\(\int \frac {1}{(a c+(b c-a d) x^2-b d x^4)^{7/2}} \, dx\) [25]

Optimal result

Integrand size = 27, antiderivative size = 722 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\frac {x \left (b^2 c^2+a^2 d^2-b d (b c-a d) x^2\right )}{5 a c (b c+a d)^2 \left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}}+\frac {x \left (4 b^4 c^4+17 a b^3 c^3 d-6 a^2 b^2 c^2 d^2+17 a^3 b c d^3+4 a^4 d^4-4 b d (b c-a d) \left (8 a b c d+(b c-a d)^2\right ) x^2\right )}{15 a^2 c^2 (b c+a d)^4 \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}}+\frac {x \left (8 b^6 c^6+49 a b^5 c^5 d+146 a^2 b^4 c^4 d^2-46 a^3 b^3 c^3 d^3+146 a^4 b^2 c^2 d^4+49 a^5 b c d^5+8 a^6 d^6-b d (b c-a d) \left (8 b^4 c^4+61 a b^3 c^3 d+234 a^2 b^2 c^2 d^2+61 a^3 b c d^3+8 a^4 d^4\right ) x^2\right )}{15 a^3 c^3 (b c+a d)^6 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {\sqrt {d} (b c-a d) \left (8 b^4 c^4+61 a b^3 c^3 d+234 a^2 b^2 c^2 d^2+61 a^3 b c d^3+8 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} (b c+a d)^6 \sqrt {a c+(b c-a d) x^2-b d x^4}}-\frac {\sqrt {d} \left (4 b^4 c^4+23 a b^3 c^3 d-150 a^2 b^2 c^2 d^2-49 a^3 b c d^3-8 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} (b c+a d)^5 \sqrt {a c+(b c-a d) x^2-b d x^4}} \] 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*x*(8*b^6*c^6+49*a*b 
^5*c^5*d+146*a^2*b^4*c^4*d^2-46*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*(-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)*x^2)/a^3/c^3/(a*d+b*c)^6/(a*c+(-a*d+b*c)*x 
^2-b*d*x^4)^(1/2)+1/15*d^(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)*(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^2/c^(5/2)/(a*d+b*c)^6/(a 
*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)-1/15*d^(1/2)*(-8*a^4*d^4-49*a^3*b*c*d^3-1 
50*a^2*b^2*c^2*d^2+23*a*b^3*c^3*d+4*b^4*c^4)*(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^2/c^(5/2)/(a*d+b*c) 
^5/(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 = 8.71 (sec) , antiderivative size = 560, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\frac {\sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )} \left (\sqrt {\frac {b}{a}} x \left (3 a^3 c^2 d^4 (b c+a d)^2 \left (a+b x^2\right )^3+a^3 c d^4 (b c+a d) (23 b c+4 a d) \left (a+b x^2\right )^3 \left (c-d x^2\right )+a^3 d^4 \left (173 b^2 c^2+53 a b c d+8 a^2 d^2\right ) \left (a+b x^2\right )^3 \left (c-d x^2\right )^2+3 a^2 b^4 c^3 (b c+a d)^2 \left (c-d x^2\right )^3+a b^4 c^3 (b c+a d) (4 b c+23 a d) \left (a+b x^2\right ) \left (c-d x^2\right )^3+b^4 c^3 \left (8 b^2 c^2+53 a b c d+173 a^2 d^2\right ) \left (a+b x^2\right )^2 \left (c-d x^2\right )^3\right )-i b c \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}} \left (c-d x^2\right )^2 \sqrt {1-\frac {d x^2}{c}} \left (\left (-8 b^5 c^5-53 a b^4 c^4 d-173 a^2 b^3 c^3 d^2+173 a^3 b^2 c^2 d^3+53 a^4 b c d^4+8 a^5 d^5\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+\left (8 b^5 c^5+57 a b^4 c^4 d+199 a^2 b^3 c^3 d^2+127 a^3 b^2 c^2 d^3-27 a^4 b c d^4-4 a^5 d^5\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )\right )}{15 a^3 \sqrt {\frac {b}{a}} c^3 (b c+a d)^6 \left (a+b x^2\right )^3 \left (c-d x^2\right )^3} \] 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*Sq 
rt[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 + 5 
7*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)
 

Rubi [A] (verified)

Time = 1.92 (sec) , antiderivative size = 711, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1405, 25, 1492, 25, 1492, 25, 27, 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.

\(\displaystyle \int \frac {1}{\left (x^2 (b c-a d)+a c-b d x^4\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}-\frac {\int -\frac {2 (2 b c+a d) (b c+2 a d)-7 b d (b c-a d) x^2}{\left (-b d x^4+(b c-a d) x^2+a c\right )^{5/2}}dx}{5 a c (a d+b c)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 (2 b c+a d) (b c+2 a d)-7 b d (b c-a d) x^2}{\left (-b d x^4+(b c-a d) x^2+a c\right )^{5/2}}dx}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}-\frac {\int -\frac {8 b^4 c^4+37 a b^3 d c^3+90 a^2 b^2 d^2 c^2+37 a^3 b d^3 c+8 a^4 d^4-12 b d (b c-a d) \left (b^2 c^2+6 a b d c+a^2 d^2\right ) x^2}{\left (-b d x^4+(b c-a d) x^2+a c\right )^{3/2}}dx}{3 a c (a d+b c)^2}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {8 b^4 c^4+37 a b^3 d c^3+90 a^2 b^2 d^2 c^2+37 a^3 b d^3 c+8 a^4 d^4-12 b d (b c-a d) \left (b^2 c^2+6 a b d c+a^2 d^2\right ) x^2}{\left (-b d x^4+(b c-a d) x^2+a c\right )^{3/2}}dx}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {\int -\frac {b d \left ((b c-a d) \left (8 b^4 c^4+61 a b^3 d c^3+234 a^2 b^2 d^2 c^2+61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4+13 a b^3 d c^3+150 a^2 b^2 d^2 c^2+13 a^3 b d^3 c+2 a^4 d^4\right )\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {b d \left ((b c-a d) \left (8 b^4 c^4+61 a b^3 d c^3+234 a^2 b^2 d^2 c^2+61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4+13 a b^3 d c^3+150 a^2 b^2 d^2 c^2+13 a^3 b d^3 c+2 a^4 d^4\right )\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}+\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {b d \int \frac {(b c-a d) \left (8 b^4 c^4+61 a b^3 d c^3+234 a^2 b^2 d^2 c^2+61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4+13 a b^3 d c^3+150 a^2 b^2 d^2 c^2+13 a^3 b d^3 c+2 a^4 d^4\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}+\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {(b c-a d) \left (8 b^4 c^4+61 a b^3 d c^3+234 a^2 b^2 d^2 c^2+61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4+13 a b^3 d c^3+150 a^2 b^2 d^2 c^2+13 a^3 b d^3 c+2 a^4 d^4\right )}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a (a d+b c) \left (-8 a^4 d^4-49 a^3 b c d^3-150 a^2 b^2 c^2 d^2+23 a b^3 c^3 d+4 b^4 c^4\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{b}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a \sqrt {c} (a d+b c) \left (-8 a^4 d^4-49 a^3 b c d^3-150 a^2 b^2 c^2 d^2+23 a b^3 c^3 d+4 b^4 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}}{5 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{5 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}}+\frac {\frac {x \left (-4 b d x^2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) (b c-a d)+2 (a d+2 b c) (2 a d+b c) \left (a^2 d^2+b^2 c^2\right )+7 a b c d (b c-a d)^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}+\frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a \sqrt {c} (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d}}-\frac {a \sqrt {c} (a d+b c) \left (-8 a^4 d^4-49 a^3 b c d^3-150 a^2 b^2 c^2 d^2+23 a b^3 c^3 d+4 b^4 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (8 a^6 d^6+49 a^5 b c d^5+146 a^4 b^2 c^2 d^4-46 a^3 b^3 c^3 d^3+146 a^2 b^4 c^4 d^2-b d x^2 (b c-a d) \left (8 a^4 d^4+61 a^3 b c d^3+234 a^2 b^2 c^2 d^2+61 a b^3 c^3 d+8 b^4 c^4\right )+49 a b^5 c^5 d+8 b^6 c^6\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}}{5 a c (a d+b c)^2}\)

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*(7*a*b*c*d*(b*c - a*d)^2 + 2*(2*b* 
c + a*d)*(b*c + 2*a*d)*(b^2*c^2 + a^2*d^2) - 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*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*((a*Sqrt[c]*(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)*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqr 
t[d]) - (a*Sqrt[c]*(b*c + a*d)*(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)*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]] 
, -((b*c)/(a*d))])/(b*Sqrt[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))/(5*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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 1405

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]
 

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]
 

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 1144, normalized size of antiderivative = 1.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1144\)
elliptic	\(\text {Expression too large to display}\)	\(1144\)

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 
+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^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+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)/(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))+1/15*d*(8*a^5*d^5+5 
3*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^2/c^3/(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/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3272 vs. \(2 (681) = 1362\).

Time = 0.73 (sec) , antiderivative size = 3272, normalized size of antiderivative = 4.53 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\text {Too large to display} \] 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
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a*c+(-a*d+b*c)*x**2-b*d*x**4)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {7}{2}}} \,d x } \] 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)
 

Giac [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {7}{2}}} \,d x } \] 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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\int \frac {1}{{\left (-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c\right )}^{7/2}} \,d x \] 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)
 

Reduce [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{7/2}} \, dx=\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{4} d^{4} x^{16}+4 a \,b^{3} d^{4} x^{14}-4 b^{4} c \,d^{3} x^{14}+6 a^{2} b^{2} d^{4} x^{12}-16 a \,b^{3} c \,d^{3} x^{12}+6 b^{4} c^{2} d^{2} x^{12}+4 a^{3} b \,d^{4} x^{10}-24 a^{2} b^{2} c \,d^{3} x^{10}+24 a \,b^{3} c^{2} d^{2} x^{10}-4 b^{4} c^{3} d \,x^{10}+a^{4} d^{4} x^{8}-16 a^{3} b c \,d^{3} x^{8}+36 a^{2} b^{2} c^{2} d^{2} x^{8}-16 a \,b^{3} c^{3} d \,x^{8}+b^{4} c^{4} x^{8}-4 a^{4} c \,d^{3} x^{6}+24 a^{3} b \,c^{2} d^{2} x^{6}-24 a^{2} b^{2} c^{3} d \,x^{6}+4 a \,b^{3} c^{4} x^{6}+6 a^{4} c^{2} d^{2} x^{4}-16 a^{3} b \,c^{3} d \,x^{4}+6 a^{2} b^{2} c^{4} x^{4}-4 a^{4} c^{3} d \,x^{2}+4 a^{3} b \,c^{4} x^{2}+a^{4} c^{4}}d x \] 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)