3.1.42 \(\int \frac {(A+B x) (b x+c x^2)^3}{x^{11}} \, dx\)

Optimal. Leaf size=75 \[ -\frac {A b^3}{7 x^7}-\frac {b^2 (3 A c+b B)}{6 x^6}-\frac {c^2 (A c+3 b B)}{4 x^4}-\frac {3 b c (A c+b B)}{5 x^5}-\frac {B c^3}{3 x^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 75, 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)}{6 x^6}-\frac {A b^3}{7 x^7}-\frac {c^2 (A c+3 b B)}{4 x^4}-\frac {3 b c (A c+b B)}{5 x^5}-\frac {B c^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^11,x]

[Out]

-(A*b^3)/(7*x^7) - (b^2*(b*B + 3*A*c))/(6*x^6) - (3*b*c*(b*B + A*c))/(5*x^5) - (c^2*(3*b*B + A*c))/(4*x^4) - (
B*c^3)/(3*x^3)

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^{11}} \, dx &=\int \left (\frac {A b^3}{x^8}+\frac {b^2 (b B+3 A c)}{x^7}+\frac {3 b c (b B+A c)}{x^6}+\frac {c^2 (3 b B+A c)}{x^5}+\frac {B c^3}{x^4}\right ) \, dx\\ &=-\frac {A b^3}{7 x^7}-\frac {b^2 (b B+3 A c)}{6 x^6}-\frac {3 b c (b B+A c)}{5 x^5}-\frac {c^2 (3 b B+A c)}{4 x^4}-\frac {B c^3}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 75, normalized size = 1.00 \begin {gather*} -\frac {3 A \left (20 b^3+70 b^2 c x+84 b c^2 x^2+35 c^3 x^3\right )+7 B x \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )}{420 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^11,x]

[Out]

-1/420*(7*B*x*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 + 20*c^3*x^3) + 3*A*(20*b^3 + 70*b^2*c*x + 84*b*c^2*x^2 + 35
*c^3*x^3))/x^7

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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^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^3)/x^11,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^3)/x^11, x]

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fricas [A]  time = 0.38, size = 73, normalized size = 0.97 \begin {gather*} -\frac {140 \, B c^{3} x^{4} + 60 \, A b^{3} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 252 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 70 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^11,x, algorithm="fricas")

[Out]

-1/420*(140*B*c^3*x^4 + 60*A*b^3 + 105*(3*B*b*c^2 + A*c^3)*x^3 + 252*(B*b^2*c + A*b*c^2)*x^2 + 70*(B*b^3 + 3*A
*b^2*c)*x)/x^7

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giac [A]  time = 0.16, size = 75, normalized size = 1.00 \begin {gather*} -\frac {140 \, B c^{3} x^{4} + 315 \, B b c^{2} x^{3} + 105 \, A c^{3} x^{3} + 252 \, B b^{2} c x^{2} + 252 \, A b c^{2} x^{2} + 70 \, B b^{3} x + 210 \, A b^{2} c x + 60 \, A b^{3}}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^11,x, algorithm="giac")

[Out]

-1/420*(140*B*c^3*x^4 + 315*B*b*c^2*x^3 + 105*A*c^3*x^3 + 252*B*b^2*c*x^2 + 252*A*b*c^2*x^2 + 70*B*b^3*x + 210
*A*b^2*c*x + 60*A*b^3)/x^7

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maple [A]  time = 0.05, size = 66, normalized size = 0.88 \begin {gather*} -\frac {B \,c^{3}}{3 x^{3}}-\frac {\left (A c +3 b B \right ) c^{2}}{4 x^{4}}-\frac {A \,b^{3}}{7 x^{7}}-\frac {3 \left (A c +b B \right ) b c}{5 x^{5}}-\frac {\left (3 A c +b B \right ) b^{2}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^11,x)

[Out]

-1/7*A*b^3/x^7-1/6*b^2*(3*A*c+B*b)/x^6-3/5*(A*c+B*b)*b*c/x^5-1/4*c^2*(A*c+3*B*b)/x^4-1/3*B*c^3/x^3

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maxima [A]  time = 0.91, size = 73, normalized size = 0.97 \begin {gather*} -\frac {140 \, B c^{3} x^{4} + 60 \, A b^{3} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 252 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 70 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^11,x, algorithm="maxima")

[Out]

-1/420*(140*B*c^3*x^4 + 60*A*b^3 + 105*(3*B*b*c^2 + A*c^3)*x^3 + 252*(B*b^2*c + A*b*c^2)*x^2 + 70*(B*b^3 + 3*A
*b^2*c)*x)/x^7

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mupad [B]  time = 0.05, size = 74, normalized size = 0.99 \begin {gather*} -\frac {x^2\,\left (\frac {3\,B\,b^2\,c}{5}+\frac {3\,A\,b\,c^2}{5}\right )+x\,\left (\frac {B\,b^3}{6}+\frac {A\,c\,b^2}{2}\right )+\frac {A\,b^3}{7}+x^3\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+\frac {B\,c^3\,x^4}{3}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^3*(A + B*x))/x^11,x)

[Out]

-(x^2*((3*A*b*c^2)/5 + (3*B*b^2*c)/5) + x*((B*b^3)/6 + (A*b^2*c)/2) + (A*b^3)/7 + x^3*((A*c^3)/4 + (3*B*b*c^2)
/4) + (B*c^3*x^4)/3)/x^7

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sympy [A]  time = 2.04, size = 82, normalized size = 1.09 \begin {gather*} \frac {- 60 A b^{3} - 140 B c^{3} x^{4} + x^{3} \left (- 105 A c^{3} - 315 B b c^{2}\right ) + x^{2} \left (- 252 A b c^{2} - 252 B b^{2} c\right ) + x \left (- 210 A b^{2} c - 70 B b^{3}\right )}{420 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**11,x)

[Out]

(-60*A*b**3 - 140*B*c**3*x**4 + x**3*(-105*A*c**3 - 315*B*b*c**2) + x**2*(-252*A*b*c**2 - 252*B*b**2*c) + x*(-
210*A*b**2*c - 70*B*b**3))/(420*x**7)

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