3.11.7 \(\int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx\) [1007]

Optimal. Leaf size=68 \[ -\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}+\frac {8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5} \]

[Out]

-1/13*(b*x^4+a)^(5/4)/a/x^13+8/117*b*(b*x^4+a)^(5/4)/a^2/x^9-32/585*b^2*(b*x^4+a)^(5/4)/a^3/x^5

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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 )^{5/4}}{585 a^3 x^5}+\frac {8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(1/4)/x^14,x]

[Out]

-1/13*(a + b*x^4)^(5/4)/(a*x^13) + (8*b*(a + b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2*(a + b*x^4)^(5/4))/(585*a^3
*x^5)

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 {\sqrt [4]{a+b x^4}}{x^{14}} \, dx &=-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}-\frac {(8 b) \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}+\frac {8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}+\frac {\left (32 b^2\right ) \int \frac {\sqrt [4]{a+b x^4}}{x^6} \, dx}{117 a^2}\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}+\frac {8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 53, normalized size = 0.78 \begin {gather*} \frac {\sqrt [4]{a+b x^4} \left (-45 a^3-5 a^2 b x^4+8 a b^2 x^8-32 b^3 x^{12}\right )}{585 a^3 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(1/4)/x^14,x]

[Out]

((a + b*x^4)^(1/4)*(-45*a^3 - 5*a^2*b*x^4 + 8*a*b^2*x^8 - 32*b^3*x^12))/(585*a^3*x^13)

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Maple [A]
time = 0.17, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}-40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\)	\(39\)
trager	\(-\frac {\left (32 b^{3} x^{12}-8 a \,b^{2} x^{8}+5 a^{2} b \,x^{4}+45 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{585 x^{13} a^{3}}\)	\(50\)
risch	\(-\frac {\left (32 b^{3} x^{12}-8 a \,b^{2} x^{8}+5 a^{2} b \,x^{4}+45 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{585 x^{13} a^{3}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(1/4)/x^14,x,method=_RETURNVERBOSE)

[Out]

-1/585*(b*x^4+a)^(5/4)*(32*b^2*x^8-40*a*b*x^4+45*a^2)/x^13/a^3

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Maxima [A]
time = 0.29, size = 52, normalized size = 0.76 \begin {gather*} -\frac {\frac {117 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2}}{x^{5}} - \frac {130 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b}{x^{9}} + \frac {45 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}}}{x^{13}}}{585 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="maxima")

[Out]

-1/585*(117*(b*x^4 + a)^(5/4)*b^2/x^5 - 130*(b*x^4 + a)^(9/4)*b/x^9 + 45*(b*x^4 + a)^(13/4)/x^13)/a^3

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Fricas [A]
time = 0.39, size = 49, normalized size = 0.72 \begin {gather*} -\frac {{\left (32 \, b^{3} x^{12} - 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} + 45 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, a^{3} x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="fricas")

[Out]

-1/585*(32*b^3*x^12 - 8*a*b^2*x^8 + 5*a^2*b*x^4 + 45*a^3)*(b*x^4 + a)^(1/4)/(a^3*x^13)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
time = 1.04, size = 520, normalized size = 7.65 \begin {gather*} \frac {45 a^{5} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {95 a^{4} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {47 a^{3} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {21 a^{2} b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {56 a b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {32 b^{\frac {37}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**14,x)

[Out]

45*a**5*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*g
amma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 95*a**4*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64
*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 47*a**3*b**
(25/4)*x**8*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-
1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 21*a**2*b**(29/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5
*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*b**(33/4)*x
**16*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) +
64*a**3*b**6*x**20*gamma(-1/4)) + 32*b**(37/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*
gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4))

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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)^(1/4)/x^14,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^14, x)

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Mupad [B]
time = 1.48, size = 73, normalized size = 1.07 \begin {gather*} \frac {8\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{585\,a^2\,x^5}-\frac {b\,{\left (b\,x^4+a\right )}^{1/4}}{117\,a\,x^9}-\frac {32\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{585\,a^3\,x}-\frac {{\left (b\,x^4+a\right )}^{1/4}}{13\,x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(1/4)/x^14,x)

[Out]

(8*b^2*(a + b*x^4)^(1/4))/(585*a^2*x^5) - (b*(a + b*x^4)^(1/4))/(117*a*x^9) - (32*b^3*(a + b*x^4)^(1/4))/(585*
a^3*x) - (a + b*x^4)^(1/4)/(13*x^13)

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