3.1158 \(\int \frac{1}{x^4 (a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{4 b x}{3 a^2 \sqrt [4]{a+b x^4}}-\frac{1}{3 a x^3 \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.007843, antiderivative size = 42, normalized size of antiderivative = 1., 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 = {271, 191} \[ -\frac{4 b x}{3 a^2 \sqrt [4]{a+b x^4}}-\frac{1}{3 a x^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^4)^(5/4)),x]

[Out]

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

Rule 271

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]

Rule 191

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

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{3 a x^3 \sqrt [4]{a+b x^4}}-\frac{(4 b) \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{3 a}\\ &=-\frac{1}{3 a x^3 \sqrt [4]{a+b x^4}}-\frac{4 b x}{3 a^2 \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.005932, size = 29, normalized size = 0.69 \[ -\frac{a+4 b x^4}{3 a^2 x^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^4)^(5/4)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 26, normalized size = 0.6 \begin{align*} -{\frac{4\,b{x}^{4}+a}{3\,{x}^{3}{a}^{2}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^4+a)^(5/4),x)

[Out]

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

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Maxima [A]  time = 0.990987, size = 46, normalized size = 1.1 \begin{align*} -\frac{b x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53167, size = 81, normalized size = 1.93 \begin{align*} -\frac{{\left (4 \, b x^{4} + a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \,{\left (a^{2} b x^{7} + a^{3} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.57327, size = 68, normalized size = 1.62 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right )}{16 a \sqrt [4]{b} x^{4} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (\frac{5}{4}\right )} + \frac{b^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a^{2} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**4+a)**(5/4),x)

[Out]

gamma(-3/4)/(16*a*b**(1/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(5/4)) + b**(3/4)*gamma(-3/4)/(4*a**2*(a/(b*x**4)
+ 1)**(1/4)*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^4), x)