Integrand size = 25, antiderivative size = 632 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\frac {x \left (b^2 c^2+a^2 d^2+b d (b c+a d) x^2\right )}{5 a c (b c-a d)^2 \left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}}+\frac {x \left (4 b^4 c^4-17 a b^3 c^3 d-6 a^2 b^2 c^2 d^2-17 a^3 b c d^3+4 a^4 d^4-4 b d (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) x^2\right )}{15 a^2 c^2 (b c-a d)^4 \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}}+\frac {d \left (4 b^4 c^4-23 a b^3 c^3 d-150 a^2 b^2 c^2 d^2+49 a^3 b c d^3-8 a^4 d^4\right ) x}{15 a^2 c^3 (b c-a d)^5 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {\sqrt {b} (b c+a d) \left (8 b^4 c^4-61 a b^3 c^3 d+234 a^2 b^2 c^2 d^2-61 a^3 b c d^3+8 a^4 d^4\right ) \left (c+d x^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} c^3 (b c-a d)^6 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {2 \sqrt {b} d \left (2 b^4 c^4-13 a b^3 c^3 d+150 a^2 b^2 c^2 d^2-13 a^3 b c d^3+2 a^4 d^4\right ) \left (c+d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 a^{3/2} c^3 (b c-a d)^6 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:
1/5*x*(b^2*c^2+a^2*d^2+b*d*(a*d+b*c)*x^2)/a/c/(-a*d+b*c)^2/(a*c+(a*d+b*c)* x^2+b*d*x^4)^(5/2)+1/15*x*(4*b^4*c^4-17*a*b^3*c^3*d-6*a^2*b^2*c^2*d^2-17*a ^3*b*c*d^3+4*a^4*d^4-4*b*d*(a*d+b*c)*(8*a*b*c*d-(a*d+b*c)^2)*x^2)/a^2/c^2/ (-a*d+b*c)^4/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(3/2)+1/15*d*(-8*a^4*d^4+49*a^3*b *c*d^3-150*a^2*b^2*c^2*d^2-23*a*b^3*c^3*d+4*b^4*c^4)*x/a^2/c^3/(-a*d+b*c)^ 5/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+1/15*b^(1/2)*(a*d+b*c)*(8*a^4*d^4-61*a ^3*b*c*d^3+234*a^2*b^2*c^2*d^2-61*a*b^3*c^3*d+8*b^4*c^4)*(d*x^2+c)*Ellipti cE(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^3/(-a* d+b*c)^6/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2) -2/15*b^(1/2)*d*(2*a^4*d^4-13*a^3*b*c*d^3+150*a^2*b^2*c^2*d^2-13*a*b^3*c^3 *d+2*b^4*c^4)*(d*x^2+c)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b /c)^(1/2))/a^(3/2)/c^3/(-a*d+b*c)^6/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(a*c+( a*d+b*c)*x^2+b*d*x^4)^(1/2)
Result contains complex when optimal does not.
Time = 8.77 (sec) , antiderivative size = 554, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\frac {\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )} \left (\sqrt {\frac {b}{a}} x \left (3 a^3 c^2 d^4 (b c-a d)^2 \left (a+b x^2\right )^3+a^3 c d^4 (-b c+a d) (-23 b c+4 a d) \left (a+b x^2\right )^3 \left (c+d x^2\right )+a^3 d^4 \left (173 b^2 c^2-53 a b c d+8 a^2 d^2\right ) \left (a+b x^2\right )^3 \left (c+d x^2\right )^2+3 a^2 b^4 c^3 (b c-a d)^2 \left (c+d x^2\right )^3+a b^4 c^3 (-b c+a d) (-4 b c+23 a d) \left (a+b x^2\right ) \left (c+d x^2\right )^3+b^4 c^3 \left (8 b^2 c^2-53 a b c d+173 a^2 d^2\right ) \left (a+b x^2\right )^2 \left (c+d x^2\right )^3\right )+i b c \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (\left (8 b^5 c^5-53 a b^4 c^4 d+173 a^2 b^3 c^3 d^2+173 a^3 b^2 c^2 d^3-53 a^4 b c d^4+8 a^5 d^5\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+\left (-8 b^5 c^5+57 a b^4 c^4 d-199 a^2 b^3 c^3 d^2+127 a^3 b^2 c^2 d^3+27 a^4 b c d^4-4 a^5 d^5\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{15 a^3 \sqrt {\frac {b}{a}} c^3 (b c-a d)^6 \left (a+b x^2\right )^3 \left (c+d x^2\right )^3} \] Input:
Integrate[(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(-7/2),x]
Output:
(Sqrt[(a + b*x^2)*(c + d*x^2)]*(Sqrt[b/a]*x*(3*a^3*c^2*d^4*(b*c - a*d)^2*( a + b*x^2)^3 + a^3*c*d^4*(-(b*c) + a*d)*(-23*b*c + 4*a*d)*(a + b*x^2)^3*(c + d*x^2) + a^3*d^4*(173*b^2*c^2 - 53*a*b*c*d + 8*a^2*d^2)*(a + b*x^2)^3*( c + d*x^2)^2 + 3*a^2*b^4*c^3*(b*c - a*d)^2*(c + d*x^2)^3 + a*b^4*c^3*(-(b* c) + a*d)*(-4*b*c + 23*a*d)*(a + b*x^2)*(c + d*x^2)^3 + b^4*c^3*(8*b^2*c^2 - 53*a*b*c*d + 173*a^2*d^2)*(a + b*x^2)^2*(c + d*x^2)^3) + I*b*c*(a + b*x ^2)^2*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*((8*b^5*c^5 - 53*a*b^4*c^4*d + 173*a^2*b^3*c^3*d^2 + 173*a^3*b^2*c^2*d^3 - 53*a^4*b*c*d^ 4 + 8*a^5*d^5)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-8*b^5*c^ 5 + 57*a*b^4*c^4*d - 199*a^2*b^3*c^3*d^2 + 127*a^3*b^2*c^2*d^3 + 27*a^4*b* c*d^4 - 4*a^5*d^5)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(15*a ^3*Sqrt[b/a]*c^3*(b*c - a*d)^6*(a + b*x^2)^3*(c + d*x^2)^3)
Time = 2.32 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.73, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1405, 25, 1492, 25, 1492, 27, 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.
| \(\displaystyle \int \frac {1}{\left (x^2 (a d+b c)+a c+b d x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}-\frac {\int -\frac {7 b d (b c+a d) x^2+2 (b c-2 a d) (2 b c-a d)}{\left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}dx}{5 a c (b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {7 b d (b c+a d) x^2+2 (b c-2 a d) (2 b c-a d)}{\left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}dx}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {x \left (4 a^4 d^4-17 a^3 b c d^3+4 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-6 a^2 b^2 c^2 d^2-17 a b^3 c^3 d+4 b^4 c^4\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}-\frac {\int -\frac {8 b^4 c^4-37 a b^3 d c^3+90 a^2 b^2 d^2 c^2-37 a^3 b d^3 c+8 a^4 d^4+12 b d (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2}{\left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}dx}{3 a c (b c-a d)^2}}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {8 b^4 c^4-37 a b^3 d c^3+90 a^2 b^2 d^2 c^2-37 a^3 b d^3 c+8 a^4 d^4+12 b d (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2}{\left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}dx}{3 a c (b c-a d)^2}+\frac {x \left (4 a^4 d^4-17 a^3 b c d^3+4 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-6 a^2 b^2 c^2 d^2-17 a b^3 c^3 d+4 b^4 c^4\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {\frac {x \left (8 a^6 d^6-49 a^5 b c d^5+146 a^4 b^2 c^2 d^4+46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2+b d x^2 (a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )-49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {\int \frac {b d \left ((b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4-13 a b^3 d c^3+150 a^2 b^2 d^2 c^2-13 a^3 b d^3 c+2 a^4 d^4\right )\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (4 a^4 d^4-17 a^3 b c d^3+4 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-6 a^2 b^2 c^2 d^2-17 a b^3 c^3 d+4 b^4 c^4\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {x \left (8 a^6 d^6-49 a^5 b c d^5+146 a^4 b^2 c^2 d^4+46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2+b d x^2 (a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )-49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \int \frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4-13 a b^3 d c^3+150 a^2 b^2 d^2 c^2-13 a^3 b d^3 c+2 a^4 d^4\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (4 a^4 d^4-17 a^3 b c d^3+4 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-6 a^2 b^2 c^2 d^2-17 a b^3 c^3 d+4 b^4 c^4\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {\frac {x \left (8 a^6 d^6-49 a^5 b c d^5+146 a^4 b^2 c^2 d^4+46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2+b d x^2 (a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )-49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \left (\sqrt {a} \sqrt {c} \left (2 \sqrt {a} \sqrt {c} \left (2 a^4 d^4-13 a^3 b c d^3+150 a^2 b^2 c^2 d^2-13 a b^3 c^3 d+2 b^4 c^4\right )+\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )}{\sqrt {b} \sqrt {d}}\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {\sqrt {a} \sqrt {c} (a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (4 a^4 d^4-17 a^3 b c d^3+4 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-6 a^2 b^2 c^2 d^2-17 a b^3 c^3 d+4 b^4 c^4\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {x \left (8 a^6 d^6-49 a^5 b c d^5+146 a^4 b^2 c^2 d^4+46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2+b d x^2 (a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )-49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {b d \left (\sqrt {a} \sqrt {c} \left (2 \sqrt {a} \sqrt {c} \left (2 a^4 d^4-13 a^3 b c d^3+150 a^2 b^2 c^2 d^2-13 a b^3 c^3 d+2 b^4 c^4\right )+\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )}{\sqrt {b} \sqrt {d}}\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}+\frac {x \left (4 a^4 d^4-17 a^3 b c d^3+4 b d x^2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )-6 a^2 b^2 c^2 d^2-17 a b^3 c^3 d+4 b^4 c^4\right )}{3 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}}{5 a c (b c-a d)^2}+\frac {x \left (a^2 d^2+b d x^2 (a d+b c)+b^2 c^2\right )}{5 a c (b c-a d)^2 \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {x \left (b^2 c^2+a^2 d^2+b d (b c+a d) x^2\right )}{5 a c (b c-a d)^2 \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}+\frac {\frac {x \left (4 b^4 c^4-17 a b^3 d c^3-6 a^2 b^2 d^2 c^2-17 a^3 b d^3 c+4 a^4 d^4+4 b d (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2\right )}{3 a c (b c-a d)^2 \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}+\frac {\frac {x \left (8 b^6 c^6-49 a b^5 d c^5+146 a^2 b^4 d^2 c^4+46 a^3 b^3 d^3 c^3+146 a^4 b^2 d^4 c^2-49 a^5 b d^5 c+8 a^6 d^6+b d (b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) x^2\right )}{a c (b c-a d)^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c} \left (2 b^4 c^4-13 a b^3 d c^3+150 a^2 b^2 d^2 c^2-13 a^3 b d^3 c+2 a^4 d^4\right )+\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right )}{\sqrt {b} \sqrt {d}}\right ) \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}}{5 a c (b c-a d)^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {x \left (b^2 c^2+a^2 d^2+b d (b c+a d) x^2\right )}{5 a c (b c-a d)^2 \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}}+\frac {\frac {x \left (4 b^4 c^4-17 a b^3 d c^3-6 a^2 b^2 d^2 c^2-17 a^3 b d^3 c+4 a^4 d^4+4 b d (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) x^2\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}}+\frac {\frac {x \left (8 b^6 c^6-49 a b^5 d c^5+146 a^2 b^4 d^2 c^4+46 a^3 b^3 d^3 c^3+146 a^4 b^2 d^4 c^2-49 a^5 b d^5 c+8 a^6 d^6+b d (b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) x^2\right )}{a c (b c-a d)^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c} \left (2 b^4 c^4-13 a b^3 d c^3+150 a^2 b^2 d^2 c^2-13 a^3 b d^3 c+2 a^4 d^4\right )+\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right )}{\sqrt {b} \sqrt {d}}\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}}-\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\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 )}{\sqrt {b} \sqrt {d}}\right )}{a c (b c-a d)^2}}{3 a c (b c-a d)^2}}{5 a c (b c-a d)^2}\) |
Input:
Int[(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(-7/2),x]
Output:
(x*(b^2*c^2 + a^2*d^2 + b*d*(b*c + a*d)*x^2))/(5*a*c*(b*c - a*d)^2*(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(5/2)) + ((x*(4*b^4*c^4 - 17*a*b^3*c^3*d - 6*a^ 2*b^2*c^2*d^2 - 17*a^3*b*c*d^3 + 4*a^4*d^4 + 4*b*d*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*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*(8*b^6*c^6 - 49*a*b^5*c^5*d + 146*a^2*b^4*c^4*d^2 + 4 6*a^3*b^3*c^3*d^3 + 146*a^4*b^2*c^2*d^4 - 49*a^5*b*c*d^5 + 8*a^6*d^6 + b*d *(b*c + a*d)*(8*b^4*c^4 - 61*a*b^3*c^3*d + 234*a^2*b^2*c^2*d^2 - 61*a^3*b* c*d^3 + 8*a^4*d^4)*x^2))/(a*c*(b*c - a*d)^2*Sqrt[a*c + (b*c + a*d)*x^2 + b *d*x^4]) - (b*d*(-(((b*c + a*d)*(8*b^4*c^4 - 61*a*b^3*c^3*d + 234*a^2*b^2* c^2*d^2 - 61*a^3*b*c*d^3 + 8*a^4*d^4)*(-((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]*Sqrt[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])))/(Sqrt[ b]*Sqrt[d])) + (a^(1/4)*c^(1/4)*(2*Sqrt[a]*Sqrt[c]*(2*b^4*c^4 - 13*a*b^3*c ^3*d + 150*a^2*b^2*c^2*d^2 - 13*a^3*b*c*d^3 + 2*a^4*d^4) + ((b*c + a*d)*(8 *b^4*c^4 - 61*a*b^3*c^3*d + 234*a^2*b^2*c^2*d^2 - 61*a^3*b*c*d^3 + 8*a^4*d ^4))/(Sqrt[b]*Sqrt[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]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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]
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]
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]
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]
Time = 0.66 (sec) , antiderivative size = 1124, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1124\) |
| elliptic | \(\text {Expression too large to display}\) | \(1124\) |
Input:
int(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(7/2),x,method=_RETURNVERBOSE)
Output:
(1/5/a/c*(a*d+b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^2/d^2*x^3+1/5*(a^2*d^2+b^ 2*c^2)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3/d^3*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)^3+(4/15*(a*d+b*c)/b/d*(a^2*d^2 -6*a*b*c*d+b^2*c^2)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2*x^3+1/15*(4*a^4* d^4-17*a^3*b*c*d^3-6*a^2*b^2*c^2*d^2-17*a*b^3*c^3*d+4*b^4*c^4)/a^2/c^2/(a^ 2*d^2-2*a*b*c*d+b^2*c^2)^2/b^2/d^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*(-1/30*(8*a^5*d^5-53*a^4*b*c*d^4+1 73*a^3*b^2*c^2*d^3+173*a^2*b^3*c^3*d^2-53*a*b^4*c^4*d+8*b^5*c^5)/a^3/c^3/( a^2*d^2-2*a*b*c*d+b^2*c^2)^3*x^3-1/30*(8*a^6*d^6-49*a^5*b*c*d^5+146*a^4*b^ 2*c^2*d^4+46*a^3*b^3*c^3*d^3+146*a^2*b^4*c^4*d^2-49*a*b^5*c^5*d+8*b^6*c^6) /a^3/c^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)^3/b/d*x)/((x^4+(a*d+b*c)/b/d*x^2+a*c/ b/d)*b*d)^(1/2)+(1/15/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2*(8*a^4*d^4-37*a^3*b*c* d^3+90*a^2*b^2*c^2*d^2-37*a*b^3*c^3*d+8*b^4*c^4)/a^3/c^3-1/15*(8*a^6*d^6-4 9*a^5*b*c*d^5+146*a^4*b^2*c^2*d^4+46*a^3*b^3*c^3*d^3+146*a^2*b^4*c^4*d^2-4 9*a*b^5*c^5*d+8*b^6*c^6)/a^3/c^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)^3)/(-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))+1/15*b*(8*a^5*d^5-53*a ^4*b*c*d^4+173*a^3*b^2*c^2*d^3+173*a^2*b^3*c^3*d^2-53*a*b^4*c^4*d+8*b^5*c^ 5)/(a^2*d^2-2*a*b*c*d+b^2*c^2)^3/a^3/c^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)...
Leaf count of result is larger than twice the leaf count of optimal. 3257 vs. \(2 (607) = 1214\).
Time = 0.69 (sec) , antiderivative size = 3257, normalized size of antiderivative = 5.15 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(7/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\int \frac {1}{\left (a c + b d x^{4} + x^{2} \left (a d + b c\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate(1/(a*c+(a*d+b*c)*x**2+b*d*x**4)**(7/2),x)
Output:
Integral((a*c + b*d*x**4 + x**2*(a*d + b*c))**(-7/2), x)
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(7/2),x, algorithm="maxima")
Output:
integrate((b*d*x^4 + (b*c + a*d)*x^2 + a*c)^(-7/2), x)
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(7/2),x, algorithm="giac")
Output:
integrate((b*d*x^4 + (b*c + a*d)*x^2 + a*c)^(-7/2), x)
Timed out. \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\int \frac {1}{{\left (b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c\right )}^{7/2}} \,d x \] Input:
int(1/(a*c + x^2*(a*d + b*c) + b*d*x^4)^(7/2),x)
Output:
int(1/(a*c + x^2*(a*d + b*c) + b*d*x^4)^(7/2), x)
\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{7/2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{4} d^{4} x^{16}+4 a \,b^{3} d^{4} x^{14}+4 b^{4} c \,d^{3} x^{14}+6 a^{2} b^{2} d^{4} x^{12}+16 a \,b^{3} c \,d^{3} x^{12}+6 b^{4} c^{2} d^{2} x^{12}+4 a^{3} b \,d^{4} x^{10}+24 a^{2} b^{2} c \,d^{3} x^{10}+24 a \,b^{3} c^{2} d^{2} x^{10}+4 b^{4} c^{3} d \,x^{10}+a^{4} d^{4} x^{8}+16 a^{3} b c \,d^{3} x^{8}+36 a^{2} b^{2} c^{2} d^{2} x^{8}+16 a \,b^{3} c^{3} d \,x^{8}+b^{4} c^{4} x^{8}+4 a^{4} c \,d^{3} x^{6}+24 a^{3} b \,c^{2} d^{2} x^{6}+24 a^{2} b^{2} c^{3} d \,x^{6}+4 a \,b^{3} c^{4} x^{6}+6 a^{4} c^{2} d^{2} x^{4}+16 a^{3} b \,c^{3} d \,x^{4}+6 a^{2} b^{2} c^{4} x^{4}+4 a^{4} c^{3} d \,x^{2}+4 a^{3} b \,c^{4} x^{2}+a^{4} c^{4}}d x \] Input:
int(1/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(7/2),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**4*c**4 + 4*a**4*c**3*d*x**2 + 6*a**4*c**2*d**2*x**4 + 4*a**4*c*d**3*x**6 + a**4*d**4*x**8 + 4*a**3*b*c** 4*x**2 + 16*a**3*b*c**3*d*x**4 + 24*a**3*b*c**2*d**2*x**6 + 16*a**3*b*c*d* *3*x**8 + 4*a**3*b*d**4*x**10 + 6*a**2*b**2*c**4*x**4 + 24*a**2*b**2*c**3* d*x**6 + 36*a**2*b**2*c**2*d**2*x**8 + 24*a**2*b**2*c*d**3*x**10 + 6*a**2* b**2*d**4*x**12 + 4*a*b**3*c**4*x**6 + 16*a*b**3*c**3*d*x**8 + 24*a*b**3*c **2*d**2*x**10 + 16*a*b**3*c*d**3*x**12 + 4*a*b**3*d**4*x**14 + b**4*c**4* x**8 + 4*b**4*c**3*d*x**10 + 6*b**4*c**2*d**2*x**12 + 4*b**4*c*d**3*x**14 + b**4*d**4*x**16),x)