3.8.86 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{x^4} \, dx\)

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

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Rubi [A]  time = 0.06, 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}{3 x^3}-\frac {A \left (2 a c+b^2\right )+2 a b B}{x}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac {a (a B+2 A b)}{2 x^2}+c x (A c+2 b B)+\frac {1}{2} B c^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^4,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 90, normalized size = 1.00 \begin {gather*} -\frac {a^2 (2 A+3 B x)}{6 x^3}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac {a (A b+2 A c x+2 b B x)}{x^2}-\frac {A b^2}{x}+A c^2 x+2 b B c x+\frac {1}{2} B c^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^4,x]

[Out]

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^2)/x^4,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^2)/x^4, 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} + 6 \, {\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} \log \relax (x) - 2 \, A a^{2} - 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 89, normalized size = 0.99 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x + {\left (B b^{2} + 2 \, B a c + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 95, normalized size = 1.06 \begin {gather*} \frac {B \,c^{2} x^{2}}{2}+2 A b c \ln \relax (x )+A \,c^{2} x +2 B a c \ln \relax (x )+B \,b^{2} \ln \relax (x )+2 B b c x -\frac {2 A a c}{x}-\frac {A \,b^{2}}{x}-\frac {2 B a b}{x}-\frac {A a b}{x^{2}}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \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^4,x)

[Out]

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

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maxima [A]  time = 0.50, size = 89, normalized size = 0.99 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} + {\left (2 \, B b c + A c^{2}\right )} x + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} \log \relax (x) - \frac {2 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 87, normalized size = 0.97 \begin {gather*} x\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {\frac {A\,a^2}{3}+x^2\,\left (A\,b^2+2\,B\,a\,b+2\,A\,a\,c\right )+x\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )}{x^3}+\ln \relax (x)\,\left (B\,b^2+2\,A\,c\,b+2\,B\,a\,c\right )+\frac {B\,c^2\,x^2}{2} \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^4,x)

[Out]

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

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sympy [A]  time = 1.07, size = 99, normalized size = 1.10 \begin {gather*} \frac {B c^{2} x^{2}}{2} + x \left (A c^{2} + 2 B b c\right ) + \left (2 A b c + 2 B a c + B b^{2}\right ) \log {\relax (x )} + \frac {- 2 A a^{2} + x^{2} \left (- 12 A a c - 6 A b^{2} - 12 B a b\right ) + x \left (- 6 A a b - 3 B a^{2}\right )}{6 x^{3}} \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**4,x)

[Out]

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

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