3.40 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^9} \, dx\)

Optimal. Leaf size=71 \[ -\frac{b^2 (3 A c+b B)}{4 x^4}-\frac{A b^3}{5 x^5}-\frac{c^2 (A c+3 b B)}{2 x^2}-\frac{b c (A c+b B)}{x^3}-\frac{B c^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0398933, antiderivative size = 71, 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} \[ -\frac{b^2 (3 A c+b B)}{4 x^4}-\frac{A b^3}{5 x^5}-\frac{c^2 (A c+3 b B)}{2 x^2}-\frac{b c (A c+b B)}{x^3}-\frac{B c^3}{x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^9,x]

[Out]

-(A*b^3)/(5*x^5) - (b^2*(b*B + 3*A*c))/(4*x^4) - (b*c*(b*B + A*c))/x^3 - (c^2*(3*b*B + A*c))/(2*x^2) - (B*c^3)
/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 )^3}{x^9} \, dx &=\int \left (\frac{A b^3}{x^6}+\frac{b^2 (b B+3 A c)}{x^5}+\frac{3 b c (b B+A c)}{x^4}+\frac{c^2 (3 b B+A c)}{x^3}+\frac{B c^3}{x^2}\right ) \, dx\\ &=-\frac{A b^3}{5 x^5}-\frac{b^2 (b B+3 A c)}{4 x^4}-\frac{b c (b B+A c)}{x^3}-\frac{c^2 (3 b B+A c)}{2 x^2}-\frac{B c^3}{x}\\ \end{align*}

Mathematica [A]  time = 0.0191064, size = 72, normalized size = 1.01 \[ -\frac{A \left (15 b^2 c x+4 b^3+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (4 b^2 c x+b^3+6 b c^2 x^2+4 c^3 x^3\right )}{20 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^9,x]

[Out]

-(5*B*x*(b^3 + 4*b^2*c*x + 6*b*c^2*x^2 + 4*c^3*x^3) + A*(4*b^3 + 15*b^2*c*x + 20*b*c^2*x^2 + 10*c^3*x^3))/(20*
x^5)

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Maple [A]  time = 0.006, size = 66, normalized size = 0.9 \begin{align*} -{\frac{A{b}^{3}}{5\,{x}^{5}}}-{\frac{{b}^{2} \left ( 3\,Ac+bB \right ) }{4\,{x}^{4}}}-{\frac{bc \left ( Ac+bB \right ) }{{x}^{3}}}-{\frac{{c}^{2} \left ( Ac+3\,bB \right ) }{2\,{x}^{2}}}-{\frac{B{c}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^9,x)

[Out]

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

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Maxima [A]  time = 1.14911, size = 99, normalized size = 1.39 \begin{align*} -\frac{20 \, B c^{3} x^{4} + 4 \, A b^{3} + 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 20 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^9,x, algorithm="maxima")

[Out]

-1/20*(20*B*c^3*x^4 + 4*A*b^3 + 10*(3*B*b*c^2 + A*c^3)*x^3 + 20*(B*b^2*c + A*b*c^2)*x^2 + 5*(B*b^3 + 3*A*b^2*c
)*x)/x^5

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Fricas [A]  time = 1.80338, size = 162, normalized size = 2.28 \begin{align*} -\frac{20 \, B c^{3} x^{4} + 4 \, A b^{3} + 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 20 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^9,x, algorithm="fricas")

[Out]

-1/20*(20*B*c^3*x^4 + 4*A*b^3 + 10*(3*B*b*c^2 + A*c^3)*x^3 + 20*(B*b^2*c + A*b*c^2)*x^2 + 5*(B*b^3 + 3*A*b^2*c
)*x)/x^5

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Sympy [A]  time = 2.31741, size = 78, normalized size = 1.1 \begin{align*} - \frac{4 A b^{3} + 20 B c^{3} x^{4} + x^{3} \left (10 A c^{3} + 30 B b c^{2}\right ) + x^{2} \left (20 A b c^{2} + 20 B b^{2} c\right ) + x \left (15 A b^{2} c + 5 B b^{3}\right )}{20 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**9,x)

[Out]

-(4*A*b**3 + 20*B*c**3*x**4 + x**3*(10*A*c**3 + 30*B*b*c**2) + x**2*(20*A*b*c**2 + 20*B*b**2*c) + x*(15*A*b**2
*c + 5*B*b**3))/(20*x**5)

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Giac [A]  time = 1.12414, size = 101, normalized size = 1.42 \begin{align*} -\frac{20 \, B c^{3} x^{4} + 30 \, B b c^{2} x^{3} + 10 \, A c^{3} x^{3} + 20 \, B b^{2} c x^{2} + 20 \, A b c^{2} x^{2} + 5 \, B b^{3} x + 15 \, A b^{2} c x + 4 \, A b^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^9,x, algorithm="giac")

[Out]

-1/20*(20*B*c^3*x^4 + 30*B*b*c^2*x^3 + 10*A*c^3*x^3 + 20*B*b^2*c*x^2 + 20*A*b*c^2*x^2 + 5*B*b^3*x + 15*A*b^2*c
*x + 4*A*b^3)/x^5