3.1038 \(\int \frac{(a+b x^4)^{3/4}}{x^{16}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}} \]

[Out]

-(a + b*x^4)^(7/4)/(15*a*x^15) + (8*b*(a + b*x^4)^(7/4))/(165*a^2*x^11) - (32*b^2*(a + b*x^4)^(7/4))/(1155*a^3
*x^7)

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Rubi [A]  time = 0.0195247, 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} \[ -\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^16,x]

[Out]

-(a + b*x^4)^(7/4)/(15*a*x^15) + (8*b*(a + b*x^4)^(7/4))/(165*a^2*x^11) - (32*b^2*(a + b*x^4)^(7/4))/(1155*a^3
*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^{16}} \, dx &=-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}}-\frac{(8 b) \int \frac{\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx}{15 a}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}+\frac{\left (32 b^2\right ) \int \frac{\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{165 a^2}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0103439, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^4\right )^{7/4} \left (77 a^2-56 a b x^4+32 b^2 x^8\right )}{1155 a^3 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^16,x]

[Out]

-((a + b*x^4)^(7/4)*(77*a^2 - 56*a*b*x^4 + 32*b^2*x^8))/(1155*a^3*x^15)

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Maple [A]  time = 0.005, size = 39, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}-56\,ab{x}^{4}+77\,{a}^{2}}{1155\,{a}^{3}{x}^{15}} \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^16,x)

[Out]

-1/1155*(b*x^4+a)^(7/4)*(32*b^2*x^8-56*a*b*x^4+77*a^2)/a^3/x^15

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Maxima [A]  time = 0.996411, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{165 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} - \frac{210 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} b}{x^{11}} + \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{x^{15}}}{1155 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="maxima")

[Out]

-1/1155*(165*(b*x^4 + a)^(7/4)*b^2/x^7 - 210*(b*x^4 + a)^(11/4)*b/x^11 + 77*(b*x^4 + a)^(15/4)/x^15)/a^3

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Fricas [A]  time = 1.76497, size = 123, normalized size = 1.81 \begin{align*} -\frac{{\left (32 \, b^{3} x^{12} - 24 \, a b^{2} x^{8} + 21 \, a^{2} b x^{4} + 77 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, a^{3} x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="fricas")

[Out]

-1/1155*(32*b^3*x^12 - 24*a*b^2*x^8 + 21*a^2*b*x^4 + 77*a^3)*(b*x^4 + a)^(3/4)/(a^3*x^15)

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Sympy [B]  time = 7.12324, size = 520, normalized size = 7.65 \begin{align*} \frac{77 a^{5} b^{\frac{19}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{175 a^{4} b^{\frac{23}{4}} x^{4} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{95 a^{3} b^{\frac{27}{4}} x^{8} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{5 a^{2} b^{\frac{31}{4}} x^{12} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{40 a b^{\frac{35}{4}} x^{16} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{32 b^{\frac{39}{4}} x^{20} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \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**16,x)

[Out]

77*a**5*b**(19/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*g
amma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 175*a**4*b**(23/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(6
4*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 95*a**3*b*
*(27/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(
-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 5*a**2*b**(31/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5
*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 40*a*b**(35/4)*x
**16*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) +
64*a**3*b**6*x**20*gamma(-3/4)) + 32*b**(39/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*
gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{16}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^16, x)