3.476 \(\int (-3+x) (-7+4 x^2) \, dx\)

Optimal. Leaf size=22 \[ -4 x^3+\frac {1}{16} \left (7-4 x^2\right )^2+21 x \]

[Out]

21*x-4*x^3+1/16*(-4*x^2+7)^2

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641} \[ -4 x^3+\frac {1}{16} \left (7-4 x^2\right )^2+21 x \]

Antiderivative was successfully verified.

[In]

Int[(-3 + x)*(-7 + 4*x^2),x]

[Out]

21*x - 4*x^3 + (7 - 4*x^2)^2/16

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (-3+x) \left (-7+4 x^2\right ) \, dx &=\frac {1}{16} \left (7-4 x^2\right )^2-3 \int \left (-7+4 x^2\right ) \, dx\\ &=21 x-4 x^3+\frac {1}{16} \left (7-4 x^2\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.86 \[ x^4-4 x^3-\frac {7 x^2}{2}+21 x \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + x)*(-7 + 4*x^2),x]

[Out]

21*x - (7*x^2)/2 - 4*x^3 + x^4

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fricas [A]  time = 0.49, size = 17, normalized size = 0.77 \[ x^{4} - 4 x^{3} - \frac {7}{2} x^{2} + 21 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)*(4*x^2-7),x, algorithm="fricas")

[Out]

x^4 - 4*x^3 - 7/2*x^2 + 21*x

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giac [A]  time = 0.36, size = 17, normalized size = 0.77 \[ x^{4} - 4 \, x^{3} - \frac {7}{2} \, x^{2} + 21 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)*(4*x^2-7),x, algorithm="giac")

[Out]

x^4 - 4*x^3 - 7/2*x^2 + 21*x

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maple [A]  time = 0.00, size = 18, normalized size = 0.82 \[ x^{4}-4 x^{3}-\frac {7}{2} x^{2}+21 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-3)*(4*x^2-7),x)

[Out]

x^4-4*x^3-7/2*x^2+21*x

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maxima [A]  time = 0.88, size = 17, normalized size = 0.77 \[ x^{4} - 4 \, x^{3} - \frac {7}{2} \, x^{2} + 21 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)*(4*x^2-7),x, algorithm="maxima")

[Out]

x^4 - 4*x^3 - 7/2*x^2 + 21*x

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mupad [B]  time = 0.03, size = 17, normalized size = 0.77 \[ x^4-4\,x^3-\frac {7\,x^2}{2}+21\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - 7)*(x - 3),x)

[Out]

21*x - (7*x^2)/2 - 4*x^3 + x^4

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sympy [A]  time = 0.06, size = 17, normalized size = 0.77 \[ x^{4} - 4 x^{3} - \frac {7 x^{2}}{2} + 21 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)*(4*x**2-7),x)

[Out]

x**4 - 4*x**3 - 7*x**2/2 + 21*x

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