Integrand size = 21, antiderivative size = 630 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=-\frac {e^2 \left (3 c d^2-2 a e^2\right ) x}{6 a d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^{3/2}}-\frac {e^2 \left (9 c^2 d^4-59 a c d^2 e^2-8 a^2 e^4\right ) x}{12 a d^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x^2}}+\frac {c x \left (d-e x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (d+e x^2\right )^{3/2} \left (a+c x^4\right )}-\frac {\sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (c d^2 e \left (c d^2+21 a e^2\right )+\left (3 c^2 d^4+15 a c d^2 e^2-8 a^2 e^4\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} d \left (c d^2+a e^2\right )^{7/2}}-\frac {\sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (c d^2 e \left (c d^2+21 a e^2\right )+\left (3 c^2 d^4+15 a c d^2 e^2-8 a^2 e^4\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} d \left (c d^2+a e^2\right )^{7/2}} \] 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)-1/16*c^( 1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(c*d^2*e*(21*a*e^2+c*d^2)+(-8*a ^2*e^4+15*a*c*d^2*e^2+3*c^2*d^4)*(e-(a*e^2+c*d^2)^(1/2)/a^(1/2)))*arctan(2 ^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^( 1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(5/4)/d/ (a*e^2+c*d^2)^(7/2)-1/16*c^(1/2)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(c *d^2*e*(21*a*e^2+c*d^2)+(-8*a^2*e^4+15*a*c*d^2*e^2+3*c^2*d^4)*(e+(a*e^2+c* d^2)^(1/2)/a^(1/2)))*arctanh(2^(1/2)*a^(1/4)*c^(1/2)*(-a^(1/2)*e+(a*e^2+c* d^2)^(1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e-(a*e^2+c*d^2)^(1/2 ))-c*d*x^2))*2^(1/2)/a^(5/4)/d/(a*e^2+c*d^2)^(7/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.36 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\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 )}{a d^2 \left (d+e x^2\right )^{3/2} \left (a+c x^4\right )}-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 {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:
((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)))/(a*d^2*(d + e*x ^2)^(3/2)*(a + c*x^4)) - 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) & ] - (3*c*e^(3/2)*Root Sum[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*S qrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 5*a*c*d*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) & ])/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]
Leaf count of result is larger than twice the leaf count of optimal. \(1381\) vs. \(2(538)=1076\).
Time = 2.85 (sec) , antiderivative size = 1382, normalized size of antiderivative = 2.19
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1382\) |
| default | \(\text {Expression too large to display}\) | \(9378\) |
Input:
int(1/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)/a^ (5/2)/(a*e^2+c*d^2)^(7/2)/(e*x^2+d)^(3/2)*(-1/4*(2*(a*(a*e^2+c*d^2))^(1/2) +2*a*e)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a *e)^(1/2)*((-(c*x^4+a)*(a^2*e^4-15/8*a*c*d^2*e^2-3/8*c^2*d^4)*(a*e^2+c*d^2 )^(1/2)-((c*e^4*x^4-9/2*c*d^2*e^2)*a^(5/2)+a^(7/2)*e^4-1/2*d^2*((9*e^2*x^4 +d^2)*a^(3/2)+c*d^2*x^4*a^(1/2))*c^2)*e)*(a*(a*e^2+c*d^2))^(1/2)+(a*(c*x^4 +a)*(a^2*e^4-15/8*a*c*d^2*e^2-3/8*c^2*d^4)*(a*e^2+c*d^2)^(1/2)-9/2*(1/9*c^ 2*d^2*(9*e^2*x^4+d^2)*a^(5/2)+c*e^2*(-2/9*e^2*x^4+d^2)*a^(7/2)+1/9*c^3*d^4 *x^4*a^(3/2)-2/9*a^(9/2)*e^4)*e)*e)*(e*x^2+d)^(3/2)*ln(((e*x^2+d)^(1/2)*(2 *(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x-(a*e^2+c*d^2)^(1/2)*x^2-a^(1/2)*(e *x^2+d))/x^2)+1/4*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(4*(a*e^2+c*d^2) ^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*((-(c*x^4+a)*(a^2*e^ 4-15/8*a*c*d^2*e^2-3/8*c^2*d^4)*(a*e^2+c*d^2)^(1/2)-((c*e^4*x^4-9/2*c*d^2* e^2)*a^(5/2)+a^(7/2)*e^4-1/2*d^2*((9*e^2*x^4+d^2)*a^(3/2)+c*d^2*x^4*a^(1/2 ))*c^2)*e)*(a*(a*e^2+c*d^2))^(1/2)+(a*(c*x^4+a)*(a^2*e^4-15/8*a*c*d^2*e^2- 3/8*c^2*d^4)*(a*e^2+c*d^2)^(1/2)-9/2*(1/9*c^2*d^2*(9*e^2*x^4+d^2)*a^(5/2)+ c*e^2*(-2/9*e^2*x^4+d^2)*a^(7/2)+1/9*c^3*d^4*x^4*a^(3/2)-2/9*a^(9/2)*e^4)* e)*e)*(e*x^2+d)^(3/2)*ln((a^(1/2)*(e*x^2+d)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c *d^2))^(1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+5*x*(-3/20*d^2*c ^2*e^2*(-59/9*e^3*x^6-19/3*d*e^2*x^4+5/3*d^2*e*x^2+d^3)*a^(5/2)+c*e^4*(...
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. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\int \frac {1}{{\left (c\,x^4+a\right )}^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \] 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)