Integrand size = 21, antiderivative size = 419 \[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d)^4 x}+\frac {\sqrt [4]{a} d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d)^4 x} \]
1/14*b*(7*a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^2+a)^(7/4)+1/42*b*(-21*a^2*d^ 2-76*a*b*c*d+20*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(b*x^2+a)^(3/4)-1/2*d*x/c/(- a*d+b*c)/(b*x^2+a)^(7/4)/(d*x^2+c)+1/42*(-21*a^2*d^2-76*a*b*c*d+20*b^2*c^2 )*(1+b*x^2/a)^(3/4)*(cos(1/2*arctan(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*a rctan(x*b^(1/2)/a^(1/2)))*EllipticF(sin(1/2*arctan(x*b^(1/2)/a^(1/2))),2^( 1/2))*b^(1/2)/a^(3/2)/c/(-a*d+b*c)^3/(b*x^2+a)^(3/4)+1/4*a^(1/4)*d^2*(-2*a *d+13*b*c)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^( 1/2),I)*(-b*x^2/a)^(1/2)/c/(-a*d+b*c)^4/x+1/4*a^(1/4)*d^2*(-2*a*d+13*b*c)* EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*(-b* x^2/a)^(1/2)/c/(-a*d+b*c)^4/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.65 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\frac {\frac {b d \left (-20 b^2 c^2+76 a b c d+21 a^2 d^2\right ) x^3 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{(-b c+a d)^3}+\frac {6 c \left (-6 a c x \left (42 a^4 d^3+63 a^3 b d^2 \left (-2 c+d x^2\right )-10 b^4 c^2 x^2 \left (3 c+2 d x^2\right )+a^2 b^2 d \left (126 c^2-38 c d x^2+21 d^2 x^4\right )+2 a b^3 c \left (-21 c^2+41 c d x^2+38 d^2 x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^3 \left (21 a^4 d^3+42 a^3 b d^3 x^2-20 b^4 c^2 x^2 \left (c+d x^2\right )+4 a b^3 c \left (-8 c^2+11 c d x^2+19 d^2 x^4\right )+a^2 b^2 d \left (88 c^2+88 c d x^2+21 d^2 x^4\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{(b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}}{252 a^2 c^2 \left (a+b x^2\right )^{3/4}} \]
((b*d*(-20*b^2*c^2 + 76*a*b*c*d + 21*a^2*d^2)*x^3*(1 + (b*x^2)/a)^(3/4)*Ap pellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(-(b*c) + a*d)^3 + ( 6*c*(-6*a*c*x*(42*a^4*d^3 + 63*a^3*b*d^2*(-2*c + d*x^2) - 10*b^4*c^2*x^2*( 3*c + 2*d*x^2) + a^2*b^2*d*(126*c^2 - 38*c*d*x^2 + 21*d^2*x^4) + 2*a*b^3*c *(-21*c^2 + 41*c*d*x^2 + 38*d^2*x^4))*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2) /a), -((d*x^2)/c)] + x^3*(21*a^4*d^3 + 42*a^3*b*d^3*x^2 - 20*b^4*c^2*x^2*( c + d*x^2) + 4*a*b^3*c*(-8*c^2 + 11*c*d*x^2 + 19*d^2*x^4) + a^2*b^2*d*(88* c^2 + 88*c*d*x^2 + 21*d^2*x^4))*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2 )/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d* x^2)/c)])))/((b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)*(6*a*c*AppellF1[1/2, 3/ 4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x ^2)/a), -((d*x^2)/c)]))))/(252*a^2*c^2*(a + b*x^2)^(3/4))
Time = 0.68 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {316, 27, 402, 27, 402, 27, 405, 231, 229, 312, 118, 25, 925, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {-9 b d x^2+4 b c-2 a d}{2 \left (b x^2+a\right )^{11/4} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 (2 b c-a d)-9 b d x^2}{\left (b x^2+a\right )^{11/4} \left (d x^2+c\right )}dx}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}-\frac {2 \int -\frac {20 b^2 c^2-56 a b d c+14 a^2 d^2+5 b d (4 b c+7 a d) x^2}{2 \left (b x^2+a\right )^{7/4} \left (d x^2+c\right )}dx}{7 a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {5 b d (4 b c+7 a d) x^2+2 \left (10 b^2 c^2-28 a b d c+7 a^2 d^2\right )}{\left (b x^2+a\right )^{7/4} \left (d x^2+c\right )}dx}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac {2 \int -\frac {20 b^3 c^3-76 a b^2 d c^2+252 a^2 b d^2 c-42 a^3 d^3+b d \left (20 b^2 c^2-76 a b d c-21 a^2 d^2\right ) x^2}{2 \left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b d \left (20 b^2 c^2-76 a b d c-21 a^2 d^2\right ) x^2+2 \left (10 b^3 c^3-38 a b^2 d c^2+126 a^2 b d^2 c-21 a^3 d^3\right )}{\left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {\frac {\frac {b \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \int \frac {1}{\left (b x^2+a\right )^{3/4}}dx+21 a^2 d^2 (13 b c-2 a d) \int \frac {1}{\left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 231 |
| \(\displaystyle \frac {\frac {\frac {\frac {b \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{3/4}}dx}{\left (a+b x^2\right )^{3/4}}+21 a^2 d^2 (13 b c-2 a d) \int \frac {1}{\left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 229 |
| \(\displaystyle \frac {\frac {\frac {21 a^2 d^2 (13 b c-2 a d) \int \frac {1}{\left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx+\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 312 |
| \(\displaystyle \frac {\frac {\frac {\frac {21 a^2 d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \int \frac {1}{\sqrt {-\frac {b x^2}{a}} \left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx^2}{2 x}+\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 118 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}-\frac {42 a^2 d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \int -\frac {1}{\sqrt {1-\frac {x^8}{a}} \left (d x^8+b c-a d\right )}d\sqrt [4]{b x^2+a}}{x}}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {42 a^2 d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \int \frac {1}{\sqrt {1-\frac {x^8}{a}} \left (d x^8+b c-a d\right )}d\sqrt [4]{b x^2+a}}{x}+\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}-\frac {42 a^2 d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \left (-\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^4}{\sqrt {a d-b c}}\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{b x^2+a}}{2 (b c-a d)}-\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^4}{\sqrt {a d-b c}}+1\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{b x^2+a}}{2 (b c-a d)}\right )}{x}}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}-\frac {42 a^2 d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \left (-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 (b c-a d)}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 (b c-a d)}\right )}{x}}{3 a (b c-a d)}+\frac {2 b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {2 b x (7 a d+4 b c)}{7 a \left (a+b x^2\right )^{7/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}\) |
-1/2*(d*x)/(c*(b*c - a*d)*(a + b*x^2)^(7/4)*(c + d*x^2)) + ((2*b*(4*b*c + 7*a*d)*x)/(7*a*(b*c - a*d)*(a + b*x^2)^(7/4)) + ((2*b*(20*b^2*c^2 - 76*a*b *c*d - 21*a^2*d^2)*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/4)) + ((2*Sqrt[a]*Sq rt[b]*(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*(1 + (b*x^2)/a)^(3/4)*Ellipti cF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(a + b*x^2)^(3/4) - (42*a^2*d^2*(13* b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*(-1/2*(a^(1/4)*EllipticPi[-((Sqrt[a]*Sqrt[ d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(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*(b*c - a*d))))/x)/(3*a*(b*c - a*d)))/(7*a* (b*c - a*d)))/(4*c*(b*c - a*d))
3.4.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 3/4)), x_] :> Simp[-4 Subst[Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] & & GtQ[-f/(d*e - c*f), 0]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( a + b*x^2)^(3/4) Int[1/(1 + b*(x^2/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim p[Sqrt[(-b)*(x^2/a)]/(2*x) Subst[Int[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)* (c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
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]
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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {11}{4}} \left (d \,x^{2}+c \right )^{2}}d x\]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {11}{4}} \left (c + d x^{2}\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {11}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {11}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{11/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]