3.14.26 \(\int (b+2 c x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{4} \left (a+b x+c x^2\right )^4 \]

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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}{4} \left (a+b x+c x^2\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

(a + b*x + c*x^2)^4/4

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 )^3 \, dx &=\frac {1}{4} \left (a+b x+c x^2\right )^4\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 51, normalized size = 3.19 \begin {gather*} \frac {1}{4} x (b+c x) \left (4 a^3+6 a^2 x (b+c x)+4 a x^2 (b+c x)^2+x^3 (b+c x)^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

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

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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 )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(b + 2*c*x)*(a + b*x + c*x^2)^3, x]

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fricas [B]  time = 0.38, size = 126, normalized size = 7.88 \begin {gather*} \frac {1}{4} x^{8} c^{4} + x^{7} c^{3} b + \frac {3}{2} x^{6} c^{2} b^{2} + x^{6} c^{3} a + x^{5} c b^{3} + 3 x^{5} c^{2} b a + \frac {1}{4} x^{4} b^{4} + 3 x^{4} c b^{2} a + \frac {3}{2} x^{4} c^{2} a^{2} + x^{3} b^{3} a + 3 x^{3} c b a^{2} + \frac {3}{2} x^{2} b^{2} a^{2} + x^{2} c a^{3} + x b a^{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)^3,x, algorithm="fricas")

[Out]

1/4*x^8*c^4 + x^7*c^3*b + 3/2*x^6*c^2*b^2 + x^6*c^3*a + x^5*c*b^3 + 3*x^5*c^2*b*a + 1/4*x^4*b^4 + 3*x^4*c*b^2*
a + 3/2*x^4*c^2*a^2 + x^3*b^3*a + 3*x^3*c*b*a^2 + 3/2*x^2*b^2*a^2 + x^2*c*a^3 + x*b*a^3

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giac [B]  time = 0.15, size = 56, normalized size = 3.50 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{4} + {\left (c x^{2} + b x\right )}^{3} a + \frac {3}{2} \, {\left (c x^{2} + b x\right )}^{2} a^{2} + {\left (c x^{2} + b x\right )} a^{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)^3,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 218, normalized size = 13.62 \begin {gather*} \frac {c^{4} x^{8}}{4}+b \,c^{3} x^{7}+\frac {\left (3 b^{2} c^{2}+2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c \right ) x^{6}}{6}+a^{3} b x +\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b +2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c \right ) x^{5}}{5}+\frac {\left (\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b +2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c \right ) x^{4}}{4}+\frac {\left (6 a^{2} b c +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b \right ) x^{3}}{3}+\frac {\left (2 c \,a^{3}+3 a^{2} 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)^3,x)

[Out]

1/4*c^4*x^8+b*c^3*x^7+1/6*(3*b^2*c^2+2*c*(a*c^2+2*b^2*c+(2*a*c+b^2)*c))*x^6+1/5*(b*(a*c^2+2*b^2*c+(2*a*c+b^2)*
c)+2*c*(4*a*b*c+(2*a*c+b^2)*b))*x^5+1/4*(b*(4*a*b*c+(2*a*c+b^2)*b)+2*c*(a^2*c+2*a*b^2+(2*a*c+b^2)*a))*x^4+1/3*
(b*(a^2*c+2*a*b^2+(2*a*c+b^2)*a)+6*a^2*b*c)*x^3+1/2*(2*a^3*c+3*a^2*b^2)*x^2+b*a^3*x

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maxima [A]  time = 0.49, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + b x + a\right )}^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

1/4*(c*x^2 + b*x + a)^4

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mupad [B]  time = 1.89, size = 112, normalized size = 7.00 \begin {gather*} x^4\,\left (\frac {3\,a^2\,c^2}{2}+3\,a\,b^2\,c+\frac {b^4}{4}\right )+\frac {c^4\,x^8}{4}+x^2\,\left (c\,a^3+\frac {3\,a^2\,b^2}{2}\right )+x^6\,\left (\frac {3\,b^2\,c^2}{2}+a\,c^3\right )+b\,c^3\,x^7+a^3\,b\,x+a\,b\,x^3\,\left (b^2+3\,a\,c\right )+b\,c\,x^5\,\left (b^2+3\,a\,c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(a + b*x + c*x^2)^3,x)

[Out]

x^4*(b^4/4 + (3*a^2*c^2)/2 + 3*a*b^2*c) + (c^4*x^8)/4 + x^2*(a^3*c + (3*a^2*b^2)/2) + x^6*(a*c^3 + (3*b^2*c^2)
/2) + b*c^3*x^7 + a^3*b*x + a*b*x^3*(3*a*c + b^2) + b*c*x^5*(3*a*c + b^2)

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sympy [B]  time = 0.10, size = 121, normalized size = 7.56 \begin {gather*} a^{3} b x + b c^{3} x^{7} + \frac {c^{4} x^{8}}{4} + x^{6} \left (a c^{3} + \frac {3 b^{2} c^{2}}{2}\right ) + x^{5} \left (3 a b c^{2} + b^{3} c\right ) + x^{4} \left (\frac {3 a^{2} c^{2}}{2} + 3 a b^{2} c + \frac {b^{4}}{4}\right ) + x^{3} \left (3 a^{2} b c + a b^{3}\right ) + x^{2} \left (a^{3} c + \frac {3 a^{2} b^{2}}{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)**3,x)

[Out]

a**3*b*x + b*c**3*x**7 + c**4*x**8/4 + x**6*(a*c**3 + 3*b**2*c**2/2) + x**5*(3*a*b*c**2 + b**3*c) + x**4*(3*a*
*2*c**2/2 + 3*a*b**2*c + b**4/4) + x**3*(3*a**2*b*c + a*b**3) + x**2*(a**3*c + 3*a**2*b**2/2)

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