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

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

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Rubi [A]  time = 0.10, antiderivative size = 32, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2056, 45, 37} \begin {gather*} \frac {4}{\sqrt [4]{x^4+x^3}}-\frac {16 (x+1)}{3 \sqrt [4]{x^4+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(x^3 + x^4)^(1/4) - (16*(1 + x))/(3*(x^3 + x^4)^(1/4))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

\begin {align*} \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{1+x}\right ) \int \frac {1}{x^{7/4} (1+x)^{5/4}} \, dx}{\sqrt [4]{x^3+x^4}}\\ &=\frac {4}{\sqrt [4]{x^3+x^4}}+\frac {\left (4 x^{3/4} \sqrt [4]{1+x}\right ) \int \frac {1}{x^{7/4} \sqrt [4]{1+x}} \, dx}{\sqrt [4]{x^3+x^4}}\\ &=\frac {4}{\sqrt [4]{x^3+x^4}}-\frac {16 (1+x)}{3 \sqrt [4]{x^3+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.71 \begin {gather*} -\frac {4 (4 x+1)}{3 \sqrt [4]{x^3 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.20, size = 28, normalized size = 1.00 \begin {gather*} -\frac {4 (1+4 x) \left (x^3+x^4\right )^{3/4}}{3 x^3 (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 16, normalized size = 0.57 \begin {gather*} -\frac {4 \, {\left (4 \, x + 1\right )}}{3 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(4*x + 1)/(x^4 + x^3)^(1/4)

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giac [A]  time = 0.66, size = 19, normalized size = 0.68 \begin {gather*} -\frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} - \frac {4}{{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 16, normalized size = 0.57

method	result	size
meijerg	\(-\frac {4 \left (1+4 x \right )}{3 \left (1+x \right )^{\frac {1}{4}} x^{\frac {3}{4}}}\)	\(16\)
gosper	\(-\frac {4 \left (1+4 x \right )}{3 \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\)	\(17\)
risch	\(-\frac {4 \left (1+4 x \right )}{3 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(17\)
trager	\(-\frac {4 \left (1+4 x \right ) \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{3 x^{3} \left (1+x \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(1+4*x)/(1+x)^(1/4)/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}} {\left (x + 1\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.17, size = 16, normalized size = 0.57 \begin {gather*} -\frac {16\,x+4}{3\,{\left (x^4+x^3\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(16*x + 4)/(3*(x^3 + x^4)^(1/4))

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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 )} \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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