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

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

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Rubi [A]  time = 0.01, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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IntegrateAlgebraic [A]  time = 1.12, size = 27, normalized size = 0.56 \[ \frac {97 x^{11}+462 x^6+1188 x}{891 \left (x^5+3\right )^{11/5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1188*x + 462*x^6 + 97*x^11)/(891*(3 + x^5)^(11/5))

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fricas [A]  time = 0.66, size = 40, normalized size = 0.83 \[ \frac {{\left (97 \, x^{11} + 462 \, x^{6} + 1188 \, x\right )} {\left (x^{5} + 3\right )}^{\frac {4}{5}}}{891 \, {\left (x^{15} + 9 \, x^{10} + 27 \, x^{5} + 27\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{5} - 2\right )}^{2}}{{\left (x^{5} + 3\right )}^{\frac {16}{5}}}\,{d x} \]

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)

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maple [A]  time = 0.32, size = 23, normalized size = 0.48

method	result	size
gosper	\(\frac {x \left (97 x^{10}+462 x^{5}+1188\right )}{891 \left (x^{5}+3\right )^{\frac {11}{5}}}\)	\(23\)
trager	\(\frac {x \left (97 x^{10}+462 x^{5}+1188\right )}{891 \left (x^{5}+3\right )^{\frac {11}{5}}}\)	\(23\)
risch	\(\frac {x \left (97 x^{10}+462 x^{5}+1188\right )}{891 \left (x^{5}+3\right )^{\frac {11}{5}}}\)	\(23\)
meijerg	\(\frac {4 \,3^{\frac {4}{5}} x \left (\frac {25}{9} x^{10}+\frac {55}{3} x^{5}+33\right )}{2673 \left (1+\frac {x^{5}}{3}\right )^{\frac {11}{5}}}+\frac {3^{\frac {4}{5}} x^{11}}{891 \left (1+\frac {x^{5}}{3}\right )^{\frac {11}{5}}}-\frac {2 \,3^{\frac {4}{5}} x^{6} \left (11+\frac {5 x^{5}}{3}\right )}{2673 \left (1+\frac {x^{5}}{3}\right )^{\frac {11}{5}}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.57, size = 73, normalized size = 1.52 \[ -\frac {4 \, x^{11} {\left (\frac {11 \, {\left (x^{5} + 3\right )}}{x^{5}} - \frac {33 \, {\left (x^{5} + 3\right )}^{2}}{x^{10}} - 3\right )}}{891 \, {\left (x^{5} + 3\right )}^{\frac {11}{5}}} - \frac {2 \, x^{11} {\left (\frac {11 \, {\left (x^{5} + 3\right )}}{x^{5}} - 6\right )}}{297 \, {\left (x^{5} + 3\right )}^{\frac {11}{5}}} + \frac {x^{11}}{33 \, {\left (x^{5} + 3\right )}^{\frac {11}{5}}} \]

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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mupad [B]  time = 0.26, size = 23, normalized size = 0.48 \[ \frac {97\,x^{11}+462\,x^6+1188\,x}{891\,{\left (x^5+3\right )}^{11/5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5 - 2)^2/(x^5 + 3)^(16/5),x)

[Out]

(1188*x + 462*x^6 + 97*x^11)/(891*(x^5 + 3)^(11/5))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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