3.1036 \(\int \frac{(a+b x^4)^{3/4}}{x^8} \, dx\)

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

[Out]

-(a + b*x^4)^(7/4)/(7*a*x^7)

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Rubi [A]  time = 0.0044587, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b x^4\right )^{7/4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^8,x]

[Out]

-(a + b*x^4)^(7/4)/(7*a*x^7)

Rule 264

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

Mathematica [A]  time = 0.0051894, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^4\right )^{7/4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^8,x]

[Out]

-(a + b*x^4)^(7/4)/(7*a*x^7)

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,a{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^8,x)

[Out]

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

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Maxima [A]  time = 0.980219, size = 23, normalized size = 1.1 \begin{align*} -\frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, a x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^8,x, algorithm="maxima")

[Out]

-1/7*(b*x^4 + a)^(7/4)/(a*x^7)

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Fricas [A]  time = 1.73672, size = 43, normalized size = 2.05 \begin{align*} -\frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, a x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^8,x, algorithm="fricas")

[Out]

-1/7*(b*x^4 + a)^(7/4)/(a*x^7)

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Sympy [B]  time = 2.05122, size = 68, normalized size = 3.24 \begin{align*} \frac{b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{4 x^{4} \Gamma \left (- \frac{3}{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 \Gamma \left (- \frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**8,x)

[Out]

b**(3/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(4*x**4*gamma(-3/4)) + b**(7/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-7/
4)/(4*a*gamma(-3/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^8,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^8, x)