Integrand size = 28, antiderivative size = 116 \[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\frac {3 d x \sqrt {d^2-e^2 x^4}}{8 \sqrt {d+e x^2}}+\frac {e x^3 \sqrt {d^2-e^2 x^4}}{4 \sqrt {d+e x^2}}+\frac {5 d^2 \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{8 \sqrt {e}} \] Output:
3/8*d*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+1/4*e*x^3*(-e^2*x^4+d^2)^(1/2 )/(e*x^2+d)^(1/2)+5/8*d^2*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 = 1.66 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\frac {1}{8} \left (\frac {x \left (3 d+2 e x^2\right ) \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}+\frac {5 i d^2 \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}\right ) \] Input:
Integrate[Sqrt[d + e*x^2]*Sqrt[d^2 - e^2*x^4],x]
Output:
((x*(3*d + 2*e*x^2)*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2] + ((5*I)*d^2*Log[ (-2*I)*Sqrt[e]*x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2]])/Sqrt[e])/8
Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1396, 299, 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.
\(\displaystyle \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \sqrt {d-e x^2} \left (e x^2+d\right )dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {5}{4} d \int \sqrt {d-e x^2}dx-\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {5}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {d-e x^2}}dx+\frac {1}{2} x \sqrt {d-e x^2}\right )-\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {5}{4} d \left (\frac {1}{2} d \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}+\frac {1}{2} x \sqrt {d-e x^2}\right )-\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {5}{4} d \left (\frac {d \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d-e x^2}\right )-\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[Sqrt[d + e*x^2]*Sqrt[d^2 - e^2*x^4],x]
Output:
(Sqrt[d^2 - e^2*x^4]*(-1/4*(x*(d - e*x^2)^(3/2)) + (5*d*((x*Sqrt[d - e*x^2 ])/2 + (d*ArcTan[(Sqrt[e]*x)/Sqrt[d - e*x^2]])/(2*Sqrt[e])))/4))/(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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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 = 96, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (2 e^{\frac {3}{2}} x^{3} \sqrt {-e \,x^{2}+d}+3 \sqrt {e}\, \sqrt {-e \,x^{2}+d}\, d x +5 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{2}\right )}{8 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\) | \(96\) |
risch | \(\frac {x \left (2 e \,x^{2}+3 d \right ) \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{8 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {5 d^{2} \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{8 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(143\) |
Input:
int((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/8*(-e^2*x^4+d^2)^(1/2)*(2*e^(3/2)*x^3*(-e*x^2+d)^(1/2)+3*e^(1/2)*(-e*x^2 +d)^(1/2)*d*x+5*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2))*d^2)/(e*x^2+d)^(1/2)/(- e*x^2+d)^(1/2)/e^(1/2)
Time = 0.08 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.19 \[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\left [-\frac {5 \, {\left (d^{2} e x^{2} + d^{3}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right ) - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (2 \, e^{2} x^{3} + 3 \, d e x\right )} \sqrt {e x^{2} + d}}{16 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {5 \, {\left (d^{2} e x^{2} + d^{3}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - \sqrt {-e^{2} x^{4} + d^{2}} {\left (2 \, e^{2} x^{3} + 3 \, d e x\right )} \sqrt {e x^{2} + d}}{8 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:
integrate((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
Output:
[-1/16*(5*(d^2*e*x^2 + d^3)*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)) - 2*sqrt(-e^2 *x^4 + d^2)*(2*e^2*x^3 + 3*d*e*x)*sqrt(e*x^2 + d))/(e^2*x^2 + d*e), -1/8*( 5*(d^2*e*x^2 + d^3)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sq rt(e)*x/(e^2*x^4 - d^2)) - sqrt(-e^2*x^4 + d^2)*(2*e^2*x^3 + 3*d*e*x)*sqrt (e*x^2 + d))/(e^2*x^2 + d*e)]
\[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\int \sqrt {- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )} \sqrt {d + e x^{2}}\, dx \] Input:
integrate((e*x**2+d)**(1/2)*(-e**2*x**4+d**2)**(1/2),x)
Output:
Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))*sqrt(d + e*x**2), x)
\[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\int { \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \,d x } \] Input:
integrate((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d), x)
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.47 \[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\frac {1}{8} \, {\left (2 \, e x^{2} + 3 \, d\right )} \sqrt {-e x^{2} + d} x - \frac {5 \, d^{2} \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{8 \, \sqrt {-e}} \] Input:
integrate((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(1/2),x, algorithm="giac")
Output:
1/8*(2*e*x^2 + 3*d)*sqrt(-e*x^2 + d)*x - 5/8*d^2*log(abs(-sqrt(-e)*x + sqr t(-e*x^2 + d)))/sqrt(-e)
Timed out. \[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\int \sqrt {d^2-e^2\,x^4}\,\sqrt {e\,x^2+d} \,d x \] 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)
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.46 \[ \int \sqrt {d+e x^2} \sqrt {d^2-e^2 x^4} \, dx=\frac {5 \sqrt {e}\, \mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}+3 \sqrt {-e \,x^{2}+d}\, d e x +2 \sqrt {-e \,x^{2}+d}\, e^{2} x^{3}}{8 e} \] Input:
int((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(1/2),x)
Output:
(5*sqrt(e)*asin((sqrt(e)*x)/sqrt(d))*d**2 + 3*sqrt(d - e*x**2)*d*e*x + 2*s qrt(d - e*x**2)*e**2*x**3)/(8*e)