3.1069 \(\int \frac{(a+b x^4)^{5/4}}{x^{18}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{32 b^2 \left (a+b x^4\right )^{9/4}}{1989 a^3 x^9}+\frac{8 b \left (a+b x^4\right )^{9/4}}{221 a^2 x^{13}}-\frac{\left (a+b x^4\right )^{9/4}}{17 a x^{17}} \]

[Out]

-(a + b*x^4)^(9/4)/(17*a*x^17) + (8*b*(a + b*x^4)^(9/4))/(221*a^2*x^13) - (32*b^2*(a + b*x^4)^(9/4))/(1989*a^3
*x^9)

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Rubi [A]  time = 0.019809, 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} \[ -\frac{32 b^2 \left (a+b x^4\right )^{9/4}}{1989 a^3 x^9}+\frac{8 b \left (a+b x^4\right )^{9/4}}{221 a^2 x^{13}}-\frac{\left (a+b x^4\right )^{9/4}}{17 a x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^18,x]

[Out]

-(a + b*x^4)^(9/4)/(17*a*x^17) + (8*b*(a + b*x^4)^(9/4))/(221*a^2*x^13) - (32*b^2*(a + b*x^4)^(9/4))/(1989*a^3
*x^9)

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 )^{5/4}}{x^{18}} \, dx &=-\frac{\left (a+b x^4\right )^{9/4}}{17 a x^{17}}-\frac{(8 b) \int \frac{\left (a+b x^4\right )^{5/4}}{x^{14}} \, dx}{17 a}\\ &=-\frac{\left (a+b x^4\right )^{9/4}}{17 a x^{17}}+\frac{8 b \left (a+b x^4\right )^{9/4}}{221 a^2 x^{13}}+\frac{\left (32 b^2\right ) \int \frac{\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx}{221 a^2}\\ &=-\frac{\left (a+b x^4\right )^{9/4}}{17 a x^{17}}+\frac{8 b \left (a+b x^4\right )^{9/4}}{221 a^2 x^{13}}-\frac{32 b^2 \left (a+b x^4\right )^{9/4}}{1989 a^3 x^9}\\ \end{align*}

Mathematica [A]  time = 0.011426, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^4\right )^{9/4} \left (117 a^2-72 a b x^4+32 b^2 x^8\right )}{1989 a^3 x^{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(5/4)/x^18,x]

[Out]

-((a + b*x^4)^(9/4)*(117*a^2 - 72*a*b*x^4 + 32*b^2*x^8))/(1989*a^3*x^17)

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Maple [A]  time = 0.005, size = 39, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}-72\,ab{x}^{4}+117\,{a}^{2}}{1989\,{x}^{17}{a}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(5/4)/x^18,x)

[Out]

-1/1989*(b*x^4+a)^(9/4)*(32*b^2*x^8-72*a*b*x^4+117*a^2)/x^17/a^3

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Maxima [A]  time = 1.0099, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{221 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b^{2}}{x^{9}} - \frac{306 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} b}{x^{13}} + \frac{117 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}}}{x^{17}}}{1989 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^18,x, algorithm="maxima")

[Out]

-1/1989*(221*(b*x^4 + a)^(9/4)*b^2/x^9 - 306*(b*x^4 + a)^(13/4)*b/x^13 + 117*(b*x^4 + a)^(17/4)/x^17)/a^3

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Fricas [A]  time = 1.98473, size = 147, normalized size = 2.16 \begin{align*} -\frac{{\left (32 \, b^{4} x^{16} - 8 \, a b^{3} x^{12} + 5 \, a^{2} b^{2} x^{8} + 162 \, a^{3} b x^{4} + 117 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{1989 \, a^{3} x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^18,x, algorithm="fricas")

[Out]

-1/1989*(32*b^4*x^16 - 8*a*b^3*x^12 + 5*a^2*b^2*x^8 + 162*a^3*b*x^4 + 117*a^4)*(b*x^4 + a)^(1/4)/(a^3*x^17)

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Sympy [B]  time = 20.5969, size = 609, normalized size = 8.96 \begin{align*} \frac{117 a^{6} b^{\frac{17}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} + \frac{396 a^{5} b^{\frac{21}{4}} x^{4} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} + \frac{446 a^{4} b^{\frac{25}{4}} x^{8} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} + \frac{164 a^{3} b^{\frac{29}{4}} x^{12} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} + \frac{21 a^{2} b^{\frac{33}{4}} x^{16} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} + \frac{56 a b^{\frac{37}{4}} x^{20} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} + \frac{32 b^{\frac{41}{4}} x^{24} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac{5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac{5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(5/4)/x**18,x)

[Out]

117*a**6*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*
gamma(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 396*a**5*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(
64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 446*a**4*
b**(25/4)*x**8*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamm
a(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 164*a**3*b**(29/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*
a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 21*a**2*b**(
33/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-
5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 56*a*b**(37/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b*
*4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 32*b**(41/4)*x**24*
(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a*
*3*b**6*x**24*gamma(-5/4))

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Giac [B]  time = 1.14563, size = 373, normalized size = 5.49 \begin{align*} -\frac{\frac{17 \,{\left (\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}}\right )} b}{a^{2}} - \frac{3 \,{\left (\frac{663 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{3}}{x} - \frac{1105 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x^{9}} + \frac{765 \,{\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}} b}{x^{13}} - \frac{195 \,{\left (b^{4} x^{16} + 4 \, a b^{3} x^{12} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{4} + a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{17}}\right )}}{a^{2}}}{9945 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^18,x, algorithm="giac")

[Out]

-1/9945*(17*(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)*b/a^2 - 3*(663*(b*x^4 + a)^(1/4)*(b
+ a/x^4)*b^3/x - 1105*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)*b^2/x^9 + 765*(b^3*x^12 + 3*a*b^2*x^8 + 3
*a^2*b*x^4 + a^3)*(b*x^4 + a)^(1/4)*b/x^13 - 195*(b^4*x^16 + 4*a*b^3*x^12 + 6*a^2*b^2*x^8 + 4*a^3*b*x^4 + a^4)
*(b*x^4 + a)^(1/4)/x^17)/a^2)/a