3.1037 \(\int \frac{(a+b x^4)^{3/4}}{x^{12}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}-\frac{\left (a+b x^4\right )^{7/4}}{11 a x^{11}} \]

[Out]

-(a + b*x^4)^(7/4)/(11*a*x^11) + (4*b*(a + b*x^4)^(7/4))/(77*a^2*x^7)

________________________________________________________________________________________

Rubi [A]  time = 0.0109337, 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} \[ \frac{4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}-\frac{\left (a+b x^4\right )^{7/4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^12,x]

[Out]

-(a + b*x^4)^(7/4)/(11*a*x^11) + (4*b*(a + b*x^4)^(7/4))/(77*a^2*x^7)

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 )^{3/4}}{x^{12}} \, dx &=-\frac{\left (a+b x^4\right )^{7/4}}{11 a x^{11}}-\frac{(4 b) \int \frac{\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{11 a}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{11 a x^{11}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0092279, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^4\right )^{7/4} \left (4 b x^4-7 a\right )}{77 a^2 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^12,x]

[Out]

((a + b*x^4)^(7/4)*(-7*a + 4*b*x^4))/(77*a^2*x^11)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-4\,b{x}^{4}+7\,a}{77\,{x}^{11}{a}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^12,x)

[Out]

-1/77*(b*x^4+a)^(7/4)*(-4*b*x^4+7*a)/x^11/a^2

________________________________________________________________________________________

Maxima [A]  time = 0.962838, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{11 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b}{x^{7}} - \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{x^{11}}}{77 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^12,x, algorithm="maxima")

[Out]

1/77*(11*(b*x^4 + a)^(7/4)*b/x^7 - 7*(b*x^4 + a)^(11/4)/x^11)/a^2

________________________________________________________________________________________

Fricas [A]  time = 1.75241, size = 90, normalized size = 2.05 \begin{align*} \frac{{\left (4 \, b^{2} x^{8} - 3 \, a b x^{4} - 7 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \, a^{2} x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^12,x, algorithm="fricas")

[Out]

1/77*(4*b^2*x^8 - 3*a*b*x^4 - 7*a^2)*(b*x^4 + a)^(3/4)/(a^2*x^11)

________________________________________________________________________________________

Sympy [B]  time = 3.69049, size = 110, normalized size = 2.5 \begin{align*} - \frac{7 b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{11}{4}\right )}{16 x^{8} \Gamma \left (- \frac{3}{4}\right )} - \frac{3 b^{\frac{7}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{11}{4}\right )}{16 a x^{4} \Gamma \left (- \frac{3}{4}\right )} + \frac{b^{\frac{11}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{11}{4}\right )}{4 a^{2} \Gamma \left (- \frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**12,x)

[Out]

-7*b**(3/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(16*x**8*gamma(-3/4)) - 3*b**(7/4)*(a/(b*x**4) + 1)**(3/4)*ga
mma(-11/4)/(16*a*x**4*gamma(-3/4)) + b**(11/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(4*a**2*gamma(-3/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^12, x)