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]

-(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]

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Rubi [A]  time = 0.0283647, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}

Mathematica [A]  time = 0.0250872, 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]

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

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Maple [A]  time = 0.007, size = 52, normalized size = 1.1 \begin{align*} B{c}^{2}\ln \left ( x \right ) -{\frac{Abc}{{x}^{2}}}-{\frac{{b}^{2}B}{2\,{x}^{2}}}-{\frac{A{c}^{2}}{x}}-2\,{\frac{Bbc}{x}}-{\frac{A{b}^{2}}{3\,{x}^{3}}} \end{align*}

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]

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

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Maxima [A]  time = 1.09453, size = 68, normalized size = 1.39 \begin{align*} B c^{2} \log \left (x\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}} \end{align*}

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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Fricas [A]  time = 1.78335, size = 122, normalized size = 2.49 \begin{align*} \frac{6 \, B c^{2} x^{3} \log \left (x\right ) - 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}} \end{align*}

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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Sympy [A]  time = 0.685877, size = 51, normalized size = 1.04 \begin{align*} B c^{2} \log{\left (x \right )} - \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}} \end{align*}

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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Giac [A]  time = 1.19538, size = 69, normalized size = 1.41 \begin{align*} 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}} \end{align*}

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