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

Optimal. Leaf size=90 \[ -\frac {a^2 A}{x}+\frac {1}{2} x^2 \left (2 a B c+2 A b c+b^2 B\right )+x \left (A \left (2 a c+b^2\right )+2 a b B\right )+a \log (x) (a B+2 A b)+\frac {1}{3} c x^3 (A c+2 b B)+\frac {1}{4} B c^2 x^4 \]

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Rubi [A]  time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} -\frac {a^2 A}{x}+\frac {1}{2} x^2 \left (2 a B c+2 A b c+b^2 B\right )+x \left (A \left (2 a c+b^2\right )+2 a b B\right )+a \log (x) (a B+2 A b)+\frac {1}{3} c x^3 (A c+2 b B)+\frac {1}{4} B c^2 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*A)/x) + (2*a*b*B + A*(b^2 + 2*a*c))*x + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^2)/2 + (c*(2*b*B + A*c)*x^3)/3 +
(B*c^2*x^4)/4 + a*(2*A*b + a*B)*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 (a+b x+c x^2\right )^2}{x^2} \, dx &=\int \left (A b^2 \left (1+\frac {2 a (b B+A c)}{A b^2}\right )+\frac {a^2 A}{x^2}+\frac {a (2 A b+a B)}{x}+\left (b^2 B+2 A b c+2 a B c\right ) x+c (2 b B+A c) x^2+B c^2 x^3\right ) \, dx\\ &=-\frac {a^2 A}{x}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x+\frac {1}{2} \left (b^2 B+2 A b c+2 a B c\right ) x^2+\frac {1}{3} c (2 b B+A c) x^3+\frac {1}{4} B c^2 x^4+a (2 A b+a B) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 87, normalized size = 0.97 \begin {gather*} -\frac {a^2 A}{x}+a x (2 A c+2 b B+B c x)+a \log (x) (a B+2 A b)+\frac {1}{12} x \left (4 A \left (3 b^2+3 b c x+c^2 x^2\right )+B x \left (6 b^2+8 b c x+3 c^2 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*A)/x) + a*x*(2*b*B + 2*A*c + B*c*x) + (x*(4*A*(3*b^2 + 3*b*c*x + c^2*x^2) + B*x*(6*b^2 + 8*b*c*x + 3*c^
2*x^2)))/12 + a*(2*A*b + a*B)*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 (a+b x+c x^2\right )^2}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 95, normalized size = 1.06 \begin {gather*} \frac {3 \, B c^{2} x^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 12 \, A a^{2} + 12 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} x \log \relax (x)}{12 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*B*c^2*x^5 + 4*(2*B*b*c + A*c^2)*x^4 + 6*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 12*A*a^2 + 12*(2*B*a*b + A*b^2
+ 2*A*a*c)*x^2 + 12*(B*a^2 + 2*A*a*b)*x*log(x))/x

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giac [A]  time = 0.20, size = 92, normalized size = 1.02 \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} + B a c x^{2} + A b c x^{2} + 2 \, B a b x + A b^{2} x + 2 \, A a c x - \frac {A a^{2}}{x} + {\left (B a^{2} + 2 \, A a b\right )} \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+a)^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 + B*a*c*x^2 + A*b*c*x^2 + 2*B*a*b*x + A*b^2*x +
2*A*a*c*x - A*a^2/x + (B*a^2 + 2*A*a*b)*log(abs(x))

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maple [A]  time = 0.05, size = 92, normalized size = 1.02 \begin {gather*} \frac {B \,c^{2} x^{4}}{4}+\frac {A \,c^{2} x^{3}}{3}+\frac {2 B b c \,x^{3}}{3}+A b c \,x^{2}+B a c \,x^{2}+\frac {B \,b^{2} x^{2}}{2}+2 A a b \ln \relax (x )+2 A a c x +A \,b^{2} x +B \,a^{2} \ln \relax (x )+2 B a b x -\frac {A \,a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 88, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, B c^{2} x^{4} + \frac {1}{3} \, {\left (2 \, B b c + A c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{2} - \frac {A a^{2}}{x} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x + {\left (B a^{2} + 2 \, A a b\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 86, normalized size = 0.96 \begin {gather*} x^3\,\left (\frac {A\,c^2}{3}+\frac {2\,B\,b\,c}{3}\right )+x\,\left (A\,b^2+2\,B\,a\,b+2\,A\,a\,c\right )+\ln \relax (x)\,\left (B\,a^2+2\,A\,b\,a\right )+x^2\,\left (\frac {B\,b^2}{2}+A\,c\,b+B\,a\,c\right )-\frac {A\,a^2}{x}+\frac {B\,c^2\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.26, size = 88, normalized size = 0.98 \begin {gather*} - \frac {A a^{2}}{x} + \frac {B c^{2} x^{4}}{4} + a \left (2 A b + B a\right ) \log {\relax (x )} + x^{3} \left (\frac {A c^{2}}{3} + \frac {2 B b c}{3}\right ) + x^{2} \left (A b c + B a c + \frac {B b^{2}}{2}\right ) + x \left (2 A a c + A b^{2} + 2 B a b\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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