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

Optimal. Leaf size=17 \[ \frac {1}{3} d \left (a+b x+c x^2\right )^3 \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {629} \begin {gather*} \frac {1}{3} d \left (a+b x+c x^2\right )^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B]  time = 0.01, size = 37, normalized size = 2.18 \begin {gather*} \frac {1}{3} d 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*d + 2*c*d*x)*(a + b*x + c*x^2)^2,x]

[Out]

(d*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 d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.37, size = 80, normalized size = 4.71 \begin {gather*} \frac {1}{3} x^{6} d c^{3} + x^{5} d c^{2} b + x^{4} d c b^{2} + x^{4} d c^{2} a + \frac {1}{3} x^{3} d b^{3} + 2 x^{3} d c b a + x^{2} d b^{2} a + x^{2} d c a^{2} + x d b a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.18, size = 52, normalized size = 3.06 \begin {gather*} {\left (c d x^{2} + b d x\right )} a^{2} + \frac {3 \, {\left (c d x^{2} + b d x\right )}^{2} a d + {\left (c d x^{2} + b d x\right )}^{3}}{3 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 95, normalized size = 5.59 \begin {gather*} \frac {c^{3} d \,x^{6}}{3}+b \,c^{2} d \,x^{5}+a^{2} b d x +\frac {\left (2 b^{2} c d +2 \left (2 a c +b^{2}\right ) c d \right ) x^{4}}{4}+\frac {\left (4 a b c d +\left (2 a c +b^{2}\right ) b d \right ) x^{3}}{3}+\frac {\left (2 c d \,a^{2}+2 a \,b^{2} d \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, {\left (c x^{2} + b x + a\right )}^{3} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c*x^2 + b*x + a)^3*d

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mupad [B]  time = 0.41, size = 67, normalized size = 3.94 \begin {gather*} \frac {c^3\,d\,x^6}{3}+a\,d\,x^2\,\left (b^2+a\,c\right )+\frac {b\,d\,x^3\,\left (b^2+6\,a\,c\right )}{3}+c\,d\,x^4\,\left (b^2+a\,c\right )+a^2\,b\,d\,x+b\,c^2\,d\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.08, size = 80, normalized size = 4.71 \begin {gather*} a^{2} b d x + b c^{2} d x^{5} + \frac {c^{3} d x^{6}}{3} + x^{4} \left (a c^{2} d + b^{2} c d\right ) + x^{3} \left (2 a b c d + \frac {b^{3} d}{3}\right ) + x^{2} \left (a^{2} c d + a b^{2} d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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