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

Optimal. Leaf size=25 \[ \frac {4 (x-1) (x+2)}{3 \sqrt [4]{(x-1)^3 (x+2)^5}} \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6719, 37} \[ -\frac {4 (1-x) (x+2)}{3 \sqrt [4]{-(1-x)^3 (x+2)^5}} \]

Antiderivative was successfully verified.

[In]

Int[((-1 + x)^3*(2 + x)^5)^(-1/4),x]

[Out]

(-4*(1 - x)*(2 + x))/(3*(-((1 - x)^3*(2 + x)^5))^(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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {4 (x-1) (x+2)}{3 \sqrt [4]{(x-1)^3 (x+2)^5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 26.03, size = 25, normalized size = 1.00 \[ \frac {4 (x-1) (x+2)}{3 \sqrt [4]{(x-1)^3 (x+2)^5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.90, size = 69, normalized size = 2.76 \[ \frac {4 \, {\left (x^{8} + 7 \, x^{7} + 13 \, x^{6} - 11 \, x^{5} - 50 \, x^{4} - 8 \, x^{3} + 64 \, x^{2} + 16 \, x - 32\right )}^{\frac {3}{4}}}{3 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} - 24 \, x^{2} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(x^8 + 7*x^7 + 13*x^6 - 11*x^5 - 50*x^4 - 8*x^3 + 64*x^2 + 16*x - 32)^(3/4)/(x^6 + 6*x^5 + 9*x^4 - 8*x^3 -
24*x^2 + 16)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 22, normalized size = 0.88

method	result	size
gosper	\(\frac {4 \left (-1+x \right ) \left (2+x \right )}{3 \left (\left (-1+x \right )^{3} \left (2+x \right )^{5}\right )^{\frac {1}{4}}}\)	\(22\)
risch	\(\frac {4 \left (-1+x \right ) \left (2+x \right )}{3 \left (\left (-1+x \right )^{3} \left (2+x \right )^{5}\right )^{\frac {1}{4}}}\)	\(22\)
trager	\(\frac {4 \left (x^{8}+7 x^{7}+13 x^{6}-11 x^{5}-50 x^{4}-8 x^{3}+64 x^{2}+16 x -32\right )^{\frac {3}{4}}}{3 \left (-1+x \right )^{2} \left (2+x \right )^{4}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.26, size = 25, normalized size = 1.00 \[ \frac {4\,{\left ({\left (x-1\right )}^3\,{\left (x+2\right )}^5\right )}^{3/4}}{3\,{\left (x-1\right )}^2\,{\left (x+2\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 1)^3*(x + 2)^5)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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