3.32 \(\int \frac{(A+B x) (b x+c x^2)^3}{x} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0439684, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{1}{4} b^2 x^4 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x,x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0105688, size = 75, normalized size = 1. \[ \frac{1}{4} b^2 x^4 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 76, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{7}}{7}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{4}}{4}}+{\frac{A{b}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02121, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{3} \, A b^{3} x^{3} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.81004, size = 163, normalized size = 2.17 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{3} \, A b^{3} x^{3} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.076226, size = 82, normalized size = 1.09 \begin{align*} \frac{A b^{3} x^{3}}{3} + \frac{B c^{3} x^{7}}{7} + x^{6} \left (\frac{A c^{3}}{6} + \frac{B b c^{2}}{2}\right ) + x^{5} \left (\frac{3 A b c^{2}}{5} + \frac{3 B b^{2} c}{5}\right ) + x^{4} \left (\frac{3 A b^{2} c}{4} + \frac{B b^{3}}{4}\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,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16755, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{2} \, B b c^{2} x^{6} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B b^{2} c x^{5} + \frac{3}{5} \, A b c^{2} x^{5} + \frac{1}{4} \, B b^{3} x^{4} + \frac{3}{4} \, A b^{2} c x^{4} + \frac{1}{3} \, A b^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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