\(\int \frac {\sqrt {d-e x} \sqrt {d+e x}}{(a+c x^2)^{7/2}} \, dx\) [30]

Optimal result

Integrand size = 31, antiderivative size = 461 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{7/2}} \, dx=\frac {x \sqrt {d-e x} \sqrt {d+e x}}{5 a \left (a+c x^2\right )^{5/2}}+\frac {\left (4 c d^2+3 a e^2\right ) x \sqrt {d-e x} \sqrt {d+e x}}{15 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\left (8 c^2 d^4+13 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt {d-e x} \sqrt {d+e x}}{15 a^3 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {d e \left (8 c^2 d^4+13 a c d^2 e^2+3 a^2 e^4\right ) \sqrt {d-e x} \sqrt {d+e x} \sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} E\left (\arcsin \left (\frac {e x}{d}\right )|-\frac {c d^2}{a e^2}\right )}{15 a^3 c \left (c d^2+a e^2\right )^2 \sqrt {1+\frac {c x^2}{a}} \left (d^2-e^2 x^2\right )}-\frac {d e \left (4 c d^2+3 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{15 a^2 c \left (c d^2+a e^2\right ) \sqrt {a+c x^2} \left (d^2-e^2 x^2\right )} \] Output:

1/5*x*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/a/(c*x^2+a)^(5/2)+1/15*(3*a*e^2+4*c*d^2 
)*x*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/a^2/(a*e^2+c*d^2)/(c*x^2+a)^(3/2)+1/15*(3 
*a^2*e^4+13*a*c*d^2*e^2+8*c^2*d^4)*x*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/a^3/(a*e 
^2+c*d^2)^2/(c*x^2+a)^(1/2)+1/15*d*e*(3*a^2*e^4+13*a*c*d^2*e^2+8*c^2*d^4)* 
(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(1-e^2*x^2/d^2)^(1/2)*Ellipti 
cE(e*x/d,(-c*d^2/a/e^2)^(1/2))/a^3/c/(a*e^2+c*d^2)^2/(1+c*x^2/a)^(1/2)/(-e 
^2*x^2+d^2)-1/15*d*e*(3*a*e^2+4*c*d^2)*(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(1+c*x 
^2/a)^(1/2)*(1-e^2*x^2/d^2)^(1/2)*EllipticF(e*x/d,(-c*d^2/a/e^2)^(1/2))/a^ 
2/c/(a*e^2+c*d^2)/(c*x^2+a)^(1/2)/(-e^2*x^2+d^2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.40 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{7/2}} \, dx=\frac {x (d-e x) (d+e x) \left (3 a^2 \left (c d^2+a e^2\right )^2+a \left (c d^2+a e^2\right ) \left (4 c d^2+3 a e^2\right ) \left (a+c x^2\right )+\left (8 c^2 d^4+13 a c d^2 e^2+3 a^2 e^4\right ) \left (a+c x^2\right )^2\right )+\frac {i d^2 \left (a+c x^2\right )^2 \sqrt {1+\frac {c x^2}{a}} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (\left (8 c^2 d^4+13 a c d^2 e^2+3 a^2 e^4\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {c}{a}} x\right )|-\frac {a e^2}{c d^2}\right )-\left (8 c^2 d^4+17 a c d^2 e^2+9 a^2 e^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {c}{a}} x\right ),-\frac {a e^2}{c d^2}\right )\right )}{\sqrt {\frac {c}{a}}}}{15 a^3 \left (c d^2+a e^2\right )^2 \sqrt {d-e x} \sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \] Input:

Integrate[(Sqrt[d - e*x]*Sqrt[d + e*x])/(a + c*x^2)^(7/2),x]
 

Output:

(x*(d - e*x)*(d + e*x)*(3*a^2*(c*d^2 + a*e^2)^2 + a*(c*d^2 + a*e^2)*(4*c*d 
^2 + 3*a*e^2)*(a + c*x^2) + (8*c^2*d^4 + 13*a*c*d^2*e^2 + 3*a^2*e^4)*(a + 
c*x^2)^2) + (I*d^2*(a + c*x^2)^2*Sqrt[1 + (c*x^2)/a]*Sqrt[1 - (e^2*x^2)/d^ 
2]*((8*c^2*d^4 + 13*a*c*d^2*e^2 + 3*a^2*e^4)*EllipticE[I*ArcSinh[Sqrt[c/a] 
*x], -((a*e^2)/(c*d^2))] - (8*c^2*d^4 + 17*a*c*d^2*e^2 + 9*a^2*e^4)*Ellipt 
icF[I*ArcSinh[Sqrt[c/a]*x], -((a*e^2)/(c*d^2))]))/Sqrt[c/a])/(15*a^3*(c*d^ 
2 + a*e^2)^2*Sqrt[d - e*x]*Sqrt[d + e*x]*(a + c*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {648, 314, 25, 402, 25, 402, 25, 27, 399, 323, 323, 321, 331, 330, 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.

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac {\int -\frac {d^2 \left (8 c d^2+9 a e^2\right )-e^2 \left (4 c d^2+3 a e^2\right ) x^2}{\left (c x^2+a\right )^{3/2} \sqrt {d^2-e^2 x^2}}dx}{3 a \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\int \frac {d^2 \left (8 c d^2+9 a e^2\right )-e^2 \left (4 c d^2+3 a e^2\right ) x^2}{\left (c x^2+a\right )^{3/2} \sqrt {d^2-e^2 x^2}}dx}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\int -\frac {e^2 \left (2 a \left (2 c d^2+3 a e^2\right ) d^2+\left (8 c^2 d^4+13 a c e^2 d^2+3 a^2 e^4\right ) x^2\right )}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{a \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {\int \frac {e^2 \left (2 a \left (2 c d^2+3 a e^2\right ) d^2+\left (8 c^2 d^4+13 a c e^2 d^2+3 a^2 e^4\right ) x^2\right )}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \int \frac {2 a \left (2 c d^2+3 a e^2\right ) d^2+\left (8 c^2 d^4+13 a c e^2 d^2+3 a^2 e^4\right ) x^2}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {\left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {\sqrt {c x^2+a}}{\sqrt {d^2-e^2 x^2}}dx}{c}-\frac {a \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{c}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {\left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {\sqrt {c x^2+a}}{\sqrt {d^2-e^2 x^2}}dx}{c}-\frac {a \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {c x^2+a} \sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{c \sqrt {d^2-e^2 x^2}}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {\left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {\sqrt {c x^2+a}}{\sqrt {d^2-e^2 x^2}}dx}{c}-\frac {a \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{c \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {\left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {\sqrt {c x^2+a}}{\sqrt {d^2-e^2 x^2}}dx}{c}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 331

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {\sqrt {c x^2+a}}{\sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{c \sqrt {d^2-e^2 x^2}}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {\sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {\sqrt {\frac {c x^2}{a}+1}}{\sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{c \sqrt {\frac {c x^2}{a}+1} \sqrt {d^2-e^2 x^2}}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\frac {\frac {e^2 \left (\frac {d \sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right ) E\left (\arcsin \left (\frac {e x}{d}\right )|-\frac {c d^2}{a e^2}\right )}{c e \sqrt {\frac {c x^2}{a}+1} \sqrt {d^2-e^2 x^2}}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a e^2+c d^2\right ) \left (3 a e^2+4 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a^2 e^4+13 a c d^2 e^2+8 c^2 d^4\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{3 a \left (a e^2+c d^2\right )}+\frac {x \sqrt {d^2-e^2 x^2} \left (3 a e^2+4 c d^2\right )}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}}{5 a}+\frac {x \sqrt {d^2-e^2 x^2}}{5 a \left (a+c x^2\right )^{5/2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

Input:

Int[(Sqrt[d - e*x]*Sqrt[d + e*x])/(a + c*x^2)^(7/2),x]
 

Output:

(Sqrt[d - e*x]*Sqrt[d + e*x]*((x*Sqrt[d^2 - e^2*x^2])/(5*a*(a + c*x^2)^(5/ 
2)) + (((4*c*d^2 + 3*a*e^2)*x*Sqrt[d^2 - e^2*x^2])/(3*a*(c*d^2 + a*e^2)*(a 
 + c*x^2)^(3/2)) + (((8*c^2*d^4 + 13*a*c*d^2*e^2 + 3*a^2*e^4)*x*Sqrt[d^2 - 
 e^2*x^2])/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + (e^2*((d*(8*c^2*d^4 + 13* 
a*c*d^2*e^2 + 3*a^2*e^4)*Sqrt[a + c*x^2]*Sqrt[1 - (e^2*x^2)/d^2]*EllipticE 
[ArcSin[(e*x)/d], -((c*d^2)/(a*e^2))])/(c*e*Sqrt[1 + (c*x^2)/a]*Sqrt[d^2 - 
 e^2*x^2]) - (a*d*(c*d^2 + a*e^2)*(4*c*d^2 + 3*a*e^2)*Sqrt[1 + (c*x^2)/a]* 
Sqrt[1 - (e^2*x^2)/d^2]*EllipticF[ArcSin[(e*x)/d], -((c*d^2)/(a*e^2))])/(c 
*e*Sqrt[a + c*x^2]*Sqrt[d^2 - e^2*x^2])))/(a*(c*d^2 + a*e^2)))/(3*a*(c*d^2 
 + a*e^2)))/(5*a)))/Sqrt[d^2 - e^2*x^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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 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 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

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 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 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 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 648

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c 
*e + d*f*x^2)^FracPart[m])   Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && 
  !(EqQ[p, 2] && LtQ[m, -1])
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {\sqrt {\left (-e^{2} x^{2}+d^{2}\right ) \left (c \,x^{2}+a \right )}\, \left (\frac {x \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}}{5 c^{3} a \left (x^{2}+\frac {a}{c}\right )^{3}}+\frac {\left (3 a \,e^{2}+4 c \,d^{2}\right ) x \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}}{15 a^{2} \left (a \,e^{2}+c \,d^{2}\right ) c^{2} \left (x^{2}+\frac {a}{c}\right )^{2}}+\frac {\left (-x^{2} c \,e^{2}+c \,d^{2}\right ) x \left (3 a^{2} e^{4}+13 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 c \,a^{3} \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (-x^{2} c \,e^{2}+c \,d^{2}\right )}}+\frac {\left (-\frac {e^{2} \left (3 a \,e^{2}+4 c \,d^{2}\right )}{15 c \,a^{2} \left (a \,e^{2}+c \,d^{2}\right )}+\frac {3 a^{2} e^{4}+13 a c \,d^{2} e^{2}+8 c^{2} d^{4}}{15 \left (a \,e^{2}+c \,d^{2}\right ) c \,a^{3}}-\frac {d^{2} \left (3 a^{2} e^{4}+13 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 a^{3} \left (a \,e^{2}+c \,d^{2}\right )^{2}}\right ) \sqrt {1-\frac {e^{2} x^{2}}{d^{2}}}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )}{\sqrt {\frac {e^{2}}{d^{2}}}\, \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}}-\frac {e^{2} \left (3 a^{2} e^{4}+13 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) \sqrt {1-\frac {e^{2} x^{2}}{d^{2}}}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )\right )}{15 a^{2} \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {\frac {e^{2}}{d^{2}}}\, \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}\, c}\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(711\)
default	\(\text {Expression too large to display}\)	\(1672\)

Input:

int((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

((-e^2*x^2+d^2)*(c*x^2+a))^(1/2)/(-e*x+d)^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1 
/2)*(1/5/c^3/a*x*(-c*e^2*x^4-a*e^2*x^2+c*d^2*x^2+a*d^2)^(1/2)/(x^2+a/c)^3+ 
1/15*(3*a*e^2+4*c*d^2)/a^2/(a*e^2+c*d^2)/c^2*x*(-c*e^2*x^4-a*e^2*x^2+c*d^2 
*x^2+a*d^2)^(1/2)/(x^2+a/c)^2+1/15*(-c*e^2*x^2+c*d^2)/c/a^3/(a*e^2+c*d^2)^ 
2*x*(3*a^2*e^4+13*a*c*d^2*e^2+8*c^2*d^4)/((x^2+a/c)*(-c*e^2*x^2+c*d^2))^(1 
/2)+(-1/15*e^2*(3*a*e^2+4*c*d^2)/c/a^2/(a*e^2+c*d^2)+1/15/(a*e^2+c*d^2)/c* 
(3*a^2*e^4+13*a*c*d^2*e^2+8*c^2*d^4)/a^3-1/15*d^2/a^3/(a*e^2+c*d^2)^2*(3*a 
^2*e^4+13*a*c*d^2*e^2+8*c^2*d^4))/(e^2/d^2)^(1/2)*(1-e^2*x^2/d^2)^(1/2)*(1 
+c/a*x^2)^(1/2)/(-c*e^2*x^4-a*e^2*x^2+c*d^2*x^2+a*d^2)^(1/2)*EllipticF(x*( 
e^2/d^2)^(1/2),(-1-(-a*e^2+c*d^2)/a/e^2)^(1/2))-1/15*e^2*(3*a^2*e^4+13*a*c 
*d^2*e^2+8*c^2*d^4)/a^2/(a*e^2+c*d^2)^2/(e^2/d^2)^(1/2)*(1-e^2*x^2/d^2)^(1 
/2)*(1+c/a*x^2)^(1/2)/(-c*e^2*x^4-a*e^2*x^2+c*d^2*x^2+a*d^2)^(1/2)/c*(Elli 
pticF(x*(e^2/d^2)^(1/2),(-1-(-a*e^2+c*d^2)/a/e^2)^(1/2))-EllipticE(x*(e^2/ 
d^2)^(1/2),(-1-(-a*e^2+c*d^2)/a/e^2)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(7/2),x, algorithm="frica 
s")
 

Output:

1/15*(((8*c^5*d^7 + 13*a*c^4*d^5*e^2 + 3*a^2*c^3*d^3*e^4)*x^5 + (20*a*c^4* 
d^7 + 33*a^2*c^3*d^5*e^2 + 9*a^3*c^2*d^3*e^4)*x^3 + (15*a^2*c^3*d^7 + 26*a 
^3*c^2*d^5*e^2 + 9*a^4*c*d^3*e^4)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d)*sqrt(-e 
*x + d) + sqrt(a*d^2)*((8*a^3*c^2*d^4*e^3 + 13*a^4*c*d^2*e^5 + 3*a^5*e^7 + 
 (8*c^5*d^4*e^3 + 13*a*c^4*d^2*e^5 + 3*a^2*c^3*e^7)*x^6 + 3*(8*a*c^4*d^4*e 
^3 + 13*a^2*c^3*d^2*e^5 + 3*a^3*c^2*e^7)*x^4 + 3*(8*a^2*c^3*d^4*e^3 + 13*a 
^3*c^2*d^2*e^5 + 3*a^4*c*e^7)*x^2)*elliptic_e(arcsin(e*x/d), -c*d^2/(a*e^2 
)) + (4*a^3*c^2*d^6*e - 13*a^4*c*d^2*e^5 - 3*a^5*e^7 + 2*(3*a^4*c - 4*a^3* 
c^2)*d^4*e^3 + (4*c^5*d^6*e - 13*a*c^4*d^2*e^5 - 3*a^2*c^3*e^7 + 2*(3*a*c^ 
4 - 4*c^5)*d^4*e^3)*x^6 + 3*(4*a*c^4*d^6*e - 13*a^2*c^3*d^2*e^5 - 3*a^3*c^ 
2*e^7 + 2*(3*a^2*c^3 - 4*a*c^4)*d^4*e^3)*x^4 + 3*(4*a^2*c^3*d^6*e - 13*a^3 
*c^2*d^2*e^5 - 3*a^4*c*e^7 + 2*(3*a^3*c^2 - 4*a^2*c^3)*d^4*e^3)*x^2)*ellip 
tic_f(arcsin(e*x/d), -c*d^2/(a*e^2))))/(a^6*c^3*d^7 + 2*a^7*c^2*d^5*e^2 + 
a^8*c*d^3*e^4 + (a^3*c^6*d^7 + 2*a^4*c^5*d^5*e^2 + a^5*c^4*d^3*e^4)*x^6 + 
3*(a^4*c^5*d^7 + 2*a^5*c^4*d^5*e^2 + a^6*c^3*d^3*e^4)*x^4 + 3*(a^5*c^4*d^7 
 + 2*a^6*c^3*d^5*e^2 + a^7*c^2*d^3*e^4)*x^2)
 

Sympy [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {d - e x} \sqrt {d + e x}}{\left (a + c x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

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

Output:

Integral(sqrt(d - e*x)*sqrt(d + e*x)/(a + c*x**2)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {e x + d} \sqrt {-e x + d}}{{\left (c x^{2} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(7/2),x, algorithm="maxim 
a")
 

Output:

integrate(sqrt(e*x + d)*sqrt(-e*x + d)/(c*x^2 + a)^(7/2), x)
 

Giac [F(-1)]

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

integrate((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(7/2),x, algorithm="giac" 
)
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(((d + e*x)^(1/2)*(d - e*x)^(1/2))/(a + c*x^2)^(7/2),x)
 

Output:

int(((d + e*x)^(1/2)*(d - e*x)^(1/2))/(a + c*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \sqrt {c \,x^{2}+a}}{c^{4} x^{8}+4 a \,c^{3} x^{6}+6 a^{2} c^{2} x^{4}+4 a^{3} c \,x^{2}+a^{4}}d x \] Input:

int((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(7/2),x)
 

Output:

int((sqrt(d + e*x)*sqrt(d - e*x)*sqrt(a + c*x**2))/(a**4 + 4*a**3*c*x**2 + 
 6*a**2*c**2*x**4 + 4*a*c**3*x**6 + c**4*x**8),x)