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

Optimal. Leaf size=46 \[ A b^2 \log (x)+\frac {1}{2} c x^2 (A c+2 b B)+b x (2 A c+b B)+\frac {1}{3} B c^2 x^3 \]

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} A b^2 \log (x)+\frac {1}{2} c x^2 (A c+2 b B)+b x (2 A c+b B)+\frac {1}{3} B c^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 43, normalized size = 0.93 \begin {gather*} A b^2 \log (x)+b c x (2 A+B x)+\frac {1}{6} c^2 x^2 (3 A+2 B x)+b^2 B x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b^2*B*x + b*c*x*(2*A + B*x) + (c^2*x^2*(3*A + 2*B*x))/6 + A*b^2*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 46, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + A b^{2} \log \relax (x) + \frac {1}{2} \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + {\left (B b^{2} + 2 \, A b c\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 46, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + B b c x^{2} + \frac {1}{2} \, A c^{2} x^{2} + B b^{2} x + 2 \, A b c x + A b^{2} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 46, normalized size = 1.00 \begin {gather*} \frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+B b c \,x^{2}+A \,b^{2} \ln \relax (x )+2 A b c x +B \,b^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 46, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + A b^{2} \log \relax (x) + \frac {1}{2} \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + {\left (B b^{2} + 2 \, A b c\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 45, normalized size = 0.98 \begin {gather*} x^2\,\left (\frac {A\,c^2}{2}+B\,b\,c\right )+x\,\left (B\,b^2+2\,A\,c\,b\right )+\frac {B\,c^2\,x^3}{3}+A\,b^2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, size = 46, normalized size = 1.00 \begin {gather*} A b^{2} \log {\relax (x )} + \frac {B c^{2} x^{3}}{3} + x^{2} \left (\frac {A c^{2}}{2} + B b c\right ) + x \left (2 A b c + 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)**2/x**3,x)

[Out]

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

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