3.5.10 \(\int \frac {1}{x^4 \sqrt [4]{x^3+x^4}} \, dx\)

Optimal. Leaf size=33 \[ \frac {4 \left (128 x^3-96 x^2+84 x-77\right ) \left (x^4+x^3\right )^{3/4}}{1155 x^6} \]

________________________________________________________________________________________

Rubi [B]  time = 0.10, antiderivative size = 73, normalized size of antiderivative = 2.21, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2016, 2014} \begin {gather*} \frac {512 \left (x^4+x^3\right )^{3/4}}{1155 x^3}-\frac {128 \left (x^4+x^3\right )^{3/4}}{385 x^4}-\frac {4 \left (x^4+x^3\right )^{3/4}}{15 x^6}+\frac {16 \left (x^4+x^3\right )^{3/4}}{55 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(x^3 + x^4)^(1/4)),x]

[Out]

(-4*(x^3 + x^4)^(3/4))/(15*x^6) + (16*(x^3 + x^4)^(3/4))/(55*x^5) - (128*(x^3 + x^4)^(3/4))/(385*x^4) + (512*(
x^3 + x^4)^(3/4))/(1155*x^3)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

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

Rubi steps

\begin {align*} \int \frac {1}{x^4 \sqrt [4]{x^3+x^4}} \, dx &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}-\frac {4}{5} \int \frac {1}{x^3 \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (x^3+x^4\right )^{3/4}}{55 x^5}+\frac {32}{55} \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (x^3+x^4\right )^{3/4}}{55 x^5}-\frac {128 \left (x^3+x^4\right )^{3/4}}{385 x^4}-\frac {128}{385} \int \frac {1}{x \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (x^3+x^4\right )^{3/4}}{55 x^5}-\frac {128 \left (x^3+x^4\right )^{3/4}}{385 x^4}+\frac {512 \left (x^3+x^4\right )^{3/4}}{1155 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3 (x+1)\right )^{3/4} \left (128 x^3-96 x^2+84 x-77\right )}{1155 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3*(1 + x))^(3/4)*(-77 + 84*x - 96*x^2 + 128*x^3))/(1155*x^6)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.22, size = 33, normalized size = 1.00 \begin {gather*} \frac {4 \left (-77+84 x-96 x^2+128 x^3\right ) \left (x^3+x^4\right )^{3/4}}{1155 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-77 + 84*x - 96*x^2 + 128*x^3)*(x^3 + x^4)^(3/4))/(1155*x^6)

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 29, normalized size = 0.88 \begin {gather*} \frac {4 \, {\left (x^{4} + x^{3}\right )}^{\frac {3}{4}} {\left (128 \, x^{3} - 96 \, x^{2} + 84 \, x - 77\right )}}{1155 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/1155*(x^4 + x^3)^(3/4)*(128*x^3 - 96*x^2 + 84*x - 77)/x^6

________________________________________________________________________________________

giac [A]  time = 0.31, size = 37, normalized size = 1.12 \begin {gather*} -\frac {4}{15} \, {\left (\frac {1}{x} + 1\right )}^{\frac {15}{4}} + \frac {12}{11} \, {\left (\frac {1}{x} + 1\right )}^{\frac {11}{4}} - \frac {12}{7} \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(1/x + 1)^(15/4) + 12/11*(1/x + 1)^(11/4) - 12/7*(1/x + 1)^(7/4) + 4/3*(1/x + 1)^(3/4)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 26, normalized size = 0.79

method	result	size
meijerg	\(-\frac {4 \left (-\frac {128}{77} x^{3}+\frac {96}{77} x^{2}-\frac {12}{11} x +1\right ) \left (1+x \right )^{\frac {3}{4}}}{15 x^{\frac {15}{4}}}\)	\(26\)
trager	\(\frac {4 \left (128 x^{3}-96 x^{2}+84 x -77\right ) \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{1155 x^{6}}\)	\(30\)
gosper	\(\frac {4 \left (1+x \right ) \left (128 x^{3}-96 x^{2}+84 x -77\right )}{1155 x^{3} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\)	\(33\)
risch	\(\frac {-\frac {4}{15}+\frac {4}{165} x -\frac {16}{385} x^{2}+\frac {128}{1155} x^{3}+\frac {512}{1155} x^{4}}{x^{3} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(-128/77*x^3+96/77*x^2-12/11*x+1)*(1+x)^(3/4)/x^(15/4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 57, normalized size = 1.73 \begin {gather*} \frac {512\,{\left (x^4+x^3\right )}^{3/4}}{1155\,x^3}-\frac {128\,{\left (x^4+x^3\right )}^{3/4}}{385\,x^4}+\frac {16\,{\left (x^4+x^3\right )}^{3/4}}{55\,x^5}-\frac {4\,{\left (x^4+x^3\right )}^{3/4}}{15\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^3 + x^4)^(1/4)),x)

[Out]

(512*(x^3 + x^4)^(3/4))/(1155*x^3) - (128*(x^3 + x^4)^(3/4))/(385*x^4) + (16*(x^3 + x^4)^(3/4))/(55*x^5) - (4*
(x^3 + x^4)^(3/4))/(15*x^6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**4*(x**3*(x + 1))**(1/4)), x)

________________________________________________________________________________________