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

Optimal. Leaf size=62 \[ -\frac{(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac{b (b+c x)^4 (b B-A c)}{4 c^3}+\frac{B (b+c x)^6}{6 c^3} \]

[Out]

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

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Rubi [A]  time = 0.0419726, antiderivative size = 62, 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} \[ -\frac{(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac{b (b+c x)^4 (b B-A c)}{4 c^3}+\frac{B (b+c x)^6}{6 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0144274, size = 67, normalized size = 1.08 \[ \frac{1}{60} x^2 \left (20 b^2 x (3 A c+b B)+30 A b^3+12 c^2 x^3 (A c+3 b B)+45 b c x^2 (A c+b B)+10 B c^3 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(30*A*b^3 + 20*b^2*(b*B + 3*A*c)*x + 45*b*c*(b*B + A*c)*x^2 + 12*c^2*(3*b*B + A*c)*x^3 + 10*B*c^3*x^4))/6
0

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Maple [A]  time = 0., size = 76, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{6}}{6}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{3}}{3}}+{\frac{A{b}^{3}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12939, size = 99, normalized size = 1.6 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{2} \, A b^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64776, size = 163, normalized size = 2.63 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{2} \, A b^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.124298, size = 80, normalized size = 1.29 \begin{align*} \frac{A b^{3} x^{2}}{2} + \frac{B c^{3} x^{6}}{6} + x^{5} \left (\frac{A c^{3}}{5} + \frac{3 B b c^{2}}{5}\right ) + x^{4} \left (\frac{3 A b c^{2}}{4} + \frac{3 B b^{2} c}{4}\right ) + x^{3} \left (A b^{2} c + \frac{B b^{3}}{3}\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**2,x)

[Out]

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

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Giac [A]  time = 1.14505, size = 103, normalized size = 1.66 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{3}{5} \, B b c^{2} x^{5} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B b^{2} c x^{4} + \frac{3}{4} \, A b c^{2} x^{4} + \frac{1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + \frac{1}{2} \, A b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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