3.1159 \(\int \frac{1}{x^8 (a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=66 \[ \frac{32 b^2 x}{21 a^3 \sqrt [4]{a+b x^4}}+\frac{8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}-\frac{1}{7 a x^7 \sqrt [4]{a+b x^4}} \]

[Out]

-1/(7*a*x^7*(a + b*x^4)^(1/4)) + (8*b)/(21*a^2*x^3*(a + b*x^4)^(1/4)) + (32*b^2*x)/(21*a^3*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0160023, antiderivative size = 66, 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, 191} \[ \frac{32 b^2 x}{21 a^3 \sqrt [4]{a+b x^4}}+\frac{8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}-\frac{1}{7 a x^7 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^4)^(5/4)),x]

[Out]

-1/(7*a*x^7*(a + b*x^4)^(1/4)) + (8*b)/(21*a^2*x^3*(a + b*x^4)^(1/4)) + (32*b^2*x)/(21*a^3*(a + b*x^4)^(1/4))

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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^8 \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{7 a x^7 \sqrt [4]{a+b x^4}}-\frac{(8 b) \int \frac{1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx}{7 a}\\ &=-\frac{1}{7 a x^7 \sqrt [4]{a+b x^4}}+\frac{8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}+\frac{\left (32 b^2\right ) \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{21 a^2}\\ &=-\frac{1}{7 a x^7 \sqrt [4]{a+b x^4}}+\frac{8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}+\frac{32 b^2 x}{21 a^3 \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0077659, size = 42, normalized size = 0.64 \[ -\frac{3 a^2-8 a b x^4-32 b^2 x^8}{21 a^3 x^7 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^4)^(5/4)),x]

[Out]

-(3*a^2 - 8*a*b*x^4 - 32*b^2*x^8)/(21*a^3*x^7*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.005, size = 39, normalized size = 0.6 \begin{align*} -{\frac{-32\,{b}^{2}{x}^{8}-8\,ab{x}^{4}+3\,{a}^{2}}{21\,{a}^{3}{x}^{7}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^4+a)^(5/4),x)

[Out]

-1/21*(-32*b^2*x^8-8*a*b*x^4+3*a^2)/x^7/(b*x^4+a)^(1/4)/a^3

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Maxima [A]  time = 1.01117, size = 72, normalized size = 1.09 \begin{align*} \frac{b^{2} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}} + \frac{\frac{14 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b}{x^{3}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{x^{7}}}{21 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

b^2*x/((b*x^4 + a)^(1/4)*a^3) + 1/21*(14*(b*x^4 + a)^(3/4)*b/x^3 - 3*(b*x^4 + a)^(7/4)/x^7)/a^3

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Fricas [A]  time = 1.51617, size = 108, normalized size = 1.64 \begin{align*} \frac{{\left (32 \, b^{2} x^{8} + 8 \, a b x^{4} - 3 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21 \,{\left (a^{3} b x^{11} + a^{4} x^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/21*(32*b^2*x^8 + 8*a*b*x^4 - 3*a^2)*(b*x^4 + a)^(3/4)/(a^3*b*x^11 + a^4*x^7)

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Sympy [B]  time = 3.05461, size = 323, normalized size = 4.89 \begin{align*} - \frac{3 a^{3} b^{\frac{19}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac{5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac{5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac{5}{4}\right )} + \frac{5 a^{2} b^{\frac{23}{4}} x^{4} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac{5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac{5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac{5}{4}\right )} + \frac{40 a b^{\frac{27}{4}} x^{8} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac{5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac{5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac{5}{4}\right )} + \frac{32 b^{\frac{31}{4}} x^{12} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac{5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac{5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**4+a)**(5/4),x)

[Out]

-3*a**3*b**(19/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(64*a**5*b**4*x**4*gamma(5/4) + 128*a**4*b**5*x**8*gamma
(5/4) + 64*a**3*b**6*x**12*gamma(5/4)) + 5*a**2*b**(23/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(64*a**5*b*
*4*x**4*gamma(5/4) + 128*a**4*b**5*x**8*gamma(5/4) + 64*a**3*b**6*x**12*gamma(5/4)) + 40*a*b**(27/4)*x**8*(a/(
b*x**4) + 1)**(3/4)*gamma(-7/4)/(64*a**5*b**4*x**4*gamma(5/4) + 128*a**4*b**5*x**8*gamma(5/4) + 64*a**3*b**6*x
**12*gamma(5/4)) + 32*b**(31/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(64*a**5*b**4*x**4*gamma(5/4) + 128*
a**4*b**5*x**8*gamma(5/4) + 64*a**3*b**6*x**12*gamma(5/4))

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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{5}{4}} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^8), x)