3.13.23 \(\int \frac {1}{x^8 \sqrt [4]{a-b x^4}} \, dx\) [1223]

Optimal. Leaf size=46 \[ -\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7}-\frac {4 b \left (a-b x^4\right )^{3/4}}{21 a^2 x^3} \]

[Out]

-1/7*(-b*x^4+a)^(3/4)/a/x^7-4/21*b*(-b*x^4+a)^(3/4)/a^2/x^3

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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270} \begin {gather*} -\frac {4 b \left (a-b x^4\right )^{3/4}}{21 a^2 x^3}-\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a - b*x^4)^(1/4)),x]

[Out]

-1/7*(a - b*x^4)^(3/4)/(a*x^7) - (4*b*(a - b*x^4)^(3/4))/(21*a^2*x^3)

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*} \int \frac {1}{x^8 \sqrt [4]{a-b x^4}} \, dx &=-\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7}+\frac {(4 b) \int \frac {1}{x^4 \sqrt [4]{a-b x^4}} \, dx}{7 a}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7}-\frac {4 b \left (a-b x^4\right )^{3/4}}{21 a^2 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 32, normalized size = 0.70 \begin {gather*} \frac {\left (-3 a-4 b x^4\right ) \left (a-b x^4\right )^{3/4}}{21 a^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a - b*x^4)^(1/4)),x]

[Out]

((-3*a - 4*b*x^4)*(a - b*x^4)^(3/4))/(21*a^2*x^7)

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Maple [A]
time = 0.17, size = 29, normalized size = 0.63

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (4 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}\)	\(29\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (4 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}\)	\(29\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (4 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(-b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/21*(-b*x^4+a)^(3/4)*(4*b*x^4+3*a)/a^2/x^7

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Maxima [A]
time = 0.30, size = 37, normalized size = 0.80 \begin {gather*} -\frac {\frac {7 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} b}{x^{3}} + \frac {3 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}}}{x^{7}}}{21 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

-1/21*(7*(-b*x^4 + a)^(3/4)*b/x^3 + 3*(-b*x^4 + a)^(7/4)/x^7)/a^2

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Fricas [A]
time = 0.35, size = 28, normalized size = 0.61 \begin {gather*} -\frac {{\left (4 \, b x^{4} + 3 \, a\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{21 \, a^{2} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/21*(4*b*x^4 + 3*a)*(-b*x^4 + a)^(3/4)/(a^2*x^7)

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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 303, normalized size = 6.59 \begin {gather*} \begin {cases} - \frac {3 b^{\frac {3}{4}} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{16 a x^{4} \Gamma \left (\frac {1}{4}\right )} - \frac {b^{\frac {7}{4}} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 a^{2} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {3 a^{2} b^{\frac {7}{4}} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 16 a^{3} b x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {a b^{\frac {11}{4}} x^{4} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 16 a^{3} b x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {4 b^{\frac {15}{4}} x^{8} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 16 a^{3} b x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-3*b**(3/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-7/4)/(16*a*x**4*gamma(1/4)) - b**(7/4)*(a/(b*x**4) - 1)*
*(3/4)*gamma(-7/4)/(4*a**2*gamma(1/4)), Abs(a/(b*x**4)) > 1), (-3*a**2*b**(7/4)*(-a/(b*x**4) + 1)**(3/4)*gamma
(-7/4)/(-16*a**3*b*x**4*exp(I*pi/4)*gamma(1/4) + 16*a**2*b**2*x**8*exp(I*pi/4)*gamma(1/4)) - a*b**(11/4)*x**4*
(-a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(-16*a**3*b*x**4*exp(I*pi/4)*gamma(1/4) + 16*a**2*b**2*x**8*exp(I*pi/4)*g
amma(1/4)) + 4*b**(15/4)*x**8*(-a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(-16*a**3*b*x**4*exp(I*pi/4)*gamma(1/4) + 1
6*a**2*b**2*x**8*exp(I*pi/4)*gamma(1/4)), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^8), x)

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Mupad [B]
time = 1.17, size = 38, normalized size = 0.83 \begin {gather*} -\frac {3\,a\,{\left (a-b\,x^4\right )}^{3/4}+4\,b\,x^4\,{\left (a-b\,x^4\right )}^{3/4}}{21\,a^2\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a - b*x^4)^(1/4)),x)

[Out]

-(3*a*(a - b*x^4)^(3/4) + 4*b*x^4*(a - b*x^4)^(3/4))/(21*a^2*x^7)

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