3.1007 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{14}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5}+\frac{8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}} \]

[Out]

-(a + b*x^4)^(5/4)/(13*a*x^13) + (8*b*(a + b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2*(a + b*x^4)^(5/4))/(585*a^3*x
^5)

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Rubi [A]  time = 0.0190968, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5}+\frac{8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(1/4)/x^14,x]

[Out]

-(a + b*x^4)^(5/4)/(13*a*x^13) + (8*b*(a + b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2*(a + b*x^4)^(5/4))/(585*a^3*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^{14}} \, dx &=-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}}-\frac{(8 b) \int \frac{\sqrt [4]{a+b x^4}}{x^{10}} \, dx}{13 a}\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}}+\frac{8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}+\frac{\left (32 b^2\right ) \int \frac{\sqrt [4]{a+b x^4}}{x^6} \, dx}{117 a^2}\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}}+\frac{8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5}\\ \end{align*}

Mathematica [A]  time = 0.010213, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^4\right )^{5/4} \left (45 a^2-40 a b x^4+32 b^2 x^8\right )}{585 a^3 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(1/4)/x^14,x]

[Out]

-((a + b*x^4)^(5/4)*(45*a^2 - 40*a*b*x^4 + 32*b^2*x^8))/(585*a^3*x^13)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}-40\,ab{x}^{4}+45\,{a}^{2}}{585\,{x}^{13}{a}^{3}} \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^14,x)

[Out]

-1/585*(b*x^4+a)^(5/4)*(32*b^2*x^8-40*a*b*x^4+45*a^2)/x^13/a^3

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Maxima [A]  time = 0.997423, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} - \frac{130 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} + \frac{45 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{585 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="maxima")

[Out]

-1/585*(117*(b*x^4 + a)^(5/4)*b^2/x^5 - 130*(b*x^4 + a)^(9/4)*b/x^9 + 45*(b*x^4 + a)^(13/4)/x^13)/a^3

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Fricas [A]  time = 1.74828, size = 119, normalized size = 1.75 \begin{align*} -\frac{{\left (32 \, b^{3} x^{12} - 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} + 45 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, a^{3} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="fricas")

[Out]

-1/585*(32*b^3*x^12 - 8*a*b^2*x^8 + 5*a^2*b*x^4 + 45*a^3)*(b*x^4 + a)^(1/4)/(a^3*x^13)

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Sympy [B]  time = 4.93775, size = 520, normalized size = 7.65 \begin{align*} \frac{45 a^{5} b^{\frac{17}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{95 a^{4} b^{\frac{21}{4}} x^{4} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{47 a^{3} b^{\frac{25}{4}} x^{8} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{21 a^{2} b^{\frac{29}{4}} x^{12} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{56 a b^{\frac{33}{4}} x^{16} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{32 b^{\frac{37}{4}} x^{20} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \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**14,x)

[Out]

45*a**5*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*g
amma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 95*a**4*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64
*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 47*a**3*b**
(25/4)*x**8*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-
1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 21*a**2*b**(29/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5
*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*b**(33/4)*x
**16*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) +
64*a**3*b**6*x**20*gamma(-1/4)) + 32*b**(37/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*
gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4))

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Giac [A]  time = 1.13708, size = 143, normalized size = 2.1 \begin{align*} -\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}}}{585 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="giac")

[Out]

-1/585*(117*(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^12 + 3*a*b^2*x^8 + 3*a^2*b*x^4 + a^3)*(b*x^4 + a)^(1/4)/x^13)/a^3