3.1068 \(\int \frac{(a+b x^4)^{5/4}}{x^{14}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}-\frac{\left (a+b x^4\right )^{9/4}}{13 a x^{13}} \]

[Out]

-(a + b*x^4)^(9/4)/(13*a*x^13) + (4*b*(a + b*x^4)^(9/4))/(117*a^2*x^9)

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Rubi [A]  time = 0.0108018, 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} \[ \frac{4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}-\frac{\left (a+b x^4\right )^{9/4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^14,x]

[Out]

-(a + b*x^4)^(9/4)/(13*a*x^13) + (4*b*(a + b*x^4)^(9/4))/(117*a^2*x^9)

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{\left (a+b x^4\right )^{5/4}}{x^{14}} \, dx &=-\frac{\left (a+b x^4\right )^{9/4}}{13 a x^{13}}-\frac{(4 b) \int \frac{\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx}{13 a}\\ &=-\frac{\left (a+b x^4\right )^{9/4}}{13 a x^{13}}+\frac{4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0106172, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^4\right )^{9/4} \left (4 b x^4-9 a\right )}{117 a^2 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(5/4)/x^14,x]

[Out]

((a + b*x^4)^(9/4)*(-9*a + 4*b*x^4))/(117*a^2*x^13)

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Maple [A]  time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-4\,b{x}^{4}+9\,a}{117\,{x}^{13}{a}^{2}} \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^14,x)

[Out]

-1/117*(b*x^4+a)^(9/4)*(-4*b*x^4+9*a)/x^13/a^2

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Maxima [A]  time = 0.989949, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{13 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} - \frac{9 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{117 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^14,x, algorithm="maxima")

[Out]

1/117*(13*(b*x^4 + a)^(9/4)*b/x^9 - 9*(b*x^4 + a)^(13/4)/x^13)/a^2

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Fricas [A]  time = 2.04774, size = 113, normalized size = 2.57 \begin{align*} \frac{{\left (4 \, b^{3} x^{12} - a b^{2} x^{8} - 14 \, a^{2} b x^{4} - 9 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{117 \, a^{2} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^14,x, algorithm="fricas")

[Out]

1/117*(4*b^3*x^12 - a*b^2*x^8 - 14*a^2*b*x^4 - 9*a^3)*(b*x^4 + a)^(1/4)/(a^2*x^13)

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Sympy [B]  time = 7.61977, size = 148, normalized size = 3.36 \begin{align*} - \frac{9 a \sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{16 x^{12} \Gamma \left (- \frac{5}{4}\right )} - \frac{7 b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{8 x^{8} \Gamma \left (- \frac{5}{4}\right )} - \frac{b^{\frac{9}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{16 a x^{4} \Gamma \left (- \frac{5}{4}\right )} + \frac{b^{\frac{13}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{4 a^{2} \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**14,x)

[Out]

-9*a*b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(16*x**12*gamma(-5/4)) - 7*b**(5/4)*(a/(b*x**4) + 1)**(1/4)
*gamma(-13/4)/(8*x**8*gamma(-5/4)) - b**(9/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(16*a*x**4*gamma(-5/4)) + b
**(13/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(4*a**2*gamma(-5/4))

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Giac [B]  time = 1.1319, size = 234, normalized size = 5.32 \begin{align*} \frac{\frac{13 \,{\left (\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}}\right )} b}{a} - \frac{\frac{117 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{2}}{x} - \frac{130 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{9}} + \frac{45 \,{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{13}}}{a}}{585 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^14,x, algorithm="giac")

[Out]

1/585*(13*(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)*b/a - (1
17*(b*x^4 + a)^(1/4)*(b + a/x^4)*b^2/x - 130*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)*b/x^9 + 45*(b^3*x^1
2 + 3*a*b^2*x^8 + 3*a^2*b*x^4 + a^3)*(b*x^4 + a)^(1/4)/x^13)/a)/a