3.11.99 \(\int \frac {1}{x^8 \sqrt [4]{a+b x^4}} \, dx\) [1099]

Optimal. Leaf size=44 \[ -\frac {\left (a+b x^4\right )^{3/4}}{7 a x^7}+\frac {4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3} \]

[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 = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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 = 0.15, size = 31, normalized size = 0.70 \begin {gather*} \frac {\left (a+b x^4\right )^{3/4} \left (-3 a+4 b x^4\right )}{21 a^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.16, size = 28, normalized size = 0.64

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-4 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}\)	\(28\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-4 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}\)	\(28\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-4 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}\)	\(28\)

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 = 0.29, size = 35, 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 = 0.38, size = 27, 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 [A]
time = 0.56, size = 70, normalized size = 1.59 \begin {gather*} - \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 )} \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]

-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)*gamm
a(-7/4)/(4*a**2*gamma(1/4))

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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 = 1.16, size = 36, normalized size = 0.82 \begin {gather*} -\frac {3\,a\,{\left (b\,x^4+a\right )}^{3/4}-4\,b\,x^4\,{\left (b\,x^4+a\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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