3.1006 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{10}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}-\frac{\left (a+b x^4\right )^{5/4}}{9 a x^9} \]

[Out]

-(a + b*x^4)^(5/4)/(9*a*x^9) + (4*b*(a + b*x^4)^(5/4))/(45*a^2*x^5)

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Rubi [A]  time = 0.0100388, antiderivative size = 44, 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, 264} \[ \frac{4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}-\frac{\left (a+b x^4\right )^{5/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(1/4)/x^10,x]

[Out]

-(a + b*x^4)^(5/4)/(9*a*x^9) + (4*b*(a + b*x^4)^(5/4))/(45*a^2*x^5)

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 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{\sqrt [4]{a+b x^4}}{x^{10}} \, dx &=-\frac{\left (a+b x^4\right )^{5/4}}{9 a x^9}-\frac{(4 b) \int \frac{\sqrt [4]{a+b x^4}}{x^6} \, dx}{9 a}\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{9 a x^9}+\frac{4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}\\ \end{align*}

Mathematica [A]  time = 0.009047, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^4\right )^{5/4} \left (4 b x^4-5 a\right )}{45 a^2 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(1/4)/x^10,x]

[Out]

((a + b*x^4)^(5/4)*(-5*a + 4*b*x^4))/(45*a^2*x^9)

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Maple [A]  time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-4\,b{x}^{4}+5\,a}{45\,{a}^{2}{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(1/4)/x^10,x)

[Out]

-1/45*(b*x^4+a)^(5/4)*(-4*b*x^4+5*a)/a^2/x^9

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Maxima [A]  time = 0.960888, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b}{x^{5}} - \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{x^{9}}}{45 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^10,x, algorithm="maxima")

[Out]

1/45*(9*(b*x^4 + a)^(5/4)*b/x^5 - 5*(b*x^4 + a)^(9/4)/x^9)/a^2

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Fricas [A]  time = 1.76902, size = 86, normalized size = 1.95 \begin{align*} \frac{{\left (4 \, b^{2} x^{8} - a b x^{4} - 5 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, a^{2} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^10,x, algorithm="fricas")

[Out]

1/45*(4*b^2*x^8 - a*b*x^4 - 5*a^2)*(b*x^4 + a)^(1/4)/(a^2*x^9)

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Sympy [B]  time = 2.37167, size = 109, normalized size = 2.48 \begin{align*} - \frac{5 \sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{16 x^{8} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{16 a x^{4} \Gamma \left (- \frac{1}{4}\right )} + \frac{b^{\frac{9}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{4 a^{2} \Gamma \left (- \frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**10,x)

[Out]

-5*b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(16*x**8*gamma(-1/4)) - b**(5/4)*(a/(b*x**4) + 1)**(1/4)*gamma
(-9/4)/(16*a*x**4*gamma(-1/4)) + b**(9/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(4*a**2*gamma(-1/4))

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Giac [A]  time = 1.19837, size = 81, normalized size = 1.84 \begin{align*} \frac{\frac{9 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b}{x} - \frac{5 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{9}}}{45 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^10,x, algorithm="giac")

[Out]

1/45*(9*(b*x^4 + a)^(1/4)*(b + a/x^4)*b/x - 5*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/x^9)/a^2