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

Optimal. Leaf size=38 \[ \frac {B (b+c x)^4}{4 c^2}-\frac {(b+c x)^3 (b B-A c)}{3 c^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, 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*} \frac {B (b+c x)^4}{4 c^2}-\frac {(b+c x)^3 (b B-A c)}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.24 \begin {gather*} \frac {1}{12} x \left (12 A b^2+4 c x^2 (A c+2 b B)+6 b x (2 A c+b B)+3 B c^2 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(12*A*b^2 + 6*b*(b*B + 2*A*c)*x + 4*c*(2*b*B + A*c)*x^2 + 3*B*c^2*x^3))/12

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IntegrateAlgebraic [A]  time = 0.03, size = 57, normalized size = 1.50 \begin {gather*} A b^2 x+A b c x^2+\frac {1}{3} A c^2 x^3+\frac {1}{2} b^2 B x^2+\frac {2}{3} b B c x^3+\frac {1}{4} B c^2 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, B c^{2} x^{4} + A b^{2} x + \frac {1}{3} \, {\left (2 \, B b c + A c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b 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)^2/x^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 49, normalized size = 1.29 \begin {gather*} \frac {1}{4} \, B c^{2} x^{4} + \frac {2}{3} \, B b c x^{3} + \frac {1}{3} \, A c^{2} x^{3} + \frac {1}{2} \, B b^{2} x^{2} + A b c x^{2} + A b^{2} 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^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 49, normalized size = 1.29 \begin {gather*} \frac {B \,c^{2} x^{4}}{4}+A \,b^{2} x +\frac {\left (A \,c^{2}+2 b B c \right ) x^{3}}{3}+\frac {\left (2 A b c +b^{2} 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)^2/x^2,x)

[Out]

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

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maxima [A]  time = 0.95, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, B c^{2} x^{4} + A b^{2} x + \frac {1}{3} \, {\left (2 \, B b c + A c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b 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)^2/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 47, normalized size = 1.24 \begin {gather*} x^2\,\left (\frac {B\,b^2}{2}+A\,c\,b\right )+x^3\,\left (\frac {A\,c^2}{3}+\frac {2\,B\,b\,c}{3}\right )+\frac {B\,c^2\,x^4}{4}+A\,b^2\,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^2,x)

[Out]

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

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sympy [A]  time = 0.08, size = 49, normalized size = 1.29 \begin {gather*} A b^{2} x + \frac {B c^{2} x^{4}}{4} + x^{3} \left (\frac {A c^{2}}{3} + \frac {2 B b c}{3}\right ) + x^{2} \left (A b c + \frac {B b^{2}}{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**2,x)

[Out]

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

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