Integrand size = 22, antiderivative size = 280 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\frac {e^2 \left (c d^2+2 a e^2\right ) x}{2 a d \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2}}+\frac {c x \left (d-e x^2\right )}{4 a \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \left (a-c x^4\right )}+\frac {3 \sqrt {c} \left (\sqrt {c} d-2 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \sqrt {c} \left (\sqrt {c} d+2 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \] Output:
1/2*e^2*(2*a*e^2+c*d^2)*x/a/d/(-a*e^2+c*d^2)^2/(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)^(1/2)/(-c*x^4+a)+3/8*c^(1/2)*(c^(1/2)*d -2*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)^(5/2)+3/8*c^(1/2)*(c^(1/2)*d+2*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)^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.75 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.53 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\frac {x \left (-4 a^2 e^4-a c e^2 \left (d^2+d e x^2-4 e^2 x^4\right )+c^2 d^2 \left (-d^2+d e x^2+2 e^2 x^4\right )\right )+2 a d e^{7/2} \sqrt {d+e x^2} \left (a-c x^4\right ) \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {17 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-6 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]+d e^{3/2} \sqrt {d+e x^2} \left (a-c x^4\right ) \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-31 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+32 a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+6 c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-12 a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+a c e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{4 a d \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \left (-a+c x^4\right )} \] Input:
Integrate[1/((d + e*x^2)^(3/2)*(a - c*x^4)^2),x]
Output:
(x*(-4*a^2*e^4 - a*c*e^2*(d^2 + d*e*x^2 - 4*e^2*x^4) + c^2*d^2*(-d^2 + d*e *x^2 + 2*e^2*x^4)) + 2*a*d*e^(7/2)*Sqrt[d + e*x^2]*(a - c*x^4)*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 & , ( 17*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 16*a*e^2*Lo g[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 6*c*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*S qrt[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) & ] + d*e^(3/2)*Sqrt[d + e*x^2]*(a - c*x^4)*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 & , (-31*a*c*d^ 2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 32*a^2*e^4*Log [d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 6*c^2*d^3*Log[d + 2*e*x ^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 12*a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + a*c*e^2*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) & ])/(4*a*d*(c*d^2 - a*e^2)^2*Sqrt[d + e*x^2]*(-a + c*x^4))
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.
\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^2 \left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^2 \left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[1/((d + e*x^2)^(3/2)*(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 = 0.69 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.32
method | result | size |
pseudoelliptic | \(\frac {\frac {3 d^{2} \left (\frac {\left (3 a \,e^{2}-c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+e^{3} a^{2}\right ) \left (-c \,x^{4}+a \right ) c \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{4}+\frac {3 \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (d^{2} \left (\frac {\left (-3 a \,e^{2}+c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+e^{3} a^{2}\right ) \left (-c \,x^{4}+a \right ) c \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+\frac {4 x \left (a^{2} e^{4}+\frac {c \,e^{2} \left (-4 e^{2} x^{4}+d e \,x^{2}+d^{2}\right ) a}{4}+\frac {c^{2} d^{2} \left (e \,x^{2}+d \right ) \left (-2 e \,x^{2}+d \right )}{4}\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}}{3}\right )}{4}}{\left (-c \,x^{4}+a \right ) \left (a \,e^{2}-c \,d^{2}\right )^{2} a \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {e \,x^{2}+d}\, d}\) | \(370\) |
default | \(\text {Expression too large to display}\) | \(4756\) |
Input:
int(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
3/4/(e*x^2+d)^(1/2)/(d^2*a*c)^(1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(d^2*( 1/2*(3*a*e^2-c*d^2)*(d^2*a*c)^(1/2)+e^3*a^2)*(-c*x^4+a)*c*((a*e+(d^2*a*c)^ (1/2))*a)^(1/2)*(e*x^2+d)^(1/2)*arctan((e*x^2+d)^(1/2)/x*a/((-a*e+(d^2*a*c )^(1/2))*a)^(1/2))+((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(d^2*(1/2*(-3*a*e^2+c* d^2)*(d^2*a*c)^(1/2)+e^3*a^2)*(-c*x^4+a)*c*(e*x^2+d)^(1/2)*arctanh((e*x^2+ d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))+4/3*x*(a^2*e^4+1/4*c*e^2*(-4 *e^2*x^4+d*e*x^2+d^2)*a+1/4*c^2*d^2*(e*x^2+d)*(-2*e*x^2+d))*((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)/(-c*x ^4+a)/(a*e^2-c*d^2)^2/a/d
Leaf count of result is larger than twice the leaf count of optimal. 7859 vs. \(2 (222) = 444\).
Time = 171.48 (sec) , antiderivative size = 7859, normalized size of antiderivative = 28.07 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x**2+d)**(3/2)/(-c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\int { \frac {1}{{\left (c x^{4} - a\right )}^{2} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate(1/((c*x^4 - a)^2*(e*x^2 + d)^(3/2)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int(1/((a - c*x^4)^2*(d + e*x^2)^(3/2)),x)
Output:
int(1/((a - c*x^4)^2*(d + e*x^2)^(3/2)), x)
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a^{2} d +\sqrt {e \,x^{2}+d}\, a^{2} e \,x^{2}-2 \sqrt {e \,x^{2}+d}\, a c d \,x^{4}-2 \sqrt {e \,x^{2}+d}\, a c e \,x^{6}+\sqrt {e \,x^{2}+d}\, c^{2} d \,x^{8}+\sqrt {e \,x^{2}+d}\, c^{2} e \,x^{10}}d x \] Input:
int(1/(e*x^2+d)^(3/2)/(-c*x^4+a)^2,x)
Output:
int(1/(sqrt(d + e*x**2)*a**2*d + sqrt(d + e*x**2)*a**2*e*x**2 - 2*sqrt(d + e*x**2)*a*c*d*x**4 - 2*sqrt(d + e*x**2)*a*c*e*x**6 + sqrt(d + e*x**2)*c** 2*d*x**8 + sqrt(d + e*x**2)*c**2*e*x**10),x)