3.318 \(\int \frac{(-2+x^5)^2}{(3+x^5)^{16/5}} \, dx\)

Optimal. Leaf size=48 \[ \frac{97 x}{891 \sqrt [5]{x^5+3}}+\frac{5 x}{297 \left (x^5+3\right )^{6/5}}-\frac{5 \left (x^5-2\right ) x}{33 \left (x^5+3\right )^{11/5}} \]

[Out]

(-5*x*(-2 + x^5))/(33*(3 + x^5)^(11/5)) + (5*x)/(297*(3 + x^5)^(6/5)) + (97*x)/(891*(3 + x^5)^(1/5))

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Rubi [A]  time = 0.0117009, antiderivative size = 59, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {378, 191} \[ \frac{x \left (2-x^5\right )^2}{33 \left (x^5+3\right )^{11/5}}+\frac{10 x \left (2-x^5\right )}{297 \left (x^5+3\right )^{6/5}}+\frac{100 x}{891 \sqrt [5]{x^5+3}} \]

Antiderivative was successfully verified.

[In]

Int[(-2 + x^5)^2/(3 + x^5)^(16/5),x]

[Out]

(x*(2 - x^5)^2)/(33*(3 + x^5)^(11/5)) + (10*x*(2 - x^5))/(297*(3 + x^5)^(6/5)) + (100*x)/(891*(3 + x^5)^(1/5))

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\left (-2+x^5\right )^2}{\left (3+x^5\right )^{16/5}} \, dx &=\frac{x \left (2-x^5\right )^2}{33 \left (3+x^5\right )^{11/5}}-\frac{20}{33} \int \frac{-2+x^5}{\left (3+x^5\right )^{11/5}} \, dx\\ &=\frac{x \left (2-x^5\right )^2}{33 \left (3+x^5\right )^{11/5}}+\frac{10 x \left (2-x^5\right )}{297 \left (3+x^5\right )^{6/5}}+\frac{100}{297} \int \frac{1}{\left (3+x^5\right )^{6/5}} \, dx\\ &=\frac{x \left (2-x^5\right )^2}{33 \left (3+x^5\right )^{11/5}}+\frac{10 x \left (2-x^5\right )}{297 \left (3+x^5\right )^{6/5}}+\frac{100 x}{891 \sqrt [5]{3+x^5}}\\ \end{align*}

Mathematica [A]  time = 0.0087394, size = 26, normalized size = 0.54 \[ \frac{x \left (97 x^{10}+462 x^5+1188\right )}{891 \left (x^5+3\right )^{11/5}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x^5)^2/(3 + x^5)^(16/5),x]

[Out]

(x*(1188 + 462*x^5 + 97*x^10))/(891*(3 + x^5)^(11/5))

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Maple [A]  time = 0.006, size = 23, normalized size = 0.5 \begin{align*}{\frac{x \left ( 97\,{x}^{10}+462\,{x}^{5}+1188 \right ) }{891} \left ({x}^{5}+3 \right ) ^{-{\frac{11}{5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5-2)^2/(x^5+3)^(16/5),x)

[Out]

1/891*x*(97*x^10+462*x^5+1188)/(x^5+3)^(11/5)

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Maxima [B]  time = 0.95903, size = 99, normalized size = 2.06 \begin{align*} -\frac{4 \, x^{11}{\left (\frac{11 \,{\left (x^{5} + 3\right )}}{x^{5}} - \frac{33 \,{\left (x^{5} + 3\right )}^{2}}{x^{10}} - 3\right )}}{891 \,{\left (x^{5} + 3\right )}^{\frac{11}{5}}} - \frac{2 \, x^{11}{\left (\frac{11 \,{\left (x^{5} + 3\right )}}{x^{5}} - 6\right )}}{297 \,{\left (x^{5} + 3\right )}^{\frac{11}{5}}} + \frac{x^{11}}{33 \,{\left (x^{5} + 3\right )}^{\frac{11}{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-2)^2/(x^5+3)^(16/5),x, algorithm="maxima")

[Out]

-4/891*x^11*(11*(x^5 + 3)/x^5 - 33*(x^5 + 3)^2/x^10 - 3)/(x^5 + 3)^(11/5) - 2/297*x^11*(11*(x^5 + 3)/x^5 - 6)/
(x^5 + 3)^(11/5) + 1/33*x^11/(x^5 + 3)^(11/5)

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Fricas [A]  time = 1.50801, size = 111, normalized size = 2.31 \begin{align*} \frac{{\left (97 \, x^{11} + 462 \, x^{6} + 1188 \, x\right )}{\left (x^{5} + 3\right )}^{\frac{4}{5}}}{891 \,{\left (x^{15} + 9 \, x^{10} + 27 \, x^{5} + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-2)^2/(x^5+3)^(16/5),x, algorithm="fricas")

[Out]

1/891*(97*x^11 + 462*x^6 + 1188*x)*(x^5 + 3)^(4/5)/(x^15 + 9*x^10 + 27*x^5 + 27)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5-2)**2/(x**5+3)**(16/5),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{5} - 2\right )}^{2}}{{\left (x^{5} + 3\right )}^{\frac{16}{5}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-2)^2/(x^5+3)^(16/5),x, algorithm="giac")

[Out]

integrate((x^5 - 2)^2/(x^5 + 3)^(16/5), x)