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

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

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Rubi [A]  time = 0.04, antiderivative size = 71, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 72, normalized size = 1.01 \begin {gather*} -\frac {A \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )}{20 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*(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))
/x^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 73, normalized size = 1.03 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.15, size = 75, normalized size = 1.06 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.05, size = 66, normalized size = 0.93 \begin {gather*} -\frac {B \,c^{3}}{x}-\frac {\left (A c +3 b B \right ) c^{2}}{2 x^{2}}-\frac {A \,b^{3}}{5 x^{5}}-\frac {\left (A c +b B \right ) b c}{x^{3}}-\frac {\left (3 A c +b B \right ) b^{2}}{4 x^{4}} \end {gather*}

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

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maxima [A]  time = 0.84, size = 73, normalized size = 1.03 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.02, size = 71, normalized size = 1.00 \begin {gather*} -\frac {x^2\,\left (B\,b^2\,c+A\,b\,c^2\right )+x\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,c\,b^2}{4}\right )+\frac {A\,b^3}{5}+x^3\,\left (\frac {A\,c^3}{2}+\frac {3\,B\,b\,c^2}{2}\right )+B\,c^3\,x^4}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.30, size = 82, normalized size = 1.15 \begin {gather*} \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 {gather*}

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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