Integrand size = 16, antiderivative size = 97 \[ \int x^4 \sqrt {a-b x^4} \, dx=-\frac {2 a x \sqrt {a-b x^4}}{21 b}+\frac {1}{7} x^5 \sqrt {a-b x^4}+\frac {2 a^{9/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 b^{5/4} \sqrt {a-b x^4}} \] Output:
-2/21*a*x*(-b*x^4+a)^(1/2)/b+1/7*x^5*(-b*x^4+a)^(1/2)+2/21*a^(9/4)*(1-b*x^ 4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/b^(5/4)/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66 \[ \int x^4 \sqrt {a-b x^4} \, dx=\frac {x \sqrt {a-b x^4} \left (-a+b x^4+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b x^4}{a}\right )}{\sqrt {1-\frac {b x^4}{a}}}\right )}{7 b} \] Input:
Integrate[x^4*Sqrt[a - b*x^4],x]
Output:
(x*Sqrt[a - b*x^4]*(-a + b*x^4 + (a*Hypergeometric2F1[-1/2, 1/4, 5/4, (b*x ^4)/a])/Sqrt[1 - (b*x^4)/a]))/(7*b)
Time = 0.37 (sec) , antiderivative size = 102, 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 = {811, 843, 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.
\(\displaystyle \int x^4 \sqrt {a-b x^4} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {2}{7} a \int \frac {x^4}{\sqrt {a-b x^4}}dx+\frac {1}{7} x^5 \sqrt {a-b x^4}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {2}{7} a \left (\frac {a \int \frac {1}{\sqrt {a-b x^4}}dx}{3 b}-\frac {x \sqrt {a-b x^4}}{3 b}\right )+\frac {1}{7} x^5 \sqrt {a-b x^4}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {2}{7} a \left (\frac {a \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{3 b \sqrt {a-b x^4}}-\frac {x \sqrt {a-b x^4}}{3 b}\right )+\frac {1}{7} x^5 \sqrt {a-b x^4}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {2}{7} a \left (\frac {a^{5/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 b^{5/4} \sqrt {a-b x^4}}-\frac {x \sqrt {a-b x^4}}{3 b}\right )+\frac {1}{7} x^5 \sqrt {a-b x^4}\) |
Input:
Int[x^4*Sqrt[a - b*x^4],x]
Output:
(x^5*Sqrt[a - b*x^4])/7 + (2*a*(-1/3*(x*Sqrt[a - b*x^4])/b + (a^(5/4)*Sqrt [1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(3*b^(5/4)*Sqr t[a - b*x^4])))/7
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Time = 0.62 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01
method | result | size |
risch | \(-\frac {x \left (-3 b \,x^{4}+2 a \right ) \sqrt {-b \,x^{4}+a}}{21 b}+\frac {2 a^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(98\) |
default | \(\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-b \,x^{4}+a}}{21 b}+\frac {2 a^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(104\) |
elliptic | \(\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-b \,x^{4}+a}}{21 b}+\frac {2 a^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(104\) |
Input:
int(x^4*(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/21*x*(-3*b*x^4+2*a)*(-b*x^4+a)^(1/2)/b+2/21/b*a^2/(1/a^(1/2)*b^(1/2))^( 1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a )^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.59 \[ \int x^4 \sqrt {a-b x^4} \, dx=\frac {2 \, a \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, b x^{5} - 2 \, a x\right )} \sqrt {-b x^{4} + a}}{21 \, b} \] Input:
integrate(x^4*(-b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/21*(2*a*sqrt(-b)*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4)/x), -1) + (3* b*x^5 - 2*a*x)*sqrt(-b*x^4 + a))/b
Time = 0.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.42 \[ \int x^4 \sqrt {a-b x^4} \, dx=\frac {\sqrt {a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate(x**4*(-b*x**4+a)**(1/2),x)
Output:
sqrt(a)*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_polar(2*I*pi )/a)/(4*gamma(9/4))
\[ \int x^4 \sqrt {a-b x^4} \, dx=\int { \sqrt {-b x^{4} + a} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^4 + a)*x^4, x)
\[ \int x^4 \sqrt {a-b x^4} \, dx=\int { \sqrt {-b x^{4} + a} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^4 + a)*x^4, x)
Timed out. \[ \int x^4 \sqrt {a-b x^4} \, dx=\int x^4\,\sqrt {a-b\,x^4} \,d x \] Input:
int(x^4*(a - b*x^4)^(1/2),x)
Output:
int(x^4*(a - b*x^4)^(1/2), x)
\[ \int x^4 \sqrt {a-b x^4} \, dx=\frac {-2 \sqrt {-b \,x^{4}+a}\, a x +3 \sqrt {-b \,x^{4}+a}\, b \,x^{5}+2 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{2}}{21 b} \] Input:
int(x^4*(-b*x^4+a)^(1/2),x)
Output:
( - 2*sqrt(a - b*x**4)*a*x + 3*sqrt(a - b*x**4)*b*x**5 + 2*int(sqrt(a - b* x**4)/(a - b*x**4),x)*a**2)/(21*b)