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

Optimal. Leaf size=75 \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{4} A b^3 x^4+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{2} b c x^6 (A c+b B)+\frac{1}{8} B c^3 x^8 \]

[Out]

(A*b^3*x^4)/4 + (b^2*(b*B + 3*A*c)*x^5)/5 + (b*c*(b*B + A*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 = 0.0551682, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{4} A b^3 x^4+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{2} b c x^6 (A c+b B)+\frac{1}{8} B c^3 x^8 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*x^4)/4 + (b^2*(b*B + 3*A*c)*x^5)/5 + (b*c*(b*B + A*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 (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^3+b^2 (b B+3 A c) x^4+3 b c (b B+A c) x^5+c^2 (3 b B+A c) x^6+B c^3 x^7\right ) \, dx\\ &=\frac{1}{4} A b^3 x^4+\frac{1}{5} b^2 (b B+3 A c) x^5+\frac{1}{2} b c (b B+A c) 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 = 0.0119994, size = 75, normalized size = 1. \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{4} A b^3 x^4+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{2} b c x^6 (A c+b B)+\frac{1}{8} B c^3 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^4)/4 + (b^2*(b*B + 3*A*c)*x^5)/5 + (b*c*(b*B + A*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., size = 76, normalized size = 1. \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}+3\,B{b}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{5}}{5}}+{\frac{A{b}^{3}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.01821, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{4} \, A b^{3} x^{4} + \frac{1}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{2} c + A b c^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63444, size = 182, normalized size = 2.43 \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} b A + \frac{1}{5} x^{5} b^{3} B + \frac{3}{5} x^{5} c b^{2} A + \frac{1}{4} x^{4} b^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^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*b*A + 1/5*x^5*b^3*B + 3/5*x^5*
c*b^2*A + 1/4*x^4*b^3*A

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Sympy [A]  time = 0.076817, size = 80, normalized size = 1.07 \begin{align*} \frac{A b^{3} x^{4}}{4} + \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 b^{2} c}{2}\right ) + x^{5} \left (\frac{3 A b^{2} c}{5} + \frac{B b^{3}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**3*x**4/4 + 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*b**2*c/2) + x**5*(3*A*b*
*2*c/5 + B*b**3/5)

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Giac [A]  time = 1.10901, size = 104, normalized size = 1.39 \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} \, A b c^{2} x^{6} + \frac{1}{5} \, B b^{3} x^{5} + \frac{3}{5} \, A b^{2} c x^{5} + \frac{1}{4} \, A b^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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