3.2.11 \(\int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx\)

Optimal. Leaf size=16 \[ \frac {4 \left (x^6+x\right )^{3/4}}{3 x^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 16, 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^6+x\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x^5)/(x^3*(x + x^6)^(1/4)),x]

[Out]

(4*(x + x^6)^(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+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx &=\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^6+x\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*x^5)/(x^3*(x + x^6)^(1/4)),x]

[Out]

(4*(x + x^6)^(3/4))/(3*x^3)

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IntegrateAlgebraic [A]  time = 0.19, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \left (x+x^6\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-3 + 2*x^5)/(x^3*(x + x^6)^(1/4)),x]

[Out]

(4*(x + x^6)^(3/4))/(3*x^3)

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fricas [A]  time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} \frac {4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5-3)/x^3/(x^6+x)^(1/4),x, algorithm="fricas")

[Out]

4/3*(x^6 + x)^(3/4)/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{5} - 3}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5-3)/x^3/(x^6+x)^(1/4),x, algorithm="giac")

[Out]

integrate((2*x^5 - 3)/((x^6 + x)^(1/4)*x^3), x)

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maple [A]  time = 0.09, size = 13, normalized size = 0.81

method	result	size
trager	\(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}}{3 x^{3}}\)	\(13\)
risch	\(\frac {\frac {4 x^{5}}{3}+\frac {4}{3}}{x^{2} \left (x \left (x^{5}+1\right )\right )^{\frac {1}{4}}}\)	\(20\)
gosper	\(\frac {4 \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}{3 x^{2} \left (x^{6}+x \right )^{\frac {1}{4}}}\)	\(32\)
meijerg	\(\frac {4 \hypergeom \left (\left [-\frac {9}{20}, \frac {1}{4}\right ], \left [\frac {11}{20}\right ], -x^{5}\right )}{3 x^{\frac {9}{4}}}+\frac {8 \hypergeom \left (\left [\frac {1}{4}, \frac {11}{20}\right ], \left [\frac {31}{20}\right ], -x^{5}\right ) x^{\frac {11}{4}}}{11}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5-3)/x^3/(x^6+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/3*(x^6+x)^(3/4)/x^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{5} - 3}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5-3)/x^3/(x^6+x)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)/((x^6 + x)^(1/4)*x^3), x)

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mupad [B]  time = 0.25, size = 12, normalized size = 0.75 \begin {gather*} \frac {4\,{\left (x^6+x\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5 - 3)/(x^3*(x + x^6)^(1/4)),x)

[Out]

(4*(x + x^6)^(3/4))/(3*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x^{5} - 3}{x^{3} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5-3)/x**3/(x**6+x)**(1/4),x)

[Out]

Integral((2*x**5 - 3)/(x**3*(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))**(1/4)), x)

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