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

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

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} x^2 (A c+b B)+A b x+\frac {1}{3} B c x^3 \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (A c+b B)+A b x+\frac {1}{3} B c x^3 \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 32, normalized size = 1.14 \begin {gather*} A b x+\frac {1}{2} A c x^2+\frac {1}{2} b B x^2+\frac {1}{3} B c x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, B c x^{3} + A b x + \frac {1}{2} \, {\left (B b + A c\right )} x^{2} \end {gather*}

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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giac [A]  time = 0.15, size = 26, normalized size = 0.93 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.05, size = 25, normalized size = 0.89 \begin {gather*} \frac {B c \,x^{3}}{3}+A b x +\frac {\left (A c +b B \right ) x^{2}}{2} \end {gather*}

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 = 0.85, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, B c x^{3} + A b x + \frac {1}{2} \, {\left (B b + A c\right )} x^{2} \end {gather*}

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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mupad [B]  time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} \frac {B\,c\,x^3}{3}+\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x^2+A\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x,x)

[Out]

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

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sympy [A]  time = 0.06, size = 26, normalized size = 0.93 \begin {gather*} A b x + \frac {B c x^{3}}{3} + x^{2} \left (\frac {A c}{2} + \frac {B b}{2}\right ) \end {gather*}

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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