3.14.16 \(\int (b+2 c x) (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{3} \left (a+b x+c x^2\right )^3 \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {629} \begin {gather*} \frac {1}{3} \left (a+b x+c x^2\right )^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(a + b*x + c*x^2)^2,x]

[Out]

(a + b*x + c*x^2)^3/3

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx &=\frac {1}{3} \left (a+b x+c x^2\right )^3\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 36, normalized size = 2.25 \begin {gather*} \frac {1}{3} x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(a + b*x + c*x^2)^2,x]

[Out]

(x*(b + c*x)*(3*a^2 + 3*a*x*(b + c*x) + x^2*(b + c*x)^2))/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(a + b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(b + 2*c*x)*(a + b*x + c*x^2)^2, x]

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fricas [B]  time = 0.37, size = 71, normalized size = 4.44 \begin {gather*} \frac {1}{3} x^{6} c^{3} + x^{5} c^{2} b + x^{4} c b^{2} + x^{4} c^{2} a + \frac {1}{3} x^{3} b^{3} + 2 x^{3} c b a + x^{2} b^{2} a + x^{2} c a^{2} + x b a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/3*x^6*c^3 + x^5*c^2*b + x^4*c*b^2 + x^4*c^2*a + 1/3*x^3*b^3 + 2*x^3*c*b*a + x^2*b^2*a + x^2*c*a^2 + x*b*a^2

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giac [B]  time = 0.16, size = 40, normalized size = 2.50 \begin {gather*} \frac {1}{3} \, {\left (c x^{2} + b x\right )}^{3} + {\left (c x^{2} + b x\right )}^{2} a + {\left (c x^{2} + b x\right )} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

1/3*(c*x^2 + b*x)^3 + (c*x^2 + b*x)^2*a + (c*x^2 + b*x)*a^2

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maple [B]  time = 0.05, size = 86, normalized size = 5.38 \begin {gather*} \frac {c^{3} x^{6}}{3}+b \,c^{2} x^{5}+a^{2} b x +\frac {\left (2 b^{2} c +2 \left (2 a c +b^{2}\right ) c \right ) x^{4}}{4}+\frac {\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) x^{3}}{3}+\frac {\left (2 c \,a^{2}+2 a \,b^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^2,x)

[Out]

1/3*c^3*x^6+b*c^2*x^5+1/4*(2*b^2*c+2*c*(2*a*c+b^2))*x^4+1/3*(b*(2*a*c+b^2)+4*a*b*c)*x^3+1/2*(2*a^2*c+2*a*b^2)*
x^2+a^2*b*x

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maxima [A]  time = 0.53, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, {\left (c x^{2} + b x + a\right )}^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

1/3*(c*x^2 + b*x + a)^3

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mupad [B]  time = 0.03, size = 62, normalized size = 3.88 \begin {gather*} x^3\,\left (\frac {b^3}{3}+2\,a\,c\,b\right )+\frac {c^3\,x^6}{3}+b\,c^2\,x^5+a\,x^2\,\left (b^2+a\,c\right )+c\,x^4\,\left (b^2+a\,c\right )+a^2\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(a + b*x + c*x^2)^2,x)

[Out]

x^3*(b^3/3 + 2*a*b*c) + (c^3*x^6)/3 + b*c^2*x^5 + a*x^2*(a*c + b^2) + c*x^4*(a*c + b^2) + a^2*b*x

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sympy [B]  time = 0.09, size = 65, normalized size = 4.06 \begin {gather*} a^{2} b x + b c^{2} x^{5} + \frac {c^{3} x^{6}}{3} + x^{4} \left (a c^{2} + b^{2} c\right ) + x^{3} \left (2 a b c + \frac {b^{3}}{3}\right ) + x^{2} \left (a^{2} c + a b^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b*x + b*c**2*x**5 + c**3*x**6/3 + x**4*(a*c**2 + b**2*c) + x**3*(2*a*b*c + b**3/3) + x**2*(a**2*c + a*b**
2)

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