3.11.69 \(\int \frac {(a+b x^4)^{5/4}}{x^{18}} \, dx\) [1069]

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

[Out]

-1/17*(b*x^4+a)^(9/4)/a/x^17+8/221*b*(b*x^4+a)^(9/4)/a^2/x^13-32/1989*b^2*(b*x^4+a)^(9/4)/a^3/x^9

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^18,x]

[Out]

-1/17*(a + b*x^4)^(9/4)/(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 270

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]

Rule 277

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]

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*}

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Mathematica [A]
time = 0.14, size = 42, normalized size = 0.62 \begin {gather*} \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}} \end {gather*}

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 = 0.18, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (32 b^{2} x^{8}-72 a b \,x^{4}+117 a^{2}\right )}{1989 x^{17} a^{3}}\)	\(39\)
trager	\(-\frac {\left (32 x^{16} b^{4}-8 a \,b^{3} x^{12}+5 a^{2} b^{2} x^{8}+162 a^{3} b \,x^{4}+117 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{1989 x^{17} a^{3}}\)	\(61\)
risch	\(-\frac {\left (32 x^{16} b^{4}-8 a \,b^{3} x^{12}+5 a^{2} b^{2} x^{8}+162 a^{3} b \,x^{4}+117 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{1989 x^{17} a^{3}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 52, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.40, size = 60, normalized size = 0.88 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (61) = 122\).
time = 2.10, size = 609, normalized size = 8.96 \begin {gather*} \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 {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^4 + a)^(5/4)/x^18, x)

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Mupad [B]
time = 2.07, size = 91, normalized size = 1.34 \begin {gather*} \frac {8\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{1989\,a^2\,x^5}-\frac {18\,b\,{\left (b\,x^4+a\right )}^{1/4}}{221\,x^{13}}-\frac {32\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{1989\,a^3\,x}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{17\,x^{17}}-\frac {5\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{1989\,a\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(5/4)/x^18,x)

[Out]

(8*b^3*(a + b*x^4)^(1/4))/(1989*a^2*x^5) - (18*b*(a + b*x^4)^(1/4))/(221*x^13) - (32*b^4*(a + b*x^4)^(1/4))/(1
989*a^3*x) - (a*(a + b*x^4)^(1/4))/(17*x^17) - (5*b^2*(a + b*x^4)^(1/4))/(1989*a*x^9)

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