\(\int \frac {1}{x^2 (a-b x^4)^{3/4}} \, dx\) [1252]

Optimal result

Integrand size = 16, antiderivative size = 20 \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{a x} \]

[Out]

-(-b*x^4+a)^(1/4)/a/x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {270} \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{a x} \]

[In]

Int[1/(x^2*(a - b*x^4)^(3/4)),x]

[Out]

-((a - b*x^4)^(1/4)/(a*x))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{a-b x^4}}{a x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{a x} \]

[In]

Integrate[1/(x^2*(a - b*x^4)^(3/4)),x]

[Out]

-((a - b*x^4)^(1/4)/(a*x))

Maple [A] (verified)

Time = 4.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a x}\)	\(19\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a x}\)	\(19\)
pseudoelliptic	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a x}\)	\(19\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} {\left (\left (-b \,x^{4}+a \right )^{3}\right )}^{\frac {1}{4}}}{a x {\left (-\left (b \,x^{4}-a \right )^{3}\right )}^{\frac {1}{4}}}\)	\(46\)

[In]

int(1/x^2/(-b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-(-b*x^4+a)^(1/4)/a/x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a x} \]

[In]

integrate(1/x^2/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

-(-b*x^4 + a)^(1/4)/(a*x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.00 \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=\begin {cases} \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {1}{4}\right )}{4 a \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {\sqrt [4]{b} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {1}{4}\right )}{4 a \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**2/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-1/4)/(4*a*gamma(3/4)), Abs(a/(b*x**4)) > 1), (-b**(1/4)*(-a
/(b*x**4) + 1)**(1/4)*exp(-3*I*pi/4)*gamma(-1/4)/(4*a*gamma(3/4)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a x} \]

[In]

integrate(1/x^2/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

-(-b*x^4 + a)^(1/4)/(a*x)

Giac [F]

\[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^2), x)

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (a-b\,x^4\right )}^{1/4}}{a\,x} \]

[In]

int(1/(x^2*(a - b*x^4)^(3/4)),x)

[Out]

-(a - b*x^4)^(1/4)/(a*x)