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

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

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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}{2 x^2}+x \left (2 a B c+2 A b c+b^2 B\right )+\log (x) \left (A \left (2 a c+b^2\right )+2 a b B\right )-\frac {a (a B+2 A b)}{x}+\frac {1}{2} c x^2 (A c+2 b B)+\frac {1}{3} B c^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 86, normalized size = 0.96 \begin {gather*} -\frac {a^2 (A+2 B x)}{2 x^2}+A \log (x) \left (2 a c+b^2\right )+a \left (2 B c x-\frac {2 A b}{x}\right )+2 a b B \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)*(a + b*x + c*x^2)^2)/x^3,x]

[Out]

b^2*B*x + b*c*x*(2*A + B*x) - (a^2*(A + 2*B*x))/(2*x^2) + (c^2*x^2*(3*A + 2*B*x))/6 + a*((-2*A*b)/x + 2*B*c*x)
+ 2*a*b*B*Log[x] + A*(b^2 + 2*a*c)*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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 95, normalized size = 1.06 \begin {gather*} \frac {2 \, B c^{2} x^{5} + 3 \, {\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} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} \log \relax (x) - 3 \, A a^{2} - 6 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 89, normalized size = 0.99 \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 \, B a c x + 2 \, A b c x + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \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^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*B*a*c*x + 2*A*b*c*x + (2*B*a*b + A*b^2 + 2*A*a*c)*log(
abs(x)) - 1/2*(A*a^2 + 2*(B*a^2 + 2*A*a*b)*x)/x^2

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maple [A]  time = 0.06, size = 92, normalized size = 1.02 \begin {gather*} \frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+B b c \,x^{2}+2 A a c \ln \relax (x )+A \,b^{2} \ln \relax (x )+2 A b c x +2 B a b \ln \relax (x )+2 B a c x +B \,b^{2} x -\frac {2 A a b}{x}-\frac {B \,a^{2}}{x}-\frac {A \,a^{2}}{2 x^{2}} \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^3,x)

[Out]

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

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maxima [A]  time = 0.53, size = 88, normalized size = 0.98 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} \log \relax (x) - \frac {A a^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.15, size = 87, normalized size = 0.97 \begin {gather*} x^2\,\left (\frac {A\,c^2}{2}+B\,b\,c\right )+x\,\left (B\,b^2+2\,A\,c\,b+2\,B\,a\,c\right )+\ln \relax (x)\,\left (A\,b^2+2\,B\,a\,b+2\,A\,a\,c\right )-\frac {\frac {A\,a^2}{2}+x\,\left (B\,a^2+2\,A\,b\,a\right )}{x^2}+\frac {B\,c^2\,x^3}{3} \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^3,x)

[Out]

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

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sympy [A]  time = 0.44, size = 94, normalized size = 1.04 \begin {gather*} \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 + 2 B a c + B b^{2}\right ) + \left (2 A a c + A b^{2} + 2 B a b\right ) \log {\relax (x )} + \frac {- A a^{2} + x \left (- 4 A a b - 2 B a^{2}\right )}{2 x^{2}} \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**3,x)

[Out]

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

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