3.2.46 \(\int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx\)

Optimal. Leaf size=18 \[ \frac {4 \left (x^5-x\right )^{3/4}}{3 x^3} \]

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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1590} \begin {gather*} \frac {4 \left (x^5-x\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + x^4)/(x^3*(-x + x^5)^(1/4)),x]

[Out]

(4*(-x + x^5)^(3/4))/(3*x^3)

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

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

Antiderivative was successfully verified.

[In]

Integrate[(3 + x^4)/(x^3*(-x + x^5)^(1/4)),x]

[Out]

(4*(x*(-1 + x^4))^(3/4))/(3*x^3)

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IntegrateAlgebraic [A]  time = 0.07, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + x^4)/(x^3*(-x + x^5)^(1/4)),x]

[Out]

(4*(-x + x^5)^(3/4))/(3*x^3)

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fricas [A]  time = 0.46, size = 14, normalized size = 0.78 \begin {gather*} \frac {4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)/x^3/(x^5-x)^(1/4),x, algorithm="fricas")

[Out]

4/3*(x^5 - x)^(3/4)/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 3}{{\left (x^{5} - x\right )}^{\frac {1}{4}} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)/x^3/(x^5-x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 + 3)/((x^5 - x)^(1/4)*x^3), x)

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maple [A]  time = 0.11, size = 15, normalized size = 0.83

method	result	size
trager	\(\frac {4 \left (x^{5}-x \right )^{\frac {3}{4}}}{3 x^{3}}\)	\(15\)
risch	\(\frac {\frac {4 x^{4}}{3}-\frac {4}{3}}{x^{2} \left (x \left (x^{4}-1\right )\right )^{\frac {1}{4}}}\)	\(20\)
gosper	\(\frac {4 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{3 x^{2} \left (x^{5}-x \right )^{\frac {1}{4}}}\)	\(26\)
meijerg	\(-\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {9}{16}, \frac {1}{4}\right ], \left [\frac {7}{16}\right ], x^{4}\right )}{3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}+\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {7}{16}\right ], \left [\frac {23}{16}\right ], x^{4}\right ) x^{\frac {7}{4}}}{7 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+3)/x^3/(x^5-x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/3*(x^5-x)^(3/4)/x^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 3}{{\left (x^{5} - x\right )}^{\frac {1}{4}} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)/x^3/(x^5-x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 + 3)/((x^5 - x)^(1/4)*x^3), x)

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mupad [B]  time = 0.25, size = 14, normalized size = 0.78 \begin {gather*} \frac {4\,{\left (x^5-x\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 3)/(x^3*(x^5 - x)^(1/4)),x)

[Out]

(4*(x^5 - x)^(3/4))/(3*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 3}{x^{3} \sqrt [4]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+3)/x**3/(x**5-x)**(1/4),x)

[Out]

Integral((x**4 + 3)/(x**3*(x*(x - 1)*(x + 1)*(x**2 + 1))**(1/4)), x)

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