3.2338 \(\int (A+B x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=158 \[ \frac{1}{2} a^2 x^2 (a B+3 A b)+a^3 A x+\frac{1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{8} B c^3 x^8 \]

[Out]

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

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Rubi [A]  time = 0.206202, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {631} \[ \frac{1}{2} a^2 x^2 (a B+3 A b)+a^3 A x+\frac{1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{8} B c^3 x^8 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + b*x + c*x^2)^3,x]

[Out]

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

Rule 631

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

Rubi steps

\begin{align*} \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A+a^2 (3 A b+a B) x+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^3+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^4+3 c \left (b^2 B+A b c+a B c\right ) x^5+c^2 (3 b B+A c) x^6+B c^3 x^7\right ) \, dx\\ &=a^3 A x+\frac{1}{2} a^2 (3 A b+a B) x^2+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac{1}{4} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\frac{1}{5} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^5+\frac{1}{2} c \left (b^2 B+A b c+a B c\right ) x^6+\frac{1}{7} c^2 (3 b B+A c) x^7+\frac{1}{8} B c^3 x^8\\ \end{align*}

Mathematica [A]  time = 0.046041, size = 158, normalized size = 1. \[ \frac{1}{2} a^2 x^2 (a B+3 A b)+a^3 A x+\frac{1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{8} B c^3 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 223, normalized size = 1.4 \begin{align*}{\frac{B{c}^{3}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Ab{c}^{2}+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\,B{a}^{2}b \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){x}^{2}}{2}}+{a}^{3}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00647, size = 219, normalized size = 1.39 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} +{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.883606, size = 444, normalized size = 2.81 \begin{align*} \frac{1}{8} x^{8} c^{3} B + \frac{3}{7} x^{7} c^{2} b B + \frac{1}{7} x^{7} c^{3} A + \frac{1}{2} x^{6} c b^{2} B + \frac{1}{2} x^{6} c^{2} a B + \frac{1}{2} x^{6} c^{2} b A + \frac{1}{5} x^{5} b^{3} B + \frac{6}{5} x^{5} c b a B + \frac{3}{5} x^{5} c b^{2} A + \frac{3}{5} x^{5} c^{2} a A + \frac{3}{4} x^{4} b^{2} a B + \frac{3}{4} x^{4} c a^{2} B + \frac{1}{4} x^{4} b^{3} A + \frac{3}{2} x^{4} c b a A + x^{3} b a^{2} B + x^{3} b^{2} a A + x^{3} c a^{2} A + \frac{1}{2} x^{2} a^{3} B + \frac{3}{2} x^{2} b a^{2} A + x a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.097507, size = 190, normalized size = 1.2 \begin{align*} A a^{3} x + \frac{B c^{3} x^{8}}{8} + x^{7} \left (\frac{A c^{3}}{7} + \frac{3 B b c^{2}}{7}\right ) + x^{6} \left (\frac{A b c^{2}}{2} + \frac{B a c^{2}}{2} + \frac{B b^{2} c}{2}\right ) + x^{5} \left (\frac{3 A a c^{2}}{5} + \frac{3 A b^{2} c}{5} + \frac{6 B a b c}{5} + \frac{B b^{3}}{5}\right ) + x^{4} \left (\frac{3 A a b c}{2} + \frac{A b^{3}}{4} + \frac{3 B a^{2} c}{4} + \frac{3 B a b^{2}}{4}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x^{2} \left (\frac{3 A a^{2} b}{2} + \frac{B a^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**3,x)

[Out]

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

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Giac [A]  time = 1.13091, size = 252, normalized size = 1.59 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{3}{7} \, B b c^{2} x^{7} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B b^{2} c x^{6} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{1}{2} \, A b c^{2} x^{6} + \frac{1}{5} \, B b^{3} x^{5} + \frac{6}{5} \, B a b c x^{5} + \frac{3}{5} \, A b^{2} c x^{5} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a b^{2} x^{4} + \frac{1}{4} \, A b^{3} x^{4} + \frac{3}{4} \, B a^{2} c x^{4} + \frac{3}{2} \, A a b c x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + \frac{1}{2} \, B a^{3} x^{2} + \frac{3}{2} \, A a^{2} b x^{2} + A a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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