3.1130 \(\int \frac{1}{x^{10} (a+b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}+\frac{8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac{\sqrt [4]{a+b x^4}}{9 a x^9} \]

[Out]

-(a + b*x^4)^(1/4)/(9*a*x^9) + (8*b*(a + b*x^4)^(1/4))/(45*a^2*x^5) - (32*b^2*(a + b*x^4)^(1/4))/(45*a^3*x)

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Rubi [A]  time = 0.0210661, 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} \[ -\frac{32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}+\frac{8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac{\sqrt [4]{a+b x^4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a + b*x^4)^(3/4)),x]

[Out]

-(a + b*x^4)^(1/4)/(9*a*x^9) + (8*b*(a + b*x^4)^(1/4))/(45*a^2*x^5) - (32*b^2*(a + b*x^4)^(1/4))/(45*a^3*x)

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{1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a+b x^4}}{9 a x^9}-\frac{(8 b) \int \frac{1}{x^6 \left (a+b x^4\right )^{3/4}} \, dx}{9 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{9 a x^9}+\frac{8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}+\frac{\left (32 b^2\right ) \int \frac{1}{x^2 \left (a+b x^4\right )^{3/4}} \, dx}{45 a^2}\\ &=-\frac{\sqrt [4]{a+b x^4}}{9 a x^9}+\frac{8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}\\ \end{align*}

Mathematica [A]  time = 0.0155905, size = 42, normalized size = 0.62 \[ -\frac{\sqrt [4]{a+b x^4} \left (5 a^2-8 a b x^4+32 b^2 x^8\right )}{45 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(a + b*x^4)^(3/4)),x]

[Out]

-((a + b*x^4)^(1/4)*(5*a^2 - 8*a*b*x^4 + 32*b^2*x^8))/(45*a^3*x^9)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}-8\,ab{x}^{4}+5\,{a}^{2}}{45\,{a}^{3}{x}^{9}}\sqrt [4]{b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(b*x^4+a)^(3/4),x)

[Out]

-1/45*(b*x^4+a)^(1/4)*(32*b^2*x^8-8*a*b*x^4+5*a^2)/a^3/x^9

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Maxima [A]  time = 1.00624, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{45 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x} - \frac{18 \,{\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^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

-1/45*(45*(b*x^4 + a)^(1/4)*b^2/x - 18*(b*x^4 + a)^(5/4)*b/x^5 + 5*(b*x^4 + a)^(9/4)/x^9)/a^3

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Fricas [A]  time = 1.50997, size = 92, normalized size = 1.35 \begin{align*} -\frac{{\left (32 \, b^{2} x^{8} - 8 \, a b x^{4} + 5 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, a^{3} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

-1/45*(32*b^2*x^8 - 8*a*b*x^4 + 5*a^2)*(b*x^4 + a)^(1/4)/(a^3*x^9)

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Sympy [B]  time = 3.14449, size = 406, normalized size = 5.97 \begin{align*} \frac{5 a^{4} b^{\frac{17}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac{3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac{3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac{3}{4}\right )} + \frac{2 a^{3} b^{\frac{21}{4}} x^{4} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac{3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac{3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac{3}{4}\right )} + \frac{21 a^{2} b^{\frac{25}{4}} x^{8} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac{3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac{3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac{3}{4}\right )} + \frac{56 a b^{\frac{29}{4}} x^{12} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac{3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac{3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac{3}{4}\right )} + \frac{32 b^{\frac{33}{4}} x^{16} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac{3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac{3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(b*x**4+a)**(3/4),x)

[Out]

5*a**4*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) + 128*a**4*b**5*x**12*gamma
(3/4) + 64*a**3*b**6*x**16*gamma(3/4)) + 2*a**3*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b*
*4*x**8*gamma(3/4) + 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)) + 21*a**2*b**(25/4)*x**8*
(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) + 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b
**6*x**16*gamma(3/4)) + 56*a*b**(29/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4)
+ 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)) + 32*b**(33/4)*x**16*(a/(b*x**4) + 1)**(1/4
)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) + 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4))

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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{3}{4}} x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^10), x)