3.209 \(\int (-3 x^{3/5}+x^{3/2})^2 (-\frac{x^{2/3}}{3}+4 x^{3/2}) \, dx\)

Optimal. Leaf size=55 \[ \frac{8 x^{11/2}}{11}-\frac{x^{14/3}}{14}-\frac{120 x^{23/5}}{23}+\frac{60 x^{113/30}}{113}+\frac{360 x^{37/10}}{37}-\frac{45 x^{43/15}}{43} \]

[Out]

(-45*x^(43/15))/43 + (360*x^(37/10))/37 + (60*x^(113/30))/113 - (120*x^(23/5))/23 - x^(14/3)/14 + (8*x^(11/2))
/11

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Rubi [A]  time = 0.237876, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1593, 1584, 1820} \[ \frac{8 x^{11/2}}{11}-\frac{x^{14/3}}{14}-\frac{120 x^{23/5}}{23}+\frac{60 x^{113/30}}{113}+\frac{360 x^{37/10}}{37}-\frac{45 x^{43/15}}{43} \]

Antiderivative was successfully verified.

[In]

Int[(-3*x^(3/5) + x^(3/2))^2*(-x^(2/3)/3 + 4*x^(3/2)),x]

[Out]

(-45*x^(43/15))/43 + (360*x^(37/10))/37 + (60*x^(113/30))/113 - (120*x^(23/5))/23 - x^(14/3)/14 + (8*x^(11/2))
/11

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac{x^{2/3}}{3}+4 x^{3/2}\right ) \, dx &=\int \left (-3+x^{9/10}\right )^2 x^{6/5} \left (-\frac{x^{2/3}}{3}+4 x^{3/2}\right ) \, dx\\ &=\int \left (-\frac{1}{3}+4 x^{5/6}\right ) \left (-3+x^{9/10}\right )^2 x^{28/15} \, dx\\ &=30 \operatorname{Subst}\left (\int x^{85} \left (-\frac{1}{3}+4 x^{25}\right ) \left (-3+x^{27}\right )^2 \, dx,x,\sqrt [30]{x}\right )\\ &=30 \operatorname{Subst}\left (\int \left (-3 x^{85}+36 x^{110}+2 x^{112}-24 x^{137}-\frac{x^{139}}{3}+4 x^{164}\right ) \, dx,x,\sqrt [30]{x}\right )\\ &=-\frac{45 x^{43/15}}{43}+\frac{360 x^{37/10}}{37}+\frac{60 x^{113/30}}{113}-\frac{120 x^{23/5}}{23}-\frac{x^{14/3}}{14}+\frac{8 x^{11/2}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0459141, size = 55, normalized size = 1. \[ \frac{8 x^{11/2}}{11}-\frac{x^{14/3}}{14}-\frac{120 x^{23/5}}{23}+\frac{60 x^{113/30}}{113}+\frac{360 x^{37/10}}{37}-\frac{45 x^{43/15}}{43} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^(3/5) + x^(3/2))^2*(-x^(2/3)/3 + 4*x^(3/2)),x]

[Out]

(-45*x^(43/15))/43 + (360*x^(37/10))/37 + (60*x^(113/30))/113 - (120*x^(23/5))/23 - x^(14/3)/14 + (8*x^(11/2))
/11

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Maple [A]  time = 0.003, size = 32, normalized size = 0.6 \begin{align*} -{\frac{45}{43}{x}^{{\frac{43}{15}}}}+{\frac{360}{37}{x}^{{\frac{37}{10}}}}+{\frac{60}{113}{x}^{{\frac{113}{30}}}}-{\frac{120}{23}{x}^{{\frac{23}{5}}}}-{\frac{1}{14}{x}^{{\frac{14}{3}}}}+{\frac{8}{11}{x}^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/43*x^(43/15)+360/37*x^(37/10)+60/113*x^(113/30)-120/23*x^(23/5)-1/14*x^(14/3)+8/11*x^(11/2)

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Maxima [A]  time = 0.940615, size = 42, normalized size = 0.76 \begin{align*} \frac{8}{11} \, x^{\frac{11}{2}} - \frac{1}{14} \, x^{\frac{14}{3}} - \frac{120}{23} \, x^{\frac{23}{5}} + \frac{60}{113} \, x^{\frac{113}{30}} + \frac{360}{37} \, x^{\frac{37}{10}} - \frac{45}{43} \, x^{\frac{43}{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x, algorithm="maxima")

[Out]

8/11*x^(11/2) - 1/14*x^(14/3) - 120/23*x^(23/5) + 60/113*x^(113/30) + 360/37*x^(37/10) - 45/43*x^(43/15)

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Fricas [A]  time = 1.6393, size = 143, normalized size = 2.6 \begin{align*} \frac{8}{11} \, x^{\frac{11}{2}} - \frac{1}{14} \, x^{\frac{14}{3}} - \frac{120}{23} \, x^{\frac{23}{5}} + \frac{60}{113} \, x^{\frac{113}{30}} + \frac{360}{37} \, x^{\frac{37}{10}} - \frac{45}{43} \, x^{\frac{43}{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x, algorithm="fricas")

[Out]

8/11*x^(11/2) - 1/14*x^(14/3) - 120/23*x^(23/5) + 60/113*x^(113/30) + 360/37*x^(37/10) - 45/43*x^(43/15)

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Sympy [A]  time = 2.03727, size = 48, normalized size = 0.87 \begin{align*} \frac{60 x^{\frac{113}{30}}}{113} - \frac{45 x^{\frac{43}{15}}}{43} + \frac{360 x^{\frac{37}{10}}}{37} - \frac{120 x^{\frac{23}{5}}}{23} - \frac{x^{\frac{14}{3}}}{14} + \frac{8 x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

60*x**(113/30)/113 - 45*x**(43/15)/43 + 360*x**(37/10)/37 - 120*x**(23/5)/23 - x**(14/3)/14 + 8*x**(11/2)/11

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Giac [A]  time = 1.07293, size = 42, normalized size = 0.76 \begin{align*} \frac{8}{11} \, x^{\frac{11}{2}} - \frac{1}{14} \, x^{\frac{14}{3}} - \frac{120}{23} \, x^{\frac{23}{5}} + \frac{60}{113} \, x^{\frac{113}{30}} + \frac{360}{37} \, x^{\frac{37}{10}} - \frac{45}{43} \, x^{\frac{43}{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x, algorithm="giac")

[Out]

8/11*x^(11/2) - 1/14*x^(14/3) - 120/23*x^(23/5) + 60/113*x^(113/30) + 360/37*x^(37/10) - 45/43*x^(43/15)