3.6.62 \(\int \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}} \, dx\)

Optimal. Leaf size=43 \[ \frac {x \sqrt [5]{\left (4 x^2-13 x+3\right )^5} \left (8 x^2-39 x+18\right )}{6 \left (4 x^2-13 x+3\right )} \]

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Rubi [B]  time = 0.06, antiderivative size = 102, normalized size of antiderivative = 2.37, number of steps used = 3, number of rules used = 2, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6688, 6720} \begin {gather*} -\frac {13 \sqrt [5]{\left (4 x^2-13 x+3\right )^5} x^2}{2 \left (4 x^2-13 x+3\right )}+\frac {3 \sqrt [5]{\left (4 x^2-13 x+3\right )^5} x}{4 x^2-13 x+3}+\frac {4 \sqrt [5]{\left (4 x^2-13 x+3\right )^5} x^3}{3 \left (4 x^2-13 x+3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 -
16640*x^9 + 1024*x^10)^(1/5),x]

[Out]

(3*x*((3 - 13*x + 4*x^2)^5)^(1/5))/(3 - 13*x + 4*x^2) - (13*x^2*((3 - 13*x + 4*x^2)^5)^(1/5))/(2*(3 - 13*x + 4
*x^2)) + (4*x^3*((3 - 13*x + 4*x^2)^5)^(1/5))/(3*(3 - 13*x + 4*x^2))

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

\begin {align*} \int \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}} \, dx &=\int \sqrt [5]{\left (3-13 x+4 x^2\right )^5} \, dx\\ &=\frac {\sqrt [5]{\left (3-13 x+4 x^2\right )^5} \int \left (3-13 x+4 x^2\right ) \, dx}{3-13 x+4 x^2}\\ &=\frac {3 x \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}{3-13 x+4 x^2}-\frac {13 x^2 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}{2 \left (3-13 x+4 x^2\right )}+\frac {4 x^3 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}{3 \left (3-13 x+4 x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} \frac {x \sqrt [5]{\left (4 x^2-13 x+3\right )^5} \left (8 x^2-39 x+18\right )}{6 \left (4 x^2-13 x+3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*
x^8 - 16640*x^9 + 1024*x^10)^(1/5),x]

[Out]

(x*((3 - 13*x + 4*x^2)^5)^(1/5)*(18 - 39*x + 8*x^2))/(6*(3 - 13*x + 4*x^2))

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IntegrateAlgebraic [A]  time = 19.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {x \sqrt [5]{\left (3-13 x+4 x^2\right )^5} \left (18-39 x+8 x^2\right )}{6 \left (3-13 x+4 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7
+ 112000*x^8 - 16640*x^9 + 1024*x^10)^(1/5),x]

[Out]

(x*((3 - 13*x + 4*x^2)^5)^(1/5)*(18 - 39*x + 8*x^2))/(6*(3 - 13*x + 4*x^2))

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fricas [A]  time = 0.43, size = 14, normalized size = 0.33 \begin {gather*} \frac {4}{3} \, x^{3} - \frac {13}{2} \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-526
5*x+243)^(1/5),x, algorithm="fricas")

[Out]

4/3*x^3 - 13/2*x^2 + 3*x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-526
5*x+243)^(1/5),x, algorithm="giac")

[Out]

integrate((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 225810*x^3
+ 47250*x^2 - 5265*x + 243)^(1/5), x)

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maple [B]  time = 0.17, size = 78, normalized size = 1.81

method	result	size
gosper	\(\frac {x \left (8 x^{2}-39 x +18\right ) \left (1024 x^{10}-16640 x^{9}+112000 x^{8}-401440 x^{7}+820340 x^{6}-954733 x^{5}+615255 x^{4}-225810 x^{3}+47250 x^{2}-5265 x +243\right )^{\frac {1}{5}}}{6 \left (-1+4 x \right ) \left (-3+x \right )}\)	\(78\)
risch	\(\frac {4 \left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}} x^{3}}{3 \left (4 x^{2}-13 x +3\right )}-\frac {13 \left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}} x^{2}}{2 \left (4 x^{2}-13 x +3\right )}+\frac {3 \left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}} x}{4 x^{2}-13 x +3}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+24
3)^(1/5),x,method=_RETURNVERBOSE)

[Out]

1/6*x*(8*x^2-39*x+18)*(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-954733*x^5+615255*x^4-225810*x^3+4
7250*x^2-5265*x+243)^(1/5)/(-1+4*x)/(-3+x)

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maxima [A]  time = 0.35, size = 14, normalized size = 0.33 \begin {gather*} \frac {4}{3} \, x^{3} - \frac {13}{2} \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-526
5*x+243)^(1/5),x, algorithm="maxima")

[Out]

4/3*x^3 - 13/2*x^2 + 3*x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (1024\,x^{10}-16640\,x^9+112000\,x^8-401440\,x^7+820340\,x^6-954733\,x^5+615255\,x^4-225810\,x^3+47250\,x^2-5265\,x+243\right )}^{1/5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((47250*x^2 - 5265*x - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*
x^9 + 1024*x^10 + 243)^(1/5),x)

[Out]

int((47250*x^2 - 5265*x - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*
x^9 + 1024*x^10 + 243)^(1/5), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [5]{1024 x^{10} - 16640 x^{9} + 112000 x^{8} - 401440 x^{7} + 820340 x^{6} - 954733 x^{5} + 615255 x^{4} - 225810 x^{3} + 47250 x^{2} - 5265 x + 243}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x**10-16640*x**9+112000*x**8-401440*x**7+820340*x**6-954733*x**5+615255*x**4-225810*x**3+47250
*x**2-5265*x+243)**(1/5),x)

[Out]

Integral((1024*x**10 - 16640*x**9 + 112000*x**8 - 401440*x**7 + 820340*x**6 - 954733*x**5 + 615255*x**4 - 2258
10*x**3 + 47250*x**2 - 5265*x + 243)**(1/5), x)

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