3*x**8),x)*e