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

Optimal. Leaf size=28 \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]

[Out]

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

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Rubi [A]  time = 0.0182864, antiderivative size = 28, 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 = {765} \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x,x]

[Out]

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

Mathematica [A]  time = 0.0044914, size = 28, normalized size = 1. \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 25, normalized size = 0.9 \begin{align*} Abx+{\frac{ \left ( Ac+bB \right ){x}^{2}}{2}}+{\frac{Bc{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)/x,x)

[Out]

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

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Maxima [A]  time = 0.96317, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, B c x^{3} + A b x + \frac{1}{2} \,{\left (B b + A c\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)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65916, size = 58, normalized size = 2.07 \begin{align*} \frac{1}{3} \, B c x^{3} + A b x + \frac{1}{2} \,{\left (B b + A c\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)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.062886, size = 26, normalized size = 0.93 \begin{align*} A b x + \frac{B c x^{3}}{3} + x^{2} \left (\frac{A c}{2} + \frac{B b}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13863, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, B b x^{2} + \frac{1}{2} \, A c x^{2} + A b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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