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

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

[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)

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Rubi [A]  time = 0.0408125, antiderivative size = 75, 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} \[ -\frac{b^2 (3 A c+b B)}{6 x^6}-\frac{A b^3}{7 x^7}-\frac{c^2 (A c+3 b B)}{4 x^4}-\frac{3 b c (A c+b B)}{5 x^5}-\frac{B c^3}{3 x^3} \]

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*}

Mathematica [A]  time = 0.0199361, size = 75, normalized size = 1. \[ -\frac{3 A \left (70 b^2 c x+20 b^3+84 b c^2 x^2+35 c^3 x^3\right )+7 B x \left (36 b^2 c x+10 b^3+45 b c^2 x^2+20 c^3 x^3\right )}{420 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/(420*x^7)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.9 \begin{align*} -{\frac{A{b}^{3}}{7\,{x}^{7}}}-{\frac{{b}^{2} \left ( 3\,Ac+bB \right ) }{6\,{x}^{6}}}-{\frac{3\,bc \left ( Ac+bB \right ) }{5\,{x}^{5}}}-{\frac{{c}^{2} \left ( Ac+3\,bB \right ) }{4\,{x}^{4}}}-{\frac{B{c}^{3}}{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)^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*b*c*(A*c+B*b)/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 = 1.08281, size = 99, normalized size = 1.32 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.72633, size = 170, normalized size = 2.27 \begin{align*} -\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{align*}

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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Sympy [A]  time = 3.8193, size = 78, normalized size = 1.04 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.16938, size = 101, normalized size = 1.35 \begin{align*} -\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{align*}

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