Integrand size = 16, antiderivative size = 95 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=-\frac {a \sqrt {a-b x^4}}{3 x^3}-\frac {1}{3} b x \sqrt {a-b x^4}-\frac {4 a^{5/4} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt {a-b x^4}} \] Output:
-1/3*a*(-b*x^4+a)^(1/2)/x^3-1/3*b*x*(-b*x^4+a)^(1/2)-4/3*a^(5/4)*b^(3/4)*( 1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.56 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=-\frac {a \sqrt {a-b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},\frac {1}{4},\frac {b x^4}{a}\right )}{3 x^3 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[(a - b*x^4)^(3/2)/x^4,x]
Output:
-1/3*(a*Sqrt[a - b*x^4]*Hypergeometric2F1[-3/2, -3/4, 1/4, (b*x^4)/a])/(x^ 3*Sqrt[1 - (b*x^4)/a])
Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {809, 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.
\(\displaystyle \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -2 b \int \sqrt {a-b x^4}dx-\frac {\left (a-b x^4\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 748 |
\(\displaystyle -2 b \left (\frac {2}{3} a \int \frac {1}{\sqrt {a-b x^4}}dx+\frac {1}{3} x \sqrt {a-b x^4}\right )-\frac {\left (a-b x^4\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle -2 b \left (\frac {2 a \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{3 \sqrt {a-b x^4}}+\frac {1}{3} x \sqrt {a-b x^4}\right )-\frac {\left (a-b x^4\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -2 b \left (\frac {2 a^{5/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{b} \sqrt {a-b x^4}}+\frac {1}{3} x \sqrt {a-b x^4}\right )-\frac {\left (a-b x^4\right )^{3/2}}{3 x^3}\) |
Input:
Int[(a - b*x^4)^(3/2)/x^4,x]
Output:
-1/3*(a - b*x^4)^(3/2)/x^3 - 2*b*((x*Sqrt[a - b*x^4])/3 + (2*a^(5/4)*Sqrt[ 1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(3*b^(1/4)*Sqrt [a - b*x^4]))
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Time = 0.62 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (b \,x^{4}+a \right )}{3 x^{3}}-\frac {4 a b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(90\) |
default | \(-\frac {a \sqrt {-b \,x^{4}+a}}{3 x^{3}}-\frac {b x \sqrt {-b \,x^{4}+a}}{3}-\frac {4 a b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(98\) |
elliptic | \(-\frac {a \sqrt {-b \,x^{4}+a}}{3 x^{3}}-\frac {b x \sqrt {-b \,x^{4}+a}}{3}-\frac {4 a b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(98\) |
Input:
int((-b*x^4+a)^(3/2)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*(-b*x^4+a)^(1/2)*(b*x^4+a)/x^3-4/3*a*b/(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)*E llipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/x^4,x, algorithm="fricas")
Output:
integral((-b*x^4 + a)^(3/2)/x^4, x)
Time = 0.59 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=- \frac {i b^{\frac {3}{2}} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((-b*x**4+a)**(3/2)/x**4,x)
Output:
-I*b**(3/2)*x**3*gamma(3/4)*hyper((-3/2, -3/4), (1/4,), a/(b*x**4))/(4*gam ma(7/4))
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/x^4,x, algorithm="maxima")
Output:
integrate((-b*x^4 + a)^(3/2)/x^4, x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/x^4,x, algorithm="giac")
Output:
integrate((-b*x^4 + a)^(3/2)/x^4, x)
Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{3/2}}{x^4} \,d x \] Input:
int((a - b*x^4)^(3/2)/x^4,x)
Output:
int((a - b*x^4)^(3/2)/x^4, x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^4} \, dx=\frac {-5 \sqrt {-b \,x^{4}+a}\, a -\sqrt {-b \,x^{4}+a}\, b \,x^{4}-12 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{8}+a \,x^{4}}d x \right ) a^{2} x^{3}}{3 x^{3}} \] Input:
int((-b*x^4+a)^(3/2)/x^4,x)
Output:
( - 5*sqrt(a - b*x**4)*a - sqrt(a - b*x**4)*b*x**4 - 12*int(sqrt(a - b*x** 4)/(a*x**4 - b*x**8),x)*a**2*x**3)/(3*x**3)