3.3.79 \(\int \frac {(-4+x^4) (1+x^4)^{3/4}}{x^{12}} \, dx\)

Optimal. Leaf size=26 \[ \frac {\left (x^4+1\right )^{3/4} \left (-27 x^8+x^4+28\right )}{77 x^{11}} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} \frac {4 \left (x^4+1\right )^{7/4}}{11 x^{11}}-\frac {27 \left (x^4+1\right )^{7/4}}{77 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4 + x^4)*(1 + x^4)^(3/4))/x^12,x]

[Out]

(4*(1 + x^4)^(7/4))/(11*x^11) - (27*(1 + x^4)^(7/4))/(77*x^7)

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]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (-4+x^4\right ) \left (1+x^4\right )^{3/4}}{x^{12}} \, dx &=\frac {4 \left (1+x^4\right )^{7/4}}{11 x^{11}}+\frac {27}{11} \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx\\ &=\frac {4 \left (1+x^4\right )^{7/4}}{11 x^{11}}-\frac {27 \left (1+x^4\right )^{7/4}}{77 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {\left (28-27 x^4\right ) \left (x^4+1\right )^{7/4}}{77 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + x^4)*(1 + x^4)^(3/4))/x^12,x]

[Out]

((28 - 27*x^4)*(1 + x^4)^(7/4))/(77*x^11)

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IntegrateAlgebraic [A]  time = 0.14, size = 23, normalized size = 0.88 \begin {gather*} \frac {\left (28-27 x^4\right ) \left (1+x^4\right )^{7/4}}{77 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4 + x^4)*(1 + x^4)^(3/4))/x^12,x]

[Out]

((28 - 27*x^4)*(1 + x^4)^(7/4))/(77*x^11)

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fricas [A]  time = 0.43, size = 24, normalized size = 0.92 \begin {gather*} -\frac {{\left (27 \, x^{8} - x^{4} - 28\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4)*(x^4+1)^(3/4)/x^12,x, algorithm="fricas")

[Out]

-1/77*(27*x^8 - x^4 - 28)*(x^4 + 1)^(3/4)/x^11

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )}}{x^{12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4)*(x^4+1)^(3/4)/x^12,x, algorithm="giac")

[Out]

integrate((x^4 + 1)^(3/4)*(x^4 - 4)/x^12, x)

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maple [A]  time = 0.08, size = 20, normalized size = 0.77

method	result	size
gosper	\(-\frac {\left (27 x^{4}-28\right ) \left (x^{4}+1\right )^{\frac {7}{4}}}{77 x^{11}}\)	\(20\)
trager	\(-\frac {\left (27 x^{8}-x^{4}-28\right ) \left (x^{4}+1\right )^{\frac {3}{4}}}{77 x^{11}}\)	\(25\)
risch	\(-\frac {27 x^{12}+26 x^{8}-29 x^{4}-28}{77 x^{11} \left (x^{4}+1\right )^{\frac {1}{4}}}\)	\(30\)
meijerg	\(\frac {4 \left (-\frac {4}{7} x^{8}+\frac {3}{7} x^{4}+1\right ) \left (x^{4}+1\right )^{\frac {3}{4}}}{11 x^{11}}-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-4)*(x^4+1)^(3/4)/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/77*(27*x^4-28)*(x^4+1)^(7/4)/x^11

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maxima [A]  time = 0.36, size = 25, normalized size = 0.96 \begin {gather*} -\frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} + \frac {4 \, {\left (x^{4} + 1\right )}^{\frac {11}{4}}}{11 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4)*(x^4+1)^(3/4)/x^12,x, algorithm="maxima")

[Out]

-5/7*(x^4 + 1)^(7/4)/x^7 + 4/11*(x^4 + 1)^(11/4)/x^11

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mupad [B]  time = 0.31, size = 38, normalized size = 1.46 \begin {gather*} \frac {28\,{\left (x^4+1\right )}^{3/4}+x^4\,{\left (x^4+1\right )}^{3/4}-27\,x^8\,{\left (x^4+1\right )}^{3/4}}{77\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 1)^(3/4)*(x^4 - 4))/x^12,x)

[Out]

(28*(x^4 + 1)^(3/4) + x^4*(x^4 + 1)^(3/4) - 27*x^8*(x^4 + 1)^(3/4))/(77*x^11)

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sympy [B]  time = 3.37, size = 136, normalized size = 5.23 \begin {gather*} \frac {\left (1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} - \frac {\left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{x^{3} \Gamma \left (- \frac {3}{4}\right )} + \frac {\left (1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} + \frac {3 \left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {7 \left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{4 x^{11} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-4)*(x**4+1)**(3/4)/x**12,x)

[Out]

(1 + x**(-4))**(3/4)*gamma(-7/4)/(4*gamma(-3/4)) - (x**4 + 1)**(3/4)*gamma(-11/4)/(x**3*gamma(-3/4)) + (1 + x*
*(-4))**(3/4)*gamma(-7/4)/(4*x**4*gamma(-3/4)) + 3*(x**4 + 1)**(3/4)*gamma(-11/4)/(4*x**7*gamma(-3/4)) + 7*(x*
*4 + 1)**(3/4)*gamma(-11/4)/(4*x**11*gamma(-3/4))

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