3.11.68 \(\int \frac {(a+b x^4)^{5/4}}{x^{14}} \, dx\) [1068]

Optimal. Leaf size=44 \[ -\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9} \]

[Out]

-1/13*(b*x^4+a)^(9/4)/a/x^13+4/117*b*(b*x^4+a)^(9/4)/a^2/x^9

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} \frac {4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^14,x]

[Out]

-1/13*(a + b*x^4)^(9/4)/(a*x^13) + (4*b*(a + b*x^4)^(9/4))/(117*a^2*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^{14}} \, dx &=-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}-\frac {(4 b) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 31, normalized size = 0.70 \begin {gather*} \frac {\left (a+b x^4\right )^{9/4} \left (-9 a+4 b x^4\right )}{117 a^2 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(5/4)/x^14,x]

[Out]

((a + b*x^4)^(9/4)*(-9*a + 4*b*x^4))/(117*a^2*x^13)

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Maple [A]
time = 0.18, size = 28, normalized size = 0.64

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-4 b \,x^{4}+9 a \right )}{117 x^{13} a^{2}}\)	\(28\)
trager	\(-\frac {\left (-4 b^{3} x^{12}+a \,b^{2} x^{8}+14 a^{2} b \,x^{4}+9 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{117 x^{13} a^{2}}\)	\(49\)
risch	\(-\frac {\left (-4 b^{3} x^{12}+a \,b^{2} x^{8}+14 a^{2} b \,x^{4}+9 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{117 x^{13} a^{2}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(5/4)/x^14,x,method=_RETURNVERBOSE)

[Out]

-1/117*(b*x^4+a)^(9/4)*(-4*b*x^4+9*a)/x^13/a^2

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Maxima [A]
time = 0.30, size = 35, normalized size = 0.80 \begin {gather*} \frac {\frac {13 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b}{x^{9}} - \frac {9 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}}}{x^{13}}}{117 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^14,x, algorithm="maxima")

[Out]

1/117*(13*(b*x^4 + a)^(9/4)*b/x^9 - 9*(b*x^4 + a)^(13/4)/x^13)/a^2

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Fricas [A]
time = 0.39, size = 49, normalized size = 1.11 \begin {gather*} \frac {{\left (4 \, b^{3} x^{12} - a b^{2} x^{8} - 14 \, a^{2} b x^{4} - 9 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{117 \, a^{2} x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^14,x, algorithm="fricas")

[Out]

1/117*(4*b^3*x^12 - a*b^2*x^8 - 14*a^2*b*x^4 - 9*a^3)*(b*x^4 + a)^(1/4)/(a^2*x^13)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (37) = 74\).
time = 1.36, size = 148, normalized size = 3.36 \begin {gather*} - \frac {9 a \sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{16 x^{12} \Gamma \left (- \frac {5}{4}\right )} - \frac {7 b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{8 x^{8} \Gamma \left (- \frac {5}{4}\right )} - \frac {b^{\frac {9}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{16 a x^{4} \Gamma \left (- \frac {5}{4}\right )} + \frac {b^{\frac {13}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{4 a^{2} \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**14,x)

[Out]

-9*a*b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(16*x**12*gamma(-5/4)) - 7*b**(5/4)*(a/(b*x**4) + 1)**(1/4)
*gamma(-13/4)/(8*x**8*gamma(-5/4)) - b**(9/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(16*a*x**4*gamma(-5/4)) + b
**(13/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(4*a**2*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^14,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^14, x)

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Mupad [B]
time = 1.74, size = 71, normalized size = 1.61 \begin {gather*} \frac {4\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{117\,a^2\,x}-\frac {14\,b\,{\left (b\,x^4+a\right )}^{1/4}}{117\,x^9}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{13\,x^{13}}-\frac {b^2\,{\left (b\,x^4+a\right )}^{1/4}}{117\,a\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(5/4)/x^14,x)

[Out]

(4*b^3*(a + b*x^4)^(1/4))/(117*a^2*x) - (14*b*(a + b*x^4)^(1/4))/(117*x^9) - (a*(a + b*x^4)^(1/4))/(13*x^13) -
(b^2*(a + b*x^4)^(1/4))/(117*a*x^5)

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