3.24 \(\int \frac {(A+B x) (b x+c x^2)^2}{x^6} \, dx\)

Optimal. Leaf size=49 \[ -\frac {A b^2}{3 x^3}-\frac {b (2 A c+b B)}{2 x^2}-\frac {c (A c+2 b B)}{x}+B c^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 49, 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} \[ -\frac {A b^2}{3 x^3}-\frac {b (2 A c+b B)}{2 x^2}-\frac {c (A c+2 b B)}{x}+B c^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^6,x]

[Out]

-(A*b^2)/(3*x^3) - (b*(b*B + 2*A*c))/(2*x^2) - (c*(2*b*B + A*c))/x + B*c^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^6} \, dx &=\int \left (\frac {A b^2}{x^4}+\frac {b (b B+2 A c)}{x^3}+\frac {c (2 b B+A c)}{x^2}+\frac {B c^2}{x}\right ) \, dx\\ &=-\frac {A b^2}{3 x^3}-\frac {b (b B+2 A c)}{2 x^2}-\frac {c (2 b B+A c)}{x}+B c^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 47, normalized size = 0.96 \[ B c^2 \log (x)-\frac {2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^6,x]

[Out]

-1/6*(3*b*B*x*(b + 4*c*x) + 2*A*(b^2 + 3*b*c*x + 3*c^2*x^2))/x^3 + B*c^2*Log[x]

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fricas [A]  time = 0.85, size = 53, normalized size = 1.08 \[ \frac {6 \, B c^{2} x^{3} \log \relax (x) - 2 \, A b^{2} - 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \, {\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^6,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 51, normalized size = 1.04 \[ B c^{2} \log \left ({\left | x \right |}\right ) - \frac {2 \, A b^{2} + 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^6,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 52, normalized size = 1.06 \[ B \,c^{2} \ln \relax (x )-\frac {A \,c^{2}}{x}-\frac {2 B b c}{x}-\frac {A b c}{x^{2}}-\frac {B \,b^{2}}{2 x^{2}}-\frac {A \,b^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^6,x)

[Out]

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

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maxima [A]  time = 0.87, size = 50, normalized size = 1.02 \[ B c^{2} \log \relax (x) - \frac {2 \, A b^{2} + 6 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^6,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.12, size = 48, normalized size = 0.98 \[ B\,c^2\,\ln \relax (x)-\frac {x^2\,\left (A\,c^2+2\,B\,b\,c\right )+\frac {A\,b^2}{3}+x\,\left (\frac {B\,b^2}{2}+A\,c\,b\right )}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^2*(A + B*x))/x^6,x)

[Out]

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

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sympy [A]  time = 0.53, size = 54, normalized size = 1.10 \[ B c^{2} \log {\relax (x )} + \frac {- 2 A b^{2} + x^{2} \left (- 6 A c^{2} - 12 B b c\right ) + x \left (- 6 A b c - 3 B b^{2}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/x**6,x)

[Out]

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

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