3.11.37 \(\int \frac {(a+b x^4)^{3/4}}{x^{12}} \, dx\) [1037]

Optimal. Leaf size=44 \[ -\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7} \]

[Out]

-1/11*(b*x^4+a)^(7/4)/a/x^11+4/77*b*(b*x^4+a)^(7/4)/a^2/x^7

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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 )^{7/4}}{77 a^2 x^7}-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^12,x]

[Out]

-1/11*(a + b*x^4)^(7/4)/(a*x^11) + (4*b*(a + b*x^4)^(7/4))/(77*a^2*x^7)

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 )^{3/4}}{x^{12}} \, dx &=-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}-\frac {(4 b) \int \frac {\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{11 a}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 31, normalized size = 0.70 \begin {gather*} \frac {\left (a+b x^4\right )^{7/4} \left (-7 a+4 b x^4\right )}{77 a^2 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^12,x]

[Out]

((a + b*x^4)^(7/4)*(-7*a + 4*b*x^4))/(77*a^2*x^11)

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Maple [A]
time = 0.16, size = 28, normalized size = 0.64

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-4 b \,x^{4}+7 a \right )}{77 x^{11} a^{2}}\)	\(28\)
trager	\(-\frac {\left (-4 b^{2} x^{8}+3 a b \,x^{4}+7 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{77 x^{11} a^{2}}\)	\(39\)
risch	\(-\frac {\left (-4 b^{2} x^{8}+3 a b \,x^{4}+7 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{77 x^{11} a^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/77*(b*x^4+a)^(7/4)*(-4*b*x^4+7*a)/x^11/a^2

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Maxima [A]
time = 0.30, size = 35, normalized size = 0.80 \begin {gather*} \frac {\frac {11 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b}{x^{7}} - \frac {7 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}}}{x^{11}}}{77 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^12,x, algorithm="maxima")

[Out]

1/77*(11*(b*x^4 + a)^(7/4)*b/x^7 - 7*(b*x^4 + a)^(11/4)/x^11)/a^2

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Fricas [A]
time = 0.40, size = 38, normalized size = 0.86 \begin {gather*} \frac {{\left (4 \, b^{2} x^{8} - 3 \, a b x^{4} - 7 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{77 \, a^{2} x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^12,x, algorithm="fricas")

[Out]

1/77*(4*b^2*x^8 - 3*a*b*x^4 - 7*a^2)*(b*x^4 + a)^(3/4)/(a^2*x^11)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (37) = 74\).
time = 0.92, size = 110, normalized size = 2.50 \begin {gather*} - \frac {7 b^{\frac {3}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right )} - \frac {3 b^{\frac {7}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 a x^{4} \Gamma \left (- \frac {3}{4}\right )} + \frac {b^{\frac {11}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{4 a^{2} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**12,x)

[Out]

-7*b**(3/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(16*x**8*gamma(-3/4)) - 3*b**(7/4)*(a/(b*x**4) + 1)**(3/4)*ga
mma(-11/4)/(16*a*x**4*gamma(-3/4)) + b**(11/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(4*a**2*gamma(-3/4))

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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)^(3/4)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^12, x)

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Mupad [B]
time = 1.33, size = 56, normalized size = 1.27 \begin {gather*} -\frac {7\,a^2\,{\left (b\,x^4+a\right )}^{3/4}-4\,b^2\,x^8\,{\left (b\,x^4+a\right )}^{3/4}+3\,a\,b\,x^4\,{\left (b\,x^4+a\right )}^{3/4}}{77\,a^2\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(3/4)/x^12,x)

[Out]

-(7*a^2*(a + b*x^4)^(3/4) - 4*b^2*x^8*(a + b*x^4)^(3/4) + 3*a*b*x^4*(a + b*x^4)^(3/4))/(77*a^2*x^11)

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