Integrand size = 28, antiderivative size = 78 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{2 \sqrt {d+e x^2}}+\frac {d \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2 \sqrt {e}} \] Output:
1/2*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+1/2*d*arctan(e^(1/2)*x*(e*x^2+d )^(1/2)/(-e^2*x^4+d^2)^(1/2))/e^(1/2)
Result contains complex when optimal does not.
Time = 1.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{2 \sqrt {d+e x^2}}+\frac {i d \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}} \] Input:
Integrate[Sqrt[d^2 - e^2*x^4]/Sqrt[d + e*x^2],x]
Output:
(x*Sqrt[d^2 - e^2*x^4])/(2*Sqrt[d + e*x^2]) + ((I/2)*d*Log[(-2*I)*Sqrt[e]* x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2]])/Sqrt[e]
Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1396, 211, 224, 216}
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.
\(\displaystyle \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \sqrt {d-e x^2}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \int \frac {1}{\sqrt {d-e x^2}}dx+\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}+\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {d \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[Sqrt[d^2 - e^2*x^4]/Sqrt[d + e*x^2],x]
Output:
(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d - e*x^2])/2 + (d*ArcTan[(Sqrt[e]*x)/Sqrt[d - e*x^2]])/(2*Sqrt[e])))/(Sqrt[d - e*x^2]*Sqrt[d + e*x^2])
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (x \sqrt {-e \,x^{2}+d}\, \sqrt {e}+d \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right )\right )}{2 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\) | \(73\) |
risch | \(\frac {x \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{2 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {d \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{2 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(131\) |
Input:
int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)*(x*(-e*x^2+d)^(1/2)*e^(1/2)+d*arc tan(e^(1/2)*x/(-e*x^2+d)^(1/2)))/(-e*x^2+d)^(1/2)/e^(1/2)
Time = 0.08 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\left [\frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - {\left (d e x^{2} + d^{2}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right )}{4 \, {\left (e^{2} x^{2} + d e\right )}}, \frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - {\left (d e x^{2} + d^{2}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right )}{2 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:
integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2),x, algorithm="fricas")
Output:
[1/4*(2*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*e*x - (d*e*x^2 + d^2)*sqrt(-e )*log(-(2*e^2*x^4 + d*e*x^2 - 2*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt( -e)*x - d^2)/(e*x^2 + d)))/(e^2*x^2 + d*e), 1/2*(sqrt(-e^2*x^4 + d^2)*sqrt (e*x^2 + d)*e*x - (d*e*x^2 + d^2)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt (e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)))/(e^2*x^2 + d*e)]
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )}}{\sqrt {d + e x^{2}}}\, dx \] Input:
integrate((-e**2*x**4+d**2)**(1/2)/(e*x**2+d)**(1/2),x)
Output:
Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/sqrt(d + e*x**2), x)
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-e^{2} x^{4} + d^{2}}}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-e^2*x^4 + d^2)/sqrt(e*x^2 + d), x)
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\frac {1}{2} \, \sqrt {-e x^{2} + d} x - \frac {d \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{2 \, \sqrt {-e}} \] Input:
integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(-e*x^2 + d)*x - 1/2*d*log(abs(-sqrt(-e)*x + sqrt(-e*x^2 + d)))/sq rt(-e)
Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {d^2-e^2\,x^4}}{\sqrt {e\,x^2+d}} \,d x \] Input:
int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^(1/2),x)
Output:
int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^(1/2), x)
\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{e \,x^{2}+d}d x \] Input:
int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4))/(d + e*x**2),x)