3.2.50 \(\int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx\)

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

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Rubi [A]  time = 0.06, 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 = {2036} \begin {gather*} \frac {4 \left (x^5+x^3\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3 + x^5)^(3/4))/(3*x^3)

Rule 2036

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*(m + j*p + 1)), x] /; FreeQ[{a, b, c, d, e,
j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && EqQ[a*d*(m + j*p + 1) - b*c*(m + n
+ p*(j + n) + 1), 0] && (GtQ[e, 0] || IntegersQ[j]) && NeQ[m + j*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx &=\frac {4 \left (x^3+x^5\right )^{3/4}}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \frac {4 \left (x^2+1\right )}{3 \sqrt [4]{x^5+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(1 + x^2))/(3*(x^3 + x^5)^(1/4))

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IntegrateAlgebraic [A]  time = 0.09, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3+x^5\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3 + x^5)^(3/4))/(3*x^3)

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fricas [A]  time = 0.44, size = 14, normalized size = 0.78 \begin {gather*} \frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(x^5 + x^3)^(3/4)/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 15, normalized size = 0.83

method	result	size
trager	\(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(15\)
gosper	\(\frac {\frac {4 x^{2}}{3}+\frac {4}{3}}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\)	\(17\)
risch	\(\frac {\frac {4 x^{2}}{3}+\frac {4}{3}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\)	\(19\)
meijerg	\(\frac {4 \hypergeom \left (\left [-\frac {3}{8}, \frac {1}{4}\right ], \left [\frac {5}{8}\right ], -x^{2}\right )}{3 x^{\frac {3}{4}}}+\frac {4 \hypergeom \left (\left [\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], -x^{2}\right ) x^{\frac {5}{4}}}{5}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} \frac {4\,{\left (x^5+x^3\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(x^3 + x^5)^(3/4))/(3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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