Integrand size = 21, antiderivative size = 395 \[ \int \frac {\sqrt {d+e x^2}}{\left (a+c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2}}{4 a \left (a+c x^4\right )}-\frac {\left (\sqrt {a} e-3 \sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} \sqrt {c} \sqrt {c d^2+a e^2}}-\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\sqrt {a} e+3 \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} \sqrt {c} \sqrt {c d^2+a e^2}} \] Output:
1/4*x*(e*x^2+d)^(1/2)/a/(c*x^4+a)-1/16*(a^(1/2)*e-3*(a*e^2+c*d^2)^(1/2))*( a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(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^(7/4)/c^(1/2)/(a*e^2+c*d^2)^(1/2)-1/1 6*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(a^(1/2)*e+3*(a*e^2+c*d^2)^(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^(7/4)/c^(1/2)/(a*e^2+c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.35 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {d+e x^2}}{\left (a+c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2}}{4 a \left (a+c x^4\right )}+\frac {8 e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c}+\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-32 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+4 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{4 a c} \] 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]
Leaf count of result is larger than twice the leaf count of optimal. \(838\) vs. \(2(311)=622\).
Time = 1.40 (sec) , antiderivative size = 839, normalized size of antiderivative = 2.12
| method | result | size |
| pseudoelliptic | \(\frac {\frac {3 \left (\left (\left (-c \,x^{4}-a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-\frac {e \left (c \,x^{4} \sqrt {a}+a^{\frac {3}{2}}\right )}{3}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (a \left (c \,x^{4}+a \right ) \sqrt {a \,e^{2}+c \,d^{2}}+\frac {e \left (c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right )}{3}\right ) e \right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{32}-\frac {3 \left (\left (\left (-c \,x^{4}-a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-\frac {e \left (c \,x^{4} \sqrt {a}+a^{\frac {3}{2}}\right )}{3}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (a \left (c \,x^{4}+a \right ) \sqrt {a \,e^{2}+c \,d^{2}}+\frac {e \left (c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right )}{3}\right ) e \right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{32}+\frac {3 d \left (\frac {2 a^{\frac {3}{2}} x \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {e \,x^{2}+d}}{3}+d \left (a \left (c \,x^{4}+a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-\frac {e \left (c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right )}{3}\right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right )\right ) c}{8}}{a^{\frac {5}{2}} \left (c \,x^{4}+a \right ) c d \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {a \,e^{2}+c \,d^{2}}}\) | \(839\) |
| default | \(\text {Expression too large to display}\) | \(5332\) |
Input:
int((e*x^2+d)^(1/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
3/8/(a*e^2+c*d^2)^(1/2)/a^(5/2)/(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)*(1/4*(((-c*x^4-a)*(a*e^2+c*d^2)^(1/2)-1/3*e*(c *x^4*a^(1/2)+a^(3/2)))*(a*(a*e^2+c*d^2))^(1/2)+(a*(c*x^4+a)*(a*e^2+c*d^2)^ (1/2)+1/3*e*(c*x^4*a^(3/2)+a^(5/2)))*e)*(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 )*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)-1/4*(((-c*x^4-a)*(a*e^2+c*d^2)^(1/2) -1/3*e*(c*x^4*a^(1/2)+a^(3/2)))*(a*(a*e^2+c*d^2))^(1/2)+(a*(c*x^4+a)*(a*e^ 2+c*d^2)^(1/2)+1/3*e*(c*x^4*a^(3/2)+a^(5/2)))*e)*(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)*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)+d*(2/3*a^(3/2)*x*(a*e^2+c*d ^2)^(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)*(e*x^2+d)^(1/2)+d*(a*(c*x^4+a)*(a*e^2+c*d^2)^(1/2)-1/3*e*(c*x^4*a^(3 /2)+a^(5/2)))*(arctan(((2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x-2*a^(1/2) *(e*x^2+d)^(1/2))/x/(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))-arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d^2))^(1 /2)+2*a*e)^(1/2)*x)/x/(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)/d/c/(c*x^4+a)
Leaf count of result is larger than twice the leaf count of optimal. 2397 vs. \(2 (313) = 626\).
Time = 3.66 (sec) , antiderivative size = 2397, normalized size of antiderivative = 6.07 \[ \int \frac {\sqrt {d+e x^2}}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] 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)*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)))/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*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^...
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. \[ \int \frac {\sqrt {d+e x^2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {\sqrt {e\,x^2+d}}{{\left (c\,x^4+a\right )}^2} \,d x \] 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)