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

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

[Out]

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

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Rubi [A]  time = 0.016036, antiderivative size = 30, normalized size of antiderivative = 1.2, 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 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]

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]

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*}

Mathematica [A]  time = 0.0125301, size = 25, normalized size = 1. \[ \frac{4 (x-1) (x+2)}{3 \sqrt [4]{(x-1)^3 (x+2)^5}} \]

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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Maple [A]  time = 0.003, size = 22, normalized size = 0.9 \begin{align*}{\frac{ \left ( -4+4\,x \right ) \left ( 2+x \right ) }{3}{\frac{1}{\sqrt [4]{ \left ( -1+x \right ) ^{3} \left ( 2+x \right ) ^{5}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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Fricas [B]  time = 1.74502, size = 169, normalized size = 6.76 \begin{align*} \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 )}} \end{align*}

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

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

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)