\(\int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx\) [839]

Optimal result

Integrand size = 16, antiderivative size = 46 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {a-b x^4}}{6 a x^6}-\frac {b \sqrt {a-b x^4}}{3 a^2 x^2} \]

[Out]

-1/6*(-b*x^4+a)^(1/2)/a/x^6-1/3*b*(-b*x^4+a)^(1/2)/a^2/x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270} \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=-\frac {b \sqrt {a-b x^4}}{3 a^2 x^2}-\frac {\sqrt {a-b x^4}}{6 a x^6} \]

[In]

Int[1/(x^7*Sqrt[a - b*x^4]),x]

[Out]

-1/6*Sqrt[a - b*x^4]/(a*x^6) - (b*Sqrt[a - b*x^4])/(3*a^2*x^2)

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]

Rule 277

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a-b x^4}}{6 a x^6}+\frac {(2 b) \int \frac {1}{x^3 \sqrt {a-b x^4}} \, dx}{3 a} \\ & = -\frac {\sqrt {a-b x^4}}{6 a x^6}-\frac {b \sqrt {a-b x^4}}{3 a^2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=\frac {\left (-a-2 b x^4\right ) \sqrt {a-b x^4}}{6 a^2 x^6} \]

[In]

Integrate[1/(x^7*Sqrt[a - b*x^4]),x]

[Out]

((-a - 2*b*x^4)*Sqrt[a - b*x^4])/(6*a^2*x^6)

Maple [A] (verified)

Time = 4.85 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.59

method	result	size
gosper	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\)	\(27\)
default	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\)	\(27\)
trager	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\)	\(27\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\)	\(27\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\)	\(27\)
pseudoelliptic	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{6 a^{2} x^{6}}\)	\(27\)

[In]

int(1/x^7/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*(-b*x^4+a)^(1/2)*(2*b*x^4+a)/a^2/x^6

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=-\frac {{\left (2 \, b x^{4} + a\right )} \sqrt {-b x^{4} + a}}{6 \, a^{2} x^{6}} \]

[In]

integrate(1/x^7/(-b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(2*b*x^4 + a)*sqrt(-b*x^4 + a)/(a^2*x^6)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.11 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=\begin {cases} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{4}} - 1}}{6 a x^{4}} - \frac {b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{4}} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {i a^{2} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} + \frac {i a b^{\frac {5}{2}} x^{4} \sqrt {- \frac {a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} - \frac {2 i b^{\frac {7}{2}} x^{8} \sqrt {- \frac {a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**7/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(6*a*x**4) - b**(3/2)*sqrt(a/(b*x**4) - 1)/(3*a**2), Abs(a/(b*x**4))
> 1), (I*a**2*b**(3/2)*sqrt(-a/(b*x**4) + 1)/(-6*a**3*b*x**4 + 6*a**2*b**2*x**8) + I*a*b**(5/2)*x**4*sqrt(-a/(
b*x**4) + 1)/(-6*a**3*b*x**4 + 6*a**2*b**2*x**8) - 2*I*b**(7/2)*x**8*sqrt(-a/(b*x**4) + 1)/(-6*a**3*b*x**4 + 6
*a**2*b**2*x**8), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=-\frac {\frac {3 \, \sqrt {-b x^{4} + a} b}{x^{2}} + \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{x^{6}}}{6 \, a^{2}} \]

[In]

integrate(1/x^7/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

-1/6*(3*sqrt(-b*x^4 + a)*b/x^2 + (-b*x^4 + a)^(3/2)/x^6)/a^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=-\frac {2 \, {\left (3 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} - a\right )} \sqrt {-b} b}{3 \, {\left ({\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} - a\right )}^{3}} \]

[In]

integrate(1/x^7/(-b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

-2/3*(3*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^2 - a)*sqrt(-b)*b/((sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^2 - a)^3

Mupad [B] (verification not implemented)

Time = 5.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {a-b\,x^4}\,\left (2\,b\,x^4+a\right )}{6\,a^2\,x^6} \]

[In]

int(1/(x^7*(a - b*x^4)^(1/2)),x)

[Out]

-((a - b*x^4)^(1/2)*(a + 2*b*x^4))/(6*a^2*x^6)