Integrand size = 22, antiderivative size = 208 \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2}}{4 a \left (a-c x^4\right )}+\frac {\left (3 \sqrt {c} d-2 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (3 \sqrt {c} d+2 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {a} e}} \] Output:
1/4*x*(e*x^2+d)^(1/2)/a/(-c*x^4+a)+1/8*(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^(1/2)/(c^(1 /2)*d-a^(1/2)*e)^(1/2)+1/8*(3*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)/(c^(1/2)*d+a^(1/ 2)*e)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2}}{4 a \left (a-c x^4\right )}+\frac {8 e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c}-\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+32 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+4 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{4 a c} \] Input:
Integrate[Sqrt[d + e*x^2]/(a - c*x^4)^2,x]
Output:
(x*Sqrt[d + e*x^2])/(4*a*(a - c*x^4)) + (8*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 & , Log[d + 2*e*x ^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]/(c*d^3 - 3*c*d^2*#1 + 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ])/c - (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 & , (c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 32*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x *Sqrt[d + e*x^2] - #1] + 4*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) & ])/(4*a*c)
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.
\(\displaystyle \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2}dx\) |
Input:
Int[Sqrt[d + e*x^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 = 0.53 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(-\frac {d \left (-\frac {\sqrt {e \,x^{2}+d}\, x}{d \left (-c \,x^{4}+a \right )}+\frac {\left (-2 a e +3 \sqrt {d^{2} a c}\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}-\frac {\left (2 a e +3 \sqrt {d^{2} a c}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{4 a}\) | \(179\) |
default | \(\text {Expression too large to display}\) | \(4714\) |
Input:
int((e*x^2+d)^(1/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*d/a*(-(e*x^2+d)^(1/2)/d*x/(-c*x^4+a)+1/2*(-2*a*e+3*(d^2*a*c)^(1/2))/( d^2*a*c)^(1/2)/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctan((e*x^2+d)^(1/2)/x*a /((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))-1/2*(2*a*e+3*(d^2*a*c)^(1/2))/(d^2*a*c) ^(1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+( d^2*a*c)^(1/2))*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2399 vs. \(2 (153) = 306\).
Time = 4.06 (sec) , antiderivative size = 2399, normalized size of antiderivative = 11.53 \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(1/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
1/32*((a*c*x^4 - a^2)*sqrt((3*c*d^2*e - 4*a*e^3 + (a^3*c^2*d^2 - a^4*c*e^2 )*sqrt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e^4)/(a^7*c^3*d^4 - 2*a^ 8*c^2*d^2*e^2 + a^9*c*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))*log((81*c^2*d^6 - 108*a*c*d^4*e^2 + 32*a^2*d^2*e^4 + (9*a^3*c^3*d^5 - 13*a^4*c^2*d^3*e^2 + 4 *a^5*c*d*e^4)*x^2*sqrt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e^4)/(a^ 7*c^3*d^4 - 2*a^8*c^2*d^2*e^2 + a^9*c*e^4)) + 2*(81*c^2*d^5*e - 108*a*c*d^ 3*e^3 + 32*a^2*d*e^5)*x^2 + 2*sqrt(e*x^2 + d)*((3*a^5*c^3*d^4 - 5*a^6*c^2* d^2*e^2 + 2*a^7*c*e^4)*x*sqrt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e ^4)/(a^7*c^3*d^4 - 2*a^8*c^2*d^2*e^2 + a^9*c*e^4)) + (9*a^2*c^2*d^4*e - 8* a^3*c*d^2*e^3)*x)*sqrt((3*c*d^2*e - 4*a*e^3 + (a^3*c^2*d^2 - a^4*c*e^2)*sq rt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e^4)/(a^7*c^3*d^4 - 2*a^8*c^ 2*d^2*e^2 + a^9*c*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2)))/x^2) - (a*c*x^4 - a^2 )*sqrt((3*c*d^2*e - 4*a*e^3 + (a^3*c^2*d^2 - a^4*c*e^2)*sqrt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e^4)/(a^7*c^3*d^4 - 2*a^8*c^2*d^2*e^2 + a^9* c*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))*log((81*c^2*d^6 - 108*a*c*d^4*e^2 + 32 *a^2*d^2*e^4 + (9*a^3*c^3*d^5 - 13*a^4*c^2*d^3*e^2 + 4*a^5*c*d*e^4)*x^2*sq rt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e^4)/(a^7*c^3*d^4 - 2*a^8*c^ 2*d^2*e^2 + a^9*c*e^4)) + 2*(81*c^2*d^5*e - 108*a*c*d^3*e^3 + 32*a^2*d*e^5 )*x^2 - 2*sqrt(e*x^2 + d)*((3*a^5*c^3*d^4 - 5*a^6*c^2*d^2*e^2 + 2*a^7*c*e^ 4)*x*sqrt((81*c^2*d^6 - 144*a*c*d^4*e^2 + 64*a^2*d^2*e^4)/(a^7*c^3*d^4 ...
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(1/2)/(-c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} - a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(1/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate(sqrt(e*x^2 + d)/(c*x^4 - a)^2, x)
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(1/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {\sqrt {e\,x^2+d}}{{\left (a-c\,x^4\right )}^2} \,d x \] Input:
int((d + e*x^2)^(1/2)/(a - c*x^4)^2,x)
Output:
int((d + e*x^2)^(1/2)/(a - c*x^4)^2, x)
\[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \] Input:
int((e*x^2+d)^(1/2)/(-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)