3.13.63 \(\int \frac {1}{25} (565+450 x+108 x^{7/5}-810 x^2+180 x^3+\sqrt [5]{x} (-216-594 x+288 x^2)) \, dx\)

Optimal. Leaf size=21 \[ x+\frac {1}{5} \left (6-3 x \left (-3+\sqrt [5]{x}+x\right )\right )^2 \]

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Rubi [B]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 2.90, number of steps used = 4, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 14} \begin {gather*} \frac {18 x^{16/5}}{5}+\frac {9 x^{12/5}}{5}-\frac {54 x^{11/5}}{5}-\frac {36 x^{6/5}}{5}+\frac {9 x^4}{5}-\frac {54 x^3}{5}+9 x^2+\frac {113 x}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(565 + 450*x + 108*x^(7/5) - 810*x^2 + 180*x^3 + x^(1/5)*(-216 - 594*x + 288*x^2))/25,x]

[Out]

(113*x)/5 - (36*x^(6/5))/5 + 9*x^2 - (54*x^(11/5))/5 + (9*x^(12/5))/5 - (54*x^3)/5 + (18*x^(16/5))/5 + (9*x^4)
/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \left (565+450 x+108 x^{7/5}-810 x^2+180 x^3+\sqrt [5]{x} \left (-216-594 x+288 x^2\right )\right ) \, dx\\ &=\frac {113 x}{5}+9 x^2+\frac {9 x^{12/5}}{5}-\frac {54 x^3}{5}+\frac {9 x^4}{5}+\frac {1}{25} \int \sqrt [5]{x} \left (-216-594 x+288 x^2\right ) \, dx\\ &=\frac {113 x}{5}+9 x^2+\frac {9 x^{12/5}}{5}-\frac {54 x^3}{5}+\frac {9 x^4}{5}+\frac {1}{25} \int \left (-216 \sqrt [5]{x}-594 x^{6/5}+288 x^{11/5}\right ) \, dx\\ &=\frac {113 x}{5}-\frac {36 x^{6/5}}{5}+9 x^2-\frac {54 x^{11/5}}{5}+\frac {9 x^{12/5}}{5}-\frac {54 x^3}{5}+\frac {18 x^{16/5}}{5}+\frac {9 x^4}{5}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 51, normalized size = 2.43 \begin {gather*} \frac {1}{25} \left (565 x-180 x^{6/5}+225 x^2-270 x^{11/5}+45 x^{12/5}-270 x^3+90 x^{16/5}+45 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(565 + 450*x + 108*x^(7/5) - 810*x^2 + 180*x^3 + x^(1/5)*(-216 - 594*x + 288*x^2))/25,x]

[Out]

(565*x - 180*x^(6/5) + 225*x^2 - 270*x^(11/5) + 45*x^(12/5) - 270*x^3 + 90*x^(16/5) + 45*x^4)/25

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fricas [B]  time = 0.76, size = 41, normalized size = 1.95 \begin {gather*} \frac {9}{5} \, x^{4} - \frac {54}{5} \, x^{3} + \frac {9}{5} \, x^{\frac {12}{5}} + 9 \, x^{2} + \frac {18}{5} \, {\left (x^{3} - 3 \, x^{2} - 2 \, x\right )} x^{\frac {1}{5}} + \frac {113}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/25*x^(7/5)+1/25*(288*x^2-594*x-216)*x^(1/5)+36/5*x^3-162/5*x^2+18*x+113/5,x, algorithm="fricas")

[Out]

9/5*x^4 - 54/5*x^3 + 9/5*x^(12/5) + 9*x^2 + 18/5*(x^3 - 3*x^2 - 2*x)*x^(1/5) + 113/5*x

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giac [B]  time = 0.37, size = 39, normalized size = 1.86 \begin {gather*} \frac {9}{5} \, x^{4} + \frac {18}{5} \, x^{\frac {16}{5}} - \frac {54}{5} \, x^{3} + \frac {9}{5} \, x^{\frac {12}{5}} - \frac {54}{5} \, x^{\frac {11}{5}} + 9 \, x^{2} - \frac {36}{5} \, x^{\frac {6}{5}} + \frac {113}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/25*x^(7/5)+1/25*(288*x^2-594*x-216)*x^(1/5)+36/5*x^3-162/5*x^2+18*x+113/5,x, algorithm="giac")

[Out]

9/5*x^4 + 18/5*x^(16/5) - 54/5*x^3 + 9/5*x^(12/5) - 54/5*x^(11/5) + 9*x^2 - 36/5*x^(6/5) + 113/5*x

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maple [A]  time = 0.06, size = 37, normalized size = 1.76

method	result	size
trager	\(\frac {\left (9 x^{3}-45 x^{2}+113\right ) \left (x -1\right )}{5}+\frac {18 x^{\frac {6}{5}} \left (x^{2}-3 x -2\right )}{5}+\frac {9 x^{\frac {12}{5}}}{5}\)	\(37\)
risch	\(\frac {9 x^{\frac {12}{5}}}{5}+\frac {18 x^{\frac {6}{5}} \left (x^{2}-3 x -2\right )}{5}+\frac {9 x^{4}}{5}-\frac {54 x^{3}}{5}+9 x^{2}+\frac {113 x}{5}\)	\(38\)
derivativedivides	\(\frac {9 x^{4}}{5}+\frac {18 x^{\frac {16}{5}}}{5}-\frac {54 x^{3}}{5}+\frac {9 x^{\frac {12}{5}}}{5}-\frac {54 x^{\frac {11}{5}}}{5}+9 x^{2}-\frac {36 x^{\frac {6}{5}}}{5}+\frac {113 x}{5}\)	\(40\)
default	\(\frac {9 x^{4}}{5}+\frac {18 x^{\frac {16}{5}}}{5}-\frac {54 x^{3}}{5}+\frac {9 x^{\frac {12}{5}}}{5}-\frac {54 x^{\frac {11}{5}}}{5}+9 x^{2}-\frac {36 x^{\frac {6}{5}}}{5}+\frac {113 x}{5}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(108/25*x^(7/5)+1/25*(288*x^2-594*x-216)*x^(1/5)+36/5*x^3-162/5*x^2+18*x+113/5,x,method=_RETURNVERBOSE)

[Out]

1/5*(9*x^3-45*x^2+113)*(x-1)+18/5*x^(6/5)*(x^2-3*x-2)+9/5*x^(12/5)

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maxima [B]  time = 0.37, size = 39, normalized size = 1.86 \begin {gather*} \frac {9}{5} \, x^{4} + \frac {18}{5} \, x^{\frac {16}{5}} - \frac {54}{5} \, x^{3} + \frac {9}{5} \, x^{\frac {12}{5}} - \frac {54}{5} \, x^{\frac {11}{5}} + 9 \, x^{2} - \frac {36}{5} \, x^{\frac {6}{5}} + \frac {113}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/25*x^(7/5)+1/25*(288*x^2-594*x-216)*x^(1/5)+36/5*x^3-162/5*x^2+18*x+113/5,x, algorithm="maxima")

[Out]

9/5*x^4 + 18/5*x^(16/5) - 54/5*x^3 + 9/5*x^(12/5) - 54/5*x^(11/5) + 9*x^2 - 36/5*x^(6/5) + 113/5*x

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mupad [B]  time = 0.04, size = 39, normalized size = 1.86 \begin {gather*} \frac {113\,x}{5}+9\,x^2-\frac {54\,x^3}{5}+\frac {9\,x^4}{5}-\frac {36\,x^{6/5}}{5}-\frac {54\,x^{11/5}}{5}+\frac {9\,x^{12/5}}{5}+\frac {18\,x^{16/5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(18*x - (x^(1/5)*(594*x - 288*x^2 + 216))/25 - (162*x^2)/5 + (36*x^3)/5 + (108*x^(7/5))/25 + 113/5,x)

[Out]

(113*x)/5 + 9*x^2 - (54*x^3)/5 + (9*x^4)/5 - (36*x^(6/5))/5 - (54*x^(11/5))/5 + (9*x^(12/5))/5 + (18*x^(16/5))
/5

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sympy [A]  time = 2.07, size = 56, normalized size = 2.67 \begin {gather*} \frac {18 x^{\frac {16}{5}}}{5} + \frac {9 x^{\frac {12}{5}}}{5} - \frac {54 x^{\frac {11}{5}}}{5} - \frac {36 x^{\frac {6}{5}}}{5} + \frac {9 x^{4}}{5} - \frac {54 x^{3}}{5} + 9 x^{2} + \frac {113 x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/25*x**(7/5)+1/25*(288*x**2-594*x-216)*x**(1/5)+36/5*x**3-162/5*x**2+18*x+113/5,x)

[Out]

18*x**(16/5)/5 + 9*x**(12/5)/5 - 54*x**(11/5)/5 - 36*x**(6/5)/5 + 9*x**4/5 - 54*x**3/5 + 9*x**2 + 113*x/5

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