3.1067 \(\int \frac{(a+b x^4)^{5/4}}{x^{10}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)

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Rubi [A]  time = 0.0045641, 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} \[ -\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^10,x]

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)

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 )^{5/4}}{x^{10}} \, dx &=-\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9}\\ \end{align*}

Mathematica [A]  time = 0.0057183, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(5/4)/x^10,x]

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{9\,a{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(5/4)/x^10,x)

[Out]

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

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Maxima [A]  time = 0.97999, size = 23, normalized size = 1.1 \begin{align*} -\frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, a x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^10,x, algorithm="maxima")

[Out]

-1/9*(b*x^4 + a)^(9/4)/(a*x^9)

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Fricas [B]  time = 2.03512, size = 81, normalized size = 3.86 \begin{align*} -\frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{9 \, a x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^10,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.51322, size = 105, normalized size = 5. \begin{align*} \frac{a \sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{4 x^{8} \Gamma \left (- \frac{5}{4}\right )} + \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{2 x^{4} \Gamma \left (- \frac{5}{4}\right )} + \frac{b^{\frac{9}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{4 a \Gamma \left (- \frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(5/4)/x**10,x)

[Out]

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

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Giac [B]  time = 1.13721, size = 47, normalized size = 2.24 \begin{align*} -\frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{9 \, a x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^10,x, algorithm="giac")

[Out]

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