Integrand size = 22, antiderivative size = 233 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\frac {e^2 x \sqrt {d+e x^2}}{4 a c}+\frac {x \left (d+e x^2\right )^{5/2}}{4 a \left (a-c x^4\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (3 \sqrt {c} d+2 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} c^{3/2}}+\frac {\left (3 \sqrt {c} d-2 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} c^{3/2}} \] Output:
1/4*e^2*x*(e*x^2+d)^(1/2)/a/c+1/4*x*(e*x^2+d)^(5/2)/a/(-c*x^4+a)+1/8*(c^(1 /2)*d-a^(1/2)*e)^(3/2)*(3*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^(3/2)+1/8*(3*c^(1/2)*d-2*a^ (1/2)*e)*(c^(1/2)*d+a^(1/2)*e)^(3/2)*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^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.60 (sec) , antiderivative size = 711, normalized size of antiderivative = 3.05 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\frac {\sqrt {d+e x^2} \left (-c d^2 x-a e^2 x-2 c d e x^3\right )}{4 a c \left (-a+c x^4\right )}+\frac {e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {49 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+10 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{2 c^2}-\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {2 c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+97 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+32 a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+20 a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 c^2 d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a c e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{4 a c^2} \] Input:
Integrate[(d + e*x^2)^(5/2)/(a - c*x^4)^2,x]
Output:
(Sqrt[d + e*x^2]*(-(c*d^2*x) - a*e^2*x - 2*c*d*e*x^3))/(4*a*c*(-a + c*x^4) ) + (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 & , (49*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* x^2] - #1] + 16*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 10*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*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*c^2) - (e^(3/2)*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 & , (2*c^2 *d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 97*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] + 2*c^2*d^3*Log[d + 2*e*x^2 - 2 *Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 20*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^2*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x* Sqrt[d + e*x^2] - #1]*#1^2 + 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*c^2)
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2}dx\) |
Input:
Int[(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 = 0.69 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.33
| method | result | size |
| pseudoelliptic | \(\frac {-d \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (-c \,x^{4}+a \right ) \left (\frac {\left (-a \,e^{2}+3 c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+a e \left (a \,e^{2}-2 c \,d^{2}\right )\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (-d \left (\frac {\left (a \,e^{2}-3 c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+a e \left (a \,e^{2}-2 c \,d^{2}\right )\right ) \left (-c \,x^{4}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+x \left (a \,e^{2}+c d \left (2 e \,x^{2}+d \right )\right ) \sqrt {e \,x^{2}+d}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}\right )}{4 \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, a c \left (-c \,x^{4}+a \right )}\) | \(310\) |
| default | \(\text {Expression too large to display}\) | \(12004\) |
Input:
int((e*x^2+d)^(5/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/4/(d^2*a*c)^(1/2)*(-d*((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(-c*x^4+a)*(1/2*(- a*e^2+3*c*d^2)*(d^2*a*c)^(1/2)+a*e*(a*e^2-2*c*d^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*(1/2*(a*e^2-3*c*d^2)*(d^2*a*c)^(1/2)+a*e*(a*e^2-2*c*d^2))*(-c*x^4+a)*arc tanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))+x*(a*e^2+c*d*(2* e*x^2+d))*(e*x^2+d)^(1/2)*((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)/a/c/(-c *x^4+a)
Leaf count of result is larger than twice the leaf count of optimal. 1935 vs. \(2 (174) = 348\).
Time = 10.33 (sec) , antiderivative size = 1935, normalized size of antiderivative = 8.30 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(5/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
1/32*((a*c^2*x^4 - a^2*c)*sqrt((15*c^2*d^4*e - 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3*sqrt((81*c^2*d^10 - 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3))) /(a^3*c^3))*log(-(81*c^3*d^10 - 162*a*c^2*d^8*e^2 + 101*a^2*c*d^6*e^4 - 20 *a^3*d^4*e^6 + (9*a^3*c^4*d^5 - 13*a^4*c^3*d^3*e^2 + 4*a^5*c^2*d*e^4)*x^2* sqrt((81*c^2*d^10 - 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) + 2*(81*c^ 3*d^9*e - 162*a*c^2*d^7*e^3 + 101*a^2*c*d^5*e^5 - 20*a^3*d^3*e^7)*x^2 + 2* sqrt(e*x^2 + d)*((3*a^5*c^4*d^2 - 2*a^6*c^3*e^2)*x*sqrt((81*c^2*d^10 - 90* a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) - (9*a^2*c^3*d^6*e - 5*a^3*c^2*d^ 4*e^3)*x)*sqrt((15*c^2*d^4*e - 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3*sqrt(( 81*c^2*d^10 - 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)))/(a^3*c^3)))/x^2 ) - (a*c^2*x^4 - a^2*c)*sqrt((15*c^2*d^4*e - 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3*sqrt((81*c^2*d^10 - 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)))/( a^3*c^3))*log(-(81*c^3*d^10 - 162*a*c^2*d^8*e^2 + 101*a^2*c*d^6*e^4 - 20*a ^3*d^4*e^6 + (9*a^3*c^4*d^5 - 13*a^4*c^3*d^3*e^2 + 4*a^5*c^2*d*e^4)*x^2*sq rt((81*c^2*d^10 - 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) + 2*(81*c^3* d^9*e - 162*a*c^2*d^7*e^3 + 101*a^2*c*d^5*e^5 - 20*a^3*d^3*e^7)*x^2 - 2*sq rt(e*x^2 + d)*((3*a^5*c^4*d^2 - 2*a^6*c^3*e^2)*x*sqrt((81*c^2*d^10 - 90*a* c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) - (9*a^2*c^3*d^6*e - 5*a^3*c^2*d^4* e^3)*x)*sqrt((15*c^2*d^4*e - 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3*sqrt((81 *c^2*d^10 - 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)))/(a^3*c^3)))/x^...
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {5}{2}}}{\left (- a + c x^{4}\right )^{2}}\, dx \] Input:
integrate((e*x**2+d)**(5/2)/(-c*x**4+a)**2,x)
Output:
Integral((d + e*x**2)**(5/2)/(-a + c*x**4)**2, x)
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (c x^{4} - a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(5/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(5/2)/(c*x^4 - a)^2, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(5/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{{\left (a-c\,x^4\right )}^2} \,d x \] Input:
int((d + e*x^2)^(5/2)/(a - c*x^4)^2,x)
Output:
int((d + e*x^2)^(5/2)/(a - c*x^4)^2, x)
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) d^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) d e \] Input:
int((e*x^2+d)^(5/2)/(-c*x^4+a)^2,x)
Output:
int(sqrt(d + e*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*d**2 + int((sqrt(d + e*x**2)*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*e**2 + 2*int((sqrt(d + e*x**2)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*d*e