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

Optimal. Leaf size=17 \[ \frac{5}{16} \left (x^4-3 x^2\right )^{8/5} \]

[Out]

(5*(-3*x^2 + x^4)^(8/5))/16

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Rubi [A]  time = 0.0102869, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1588} \[ \frac{5}{16} \left (x^4-3 x^2\right )^{8/5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(-3*x^2 + x^4)^(8/5))/16

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (-3 x+2 x^3\right ) \left (-3 x^2+x^4\right )^{3/5} \, dx &=\frac{5}{16} \left (-3 x^2+x^4\right )^{8/5}\\ \end{align*}

Mathematica [C]  time = 0.0545341, size = 75, normalized size = 4.41 \[ \frac{5 \left (x^2 \left (x^2-3\right )\right )^{3/5} \left (16 x^4 \, _2F_1\left (-\frac{3}{5},\frac{13}{5};\frac{18}{5};\frac{x^2}{3}\right )-39 x^2 \, _2F_1\left (-\frac{3}{5},\frac{8}{5};\frac{13}{5};\frac{x^2}{3}\right )\right )}{208 \left (1-\frac{x^2}{3}\right )^{3/5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(x^2*(-3 + x^2))^(3/5)*(-39*x^2*Hypergeometric2F1[-3/5, 8/5, 13/5, x^2/3] + 16*x^4*Hypergeometric2F1[-3/5,
13/5, 18/5, x^2/3]))/(208*(1 - x^2/3)^(3/5))

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Maple [A]  time = 0.005, size = 22, normalized size = 1.3 \begin{align*}{\frac{5\,{x}^{2} \left ({x}^{2}-3 \right ) }{16} \left ({x}^{4}-3\,{x}^{2} \right ) ^{{\frac{3}{5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*(x^4-3*x^2)^(3/5)*x^2*(x^2-3)

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Maxima [A]  time = 0.964498, size = 18, normalized size = 1.06 \begin{align*} \frac{5}{16} \,{\left (x^{4} - 3 \, x^{2}\right )}^{\frac{8}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*(x^4 - 3*x^2)^(8/5)

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Fricas [A]  time = 1.9955, size = 35, normalized size = 2.06 \begin{align*} \frac{5}{16} \,{\left (x^{4} - 3 \, x^{2}\right )}^{\frac{8}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*(x^4 - 3*x^2)^(8/5)

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Sympy [B]  time = 1.24886, size = 36, normalized size = 2.12 \begin{align*} \frac{5 x^{4} \left (x^{4} - 3 x^{2}\right )^{\frac{3}{5}}}{16} - \frac{15 x^{2} \left (x^{4} - 3 x^{2}\right )^{\frac{3}{5}}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x**4*(x**4 - 3*x**2)**(3/5)/16 - 15*x**2*(x**4 - 3*x**2)**(3/5)/16

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Giac [A]  time = 1.08591, size = 18, normalized size = 1.06 \begin{align*} \frac{5}{16} \,{\left (x^{4} - 3 \, x^{2}\right )}^{\frac{8}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*(x^4 - 3*x^2)^(8/5)