Integrand size = 22, antiderivative size = 341 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\frac {e^2 \left (3 c d^2+2 a e^2\right ) x}{6 a d \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}+\frac {e^2 \left (9 c^2 d^4+59 a c d^2 e^2-8 a^2 e^4\right ) x}{12 a d^2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x^2}}+\frac {c x \left (d-e x^2\right )}{4 a \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )}+\frac {c \left (3 \sqrt {c} d-8 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2}}+\frac {c \left (3 \sqrt {c} d+8 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{7/2}} \] Output:
1/6*e^2*(2*a*e^2+3*c*d^2)*x/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)^(3/2)+1/12*e^2* (-8*a^2*e^4+59*a*c*d^2*e^2+9*c^2*d^4)*x/a/d^2/(-a*e^2+c*d^2)^3/(e*x^2+d)^( 1/2)+1/4*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(e*x^2+d)^(3/2)/(-c*x^4+a)+1/8*c* (3*c^(1/2)*d-8*a^(1/2)*e)*arctan((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e* x^2+d)^(1/2))/a^(7/4)/(c^(1/2)*d-a^(1/2)*e)^(7/2)+1/8*c*(3*c^(1/2)*d+8*a^( 1/2)*e)*arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^( 7/4)/(c^(1/2)*d+a^(1/2)*e)^(7/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.58 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\frac {-48 c e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {9 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-8 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-5 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]+\frac {\frac {2 x \left (-3 c^3 d^4 \left (d-3 e x^2\right ) \left (d+e x^2\right )^2+4 a^3 e^6 \left (3 d+2 e x^2\right )-4 a^2 c e^4 \left (15 d^3+14 d^2 e x^2+3 d e^2 x^4+2 e^3 x^6\right )+a c^2 d^2 e^2 \left (-9 d^3-15 d^2 e x^2+57 d e^2 x^4+59 e^3 x^6\right )\right )}{d^2 \left (d+e x^2\right )^{3/2} \left (-a+c x^4\right )}+3 c e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c^2 d^5 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+123 a c d^3 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-128 a^2 d e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-14 c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+22 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+16 a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-5 a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{a}}{24 \left (c d^2-a e^2\right )^3} \] Input:
Integrate[1/((d + e*x^2)^(5/2)*(a - c*x^4)^2),x]
Output:
(-48*c*e^(7/2)*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 - 16*a*e^2*#1^2 - 4*c*d*#1^3 + c*#1^4 & , (9*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e *x^2] - #1] - 8*a*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1 ] - 5*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - a*e^2 *Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*d*Log[d + 2*e* x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 + 8*a*e^ 2*#1 + 3*c*d*#1^2 - c*#1^3) & ] + ((2*x*(-3*c^3*d^4*(d - 3*e*x^2)*(d + e*x ^2)^2 + 4*a^3*e^6*(3*d + 2*e*x^2) - 4*a^2*c*e^4*(15*d^3 + 14*d^2*e*x^2 + 3 *d*e^2*x^4 + 2*e^3*x^6) + a*c^2*d^2*e^2*(-9*d^3 - 15*d^2*e*x^2 + 57*d*e^2* x^4 + 59*e^3*x^6)))/(d^2*(d + e*x^2)^(3/2)*(-a + c*x^4)) + 3*c*e^(3/2)*Roo tSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 - 16*a*e^2*#1^2 - 4*c*d*#1^3 + c*#1 ^4 & , (c^2*d^5*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 123* a*c*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 128*a^2* d*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 14*c^2*d^4*Log [d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 22*a*c*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 16*a^2*e^4*Log[d + 2*e *x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c^2*d^3*Log[d + 2*e*x^2 - 2* Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 5*a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqr t[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 + 8*a*e^2*#1 + 3*c* d*#1^2 - c*#1^3) & ])/a)/(24*(c*d^2 - a*e^2)^3)
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.
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^2 \left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {1}{\left (a-c x^4\right )^2 \left (d+e x^2\right )^{5/2}}dx\) |
Input:
Int[1/((d + e*x^2)^(5/2)*(a - c*x^4)^2),x]
Output:
$Aborted
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
Time = 1.08 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.43
| method | result | size |
| pseudoelliptic | \(-\frac {21 \left (\left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{2} \left (\frac {\left (\frac {8}{3} a^{2} e^{4}+5 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {d^{2} a c}}{7}+a c \,d^{2} e \left (a \,e^{2}-\frac {c \,d^{2}}{21}\right )\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (-c \,x^{4}+a \right ) c \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\left (\left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{2} \left (\frac {\left (-\frac {8}{3} a^{2} e^{4}-5 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {d^{2} a c}}{7}+a c \,d^{2} e \left (a \,e^{2}-\frac {c \,d^{2}}{21}\right )\right ) \left (-c \,x^{4}+a \right ) c \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )-\frac {8 x \left (e^{6} \left (\frac {2 e \,x^{2}}{3}+d \right ) a^{3}-5 \left (\frac {2}{15} e^{3} x^{6}+\frac {1}{5} d \,e^{2} x^{4}+\frac {14}{15} d^{2} e \,x^{2}+d^{3}\right ) c \,e^{4} a^{2}-\frac {3 d^{2} \left (-\frac {59}{9} e^{3} x^{6}-\frac {19}{3} d \,e^{2} x^{4}+\frac {5}{3} d^{2} e \,x^{2}+d^{3}\right ) c^{2} e^{2} a}{4}-\frac {c^{3} d^{4} \left (-3 e \,x^{2}+d \right ) \left (e \,x^{2}+d \right )^{2}}{4}\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}}{21}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\right )}{8 \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (-c \,x^{4}+a \right ) \left (a \,e^{2}-c \,d^{2}\right )^{3} a \,d^{2}}\) | \(488\) |
| default | \(\text {Expression too large to display}\) | \(8296\) |
Input:
int(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-21/8/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)/(e*x^2+d)^(3/2)/(d^2*a*c)^(1/2)*((e* x^2+d)^(3/2)*d^2*(1/7*(8/3*a^2*e^4+5*a*c*d^2*e^2-c^2*d^4)*(d^2*a*c)^(1/2)+ a*c*d^2*e*(a*e^2-1/21*c*d^2))*((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(-c*x^4+a)*c *arctan((e*x^2+d)^(1/2)/x*a/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))+((e*x^2+d)^( 3/2)*d^2*(1/7*(-8/3*a^2*e^4-5*a*c*d^2*e^2+c^2*d^4)*(d^2*a*c)^(1/2)+a*c*d^2 *e*(a*e^2-1/21*c*d^2))*(-c*x^4+a)*c*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2 *a*c)^(1/2))*a)^(1/2))-8/21*x*(e^6*(2/3*e*x^2+d)*a^3-5*(2/15*e^3*x^6+1/5*d *e^2*x^4+14/15*d^2*e*x^2+d^3)*c*e^4*a^2-3/4*d^2*(-59/9*e^3*x^6-19/3*d*e^2* x^4+5/3*d^2*e*x^2+d^3)*c^2*e^2*a-1/4*c^3*d^4*(-3*e*x^2+d)*(e*x^2+d)^2)*((a *e+(d^2*a*c)^(1/2))*a)^(1/2)*(d^2*a*c)^(1/2))*((-a*e+(d^2*a*c)^(1/2))*a)^( 1/2))/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)/(-c*x^4+a)/(a*e^2-c*d^2)^3/a/d^2
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x**2+d)**(5/2)/(-c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\int { \frac {1}{{\left (c x^{4} - a\right )}^{2} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate(1/((c*x^4 - a)^2*(e*x^2 + d)^(5/2)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:
int(1/((a - c*x^4)^2*(d + e*x^2)^(5/2)),x)
Output:
int(1/((a - c*x^4)^2*(d + e*x^2)^(5/2)), x)
\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^2} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a^{2} d^{2}+2 \sqrt {e \,x^{2}+d}\, a^{2} d e \,x^{2}+\sqrt {e \,x^{2}+d}\, a^{2} e^{2} x^{4}-2 \sqrt {e \,x^{2}+d}\, a c \,d^{2} x^{4}-4 \sqrt {e \,x^{2}+d}\, a c d e \,x^{6}-2 \sqrt {e \,x^{2}+d}\, a c \,e^{2} x^{8}+\sqrt {e \,x^{2}+d}\, c^{2} d^{2} x^{8}+2 \sqrt {e \,x^{2}+d}\, c^{2} d e \,x^{10}+\sqrt {e \,x^{2}+d}\, c^{2} e^{2} x^{12}}d x \] Input:
int(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^2,x)
Output:
int(1/(sqrt(d + e*x**2)*a**2*d**2 + 2*sqrt(d + e*x**2)*a**2*d*e*x**2 + sqr t(d + e*x**2)*a**2*e**2*x**4 - 2*sqrt(d + e*x**2)*a*c*d**2*x**4 - 4*sqrt(d + e*x**2)*a*c*d*e*x**6 - 2*sqrt(d + e*x**2)*a*c*e**2*x**8 + sqrt(d + e*x* *2)*c**2*d**2*x**8 + 2*sqrt(d + e*x**2)*c**2*d*e*x**10 + sqrt(d + e*x**2)* c**2*e**2*x**12),x)