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

Optimal. Leaf size=18 \[ -\frac {4 \left (x^4+x^3\right )^{3/4}}{3 x^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2014} \begin {gather*} -\frac {4 \left (x^4+x^3\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(x^3 + x^4)^(1/4)),x]

[Out]

(-4*(x^3 + x^4)^(3/4))/(3*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])

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt [4]{x^3+x^4}} \, dx &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4 (x+1)}{3 \sqrt [4]{x^3 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(x^3 + x^4)^(1/4)),x]

[Out]

(-4*(1 + x))/(3*(x^3*(1 + x))^(1/4))

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IntegrateAlgebraic [A]  time = 0.19, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4 \left (x^3+x^4\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(x^3 + x^4)^(3/4))/(3*x^3)

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fricas [A]  time = 0.45, size = 14, normalized size = 0.78 \begin {gather*} -\frac {4 \, {\left (x^{4} + x^{3}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(x^4 + x^3)^(3/4)/x^3

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giac [A]  time = 0.58, size = 9, normalized size = 0.50 \begin {gather*} -\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/(x^4+x^3)^(1/4),x, algorithm="giac")

[Out]

-4/3*(1/x + 1)^(3/4)

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maple [A]  time = 0.06, size = 11, normalized size = 0.61

method	result	size
meijerg	\(-\frac {4 \left (1+x \right )^{\frac {3}{4}}}{3 x^{\frac {3}{4}}}\)	\(11\)
gosper	\(-\frac {4 \left (1+x \right )}{3 \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\)	\(15\)
trager	\(-\frac {4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(15\)
risch	\(-\frac {4 \left (1+x \right )}{3 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(1+x)^(3/4)/x^(3/4)

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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}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^4 + x^3)^(1/4)*x), x)

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mupad [B]  time = 0.26, size = 14, normalized size = 0.78 \begin {gather*} -\frac {4\,{\left (x^4+x^3\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^3 + x^4)^(1/4)),x)

[Out]

-(4*(x^3 + x^4)^(3/4))/(3*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \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/(x**4+x**3)**(1/4),x)

[Out]

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

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