3.26 \(\int \frac{(A+B x) (b x+c x^2)^2}{x^8} \, dx\)

Optimal. Leaf size=55 \[ -\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.027832, antiderivative size = 55, 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} \[ -\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^8,x]

[Out]

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

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^8} \, dx &=\int \left (\frac{A b^2}{x^6}+\frac{b (b B+2 A c)}{x^5}+\frac{c (2 b B+A c)}{x^4}+\frac{B c^2}{x^3}\right ) \, dx\\ &=-\frac{A b^2}{5 x^5}-\frac{b (b B+2 A c)}{4 x^4}-\frac{c (2 b B+A c)}{3 x^3}-\frac{B c^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0145733, size = 53, normalized size = 0.96 \[ -\frac{2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )}{60 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^8,x]

[Out]

-(5*B*x*(3*b^2 + 8*b*c*x + 6*c^2*x^2) + 2*A*(6*b^2 + 15*b*c*x + 10*c^2*x^2))/(60*x^5)

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Maple [A]  time = 0.004, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{b}^{2}}{5\,{x}^{5}}}-{\frac{b \left ( 2\,Ac+bB \right ) }{4\,{x}^{4}}}-{\frac{c \left ( Ac+2\,bB \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^8,x)

[Out]

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

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Maxima [A]  time = 1.05425, size = 69, normalized size = 1.25 \begin{align*} -\frac{30 \, B c^{2} x^{3} + 12 \, A b^{2} + 20 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^8,x, algorithm="maxima")

[Out]

-1/60*(30*B*c^2*x^3 + 12*A*b^2 + 20*(2*B*b*c + A*c^2)*x^2 + 15*(B*b^2 + 2*A*b*c)*x)/x^5

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Fricas [A]  time = 1.64742, size = 120, normalized size = 2.18 \begin{align*} -\frac{30 \, B c^{2} x^{3} + 12 \, A b^{2} + 20 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^8,x, algorithm="fricas")

[Out]

-1/60*(30*B*c^2*x^3 + 12*A*b^2 + 20*(2*B*b*c + A*c^2)*x^2 + 15*(B*b^2 + 2*A*b*c)*x)/x^5

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Sympy [A]  time = 1.10666, size = 54, normalized size = 0.98 \begin{align*} - \frac{12 A b^{2} + 30 B c^{2} x^{3} + x^{2} \left (20 A c^{2} + 40 B b c\right ) + x \left (30 A b c + 15 B b^{2}\right )}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/x**8,x)

[Out]

-(12*A*b**2 + 30*B*c**2*x**3 + x**2*(20*A*c**2 + 40*B*b*c) + x*(30*A*b*c + 15*B*b**2))/(60*x**5)

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Giac [A]  time = 1.1173, size = 69, normalized size = 1.25 \begin{align*} -\frac{30 \, B c^{2} x^{3} + 40 \, B b c x^{2} + 20 \, A c^{2} x^{2} + 15 \, B b^{2} x + 30 \, A b c x + 12 \, A b^{2}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^8,x, algorithm="giac")

[Out]

-1/60*(30*B*c^2*x^3 + 40*B*b*c*x^2 + 20*A*c^2*x^2 + 15*B*b^2*x + 30*A*b*c*x + 12*A*b^2)/x^5