Integrand size = 16, antiderivative size = 108 \[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {2}{7} d^2 x \sqrt {d^2-e^2 x^4}+\frac {1}{7} x \left (d^2-e^2 x^4\right )^{3/2}+\frac {4 d^{9/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{7 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/7*d^2*x*(-e^2*x^4+d^2)^(1/2)+1/7*x*(-e^2*x^4+d^2)^(3/2)+4/7*d^(9/2)*(1-e ^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1 /2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.54 \[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {d^2 x \sqrt {d^2-e^2 x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )}{\sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:
Integrate[(d^2 - e^2*x^4)^(3/2),x]
Output:
(d^2*x*Sqrt[d^2 - e^2*x^4]*Hypergeometric2F1[-3/2, 1/4, 5/4, (e^2*x^4)/d^2 ])/Sqrt[1 - (e^2*x^4)/d^2]
Time = 0.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {748, 748, 765, 762}
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.
\(\displaystyle \int \left (d^2-e^2 x^4\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 748 |
\(\displaystyle \frac {6}{7} d^2 \int \sqrt {d^2-e^2 x^4}dx+\frac {1}{7} x \left (d^2-e^2 x^4\right )^{3/2}\) |
\(\Big \downarrow \) 748 |
\(\displaystyle \frac {6}{7} d^2 \left (\frac {2}{3} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx+\frac {1}{3} x \sqrt {d^2-e^2 x^4}\right )+\frac {1}{7} x \left (d^2-e^2 x^4\right )^{3/2}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {6}{7} d^2 \left (\frac {2 d^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{3 \sqrt {d^2-e^2 x^4}}+\frac {1}{3} x \sqrt {d^2-e^2 x^4}\right )+\frac {1}{7} x \left (d^2-e^2 x^4\right )^{3/2}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {6}{7} d^2 \left (\frac {2 d^{5/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {1}{3} x \sqrt {d^2-e^2 x^4}\right )+\frac {1}{7} x \left (d^2-e^2 x^4\right )^{3/2}\) |
Input:
Int[(d^2 - e^2*x^4)^(3/2),x]
Output:
(x*(d^2 - e^2*x^4)^(3/2))/7 + (6*d^2*((x*Sqrt[d^2 - e^2*x^4])/3 + (2*d^(5/ 2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(3* Sqrt[e]*Sqrt[d^2 - e^2*x^4])))/7
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Time = 0.95 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {x \left (-e^{2} x^{4}+3 d^{2}\right ) \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(96\) |
default | \(-\frac {e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {3 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(107\) |
elliptic | \(-\frac {e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {3 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(107\) |
Input:
int((-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/7*x*(-e^2*x^4+3*d^2)*(-e^2*x^4+d^2)^(1/2)+4/7*d^4/(e/d)^(1/2)*(1-e*x^2/d )^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2),I)
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65 \[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {4 \, \sqrt {-e^{2}} d^{3} \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left (e^{3} x^{5} - 3 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{7 \, e} \] Input:
integrate((-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
1/7*(4*sqrt(-e^2)*d^3*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), -1) - (e^3 *x^5 - 3*d^2*e*x)*sqrt(-e^2*x^4 + d^2))/e
Time = 0.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.38 \[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-e**2*x**4+d**2)**(3/2),x)
Output:
d**3*x*gamma(1/4)*hyper((-3/2, 1/4), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d **2)/(4*gamma(5/4))
\[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-e^2*x^4 + d^2)^(3/2), x)
\[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
integrate((-e^2*x^4 + d^2)^(3/2), x)
Time = 17.42 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.43 \[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {x\,{\left (d^2-e^2\,x^4\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ \frac {e^2\,x^4}{d^2}\right )}{{\left (1-\frac {e^2\,x^4}{d^2}\right )}^{3/2}} \] Input:
int((d^2 - e^2*x^4)^(3/2),x)
Output:
(x*(d^2 - e^2*x^4)^(3/2)*hypergeom([-3/2, 1/4], 5/4, (e^2*x^4)/d^2))/(1 - (e^2*x^4)/d^2)^(3/2)
\[ \int \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {3 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} x}{7}-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{2} x^{5}}{7}+\frac {4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{4}}{7} \] Input:
int((-e^2*x^4+d^2)^(3/2),x)
Output:
(3*sqrt(d**2 - e**2*x**4)*d**2*x - sqrt(d**2 - e**2*x**4)*e**2*x**5 + 4*in t(sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**4)/7