3.41 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^{10}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0374368, 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} \[ -\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{A b^3}{6 x^6}-\frac{c^2 (A c+3 b B)}{3 x^3}-\frac{3 b c (A c+b B)}{4 x^4}-\frac{B c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^10,x]

[Out]

-(A*b^3)/(6*x^6) - (b^2*(b*B + 3*A*c))/(5*x^5) - (3*b*c*(b*B + A*c))/(4*x^4) - (c^2*(3*b*B + A*c))/(3*x^3) - (
B*c^3)/(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 )^3}{x^{10}} \, dx &=\int \left (\frac{A b^3}{x^7}+\frac{b^2 (b B+3 A c)}{x^6}+\frac{3 b c (b B+A c)}{x^5}+\frac{c^2 (3 b B+A c)}{x^4}+\frac{B c^3}{x^3}\right ) \, dx\\ &=-\frac{A b^3}{6 x^6}-\frac{b^2 (b B+3 A c)}{5 x^5}-\frac{3 b c (b B+A c)}{4 x^4}-\frac{c^2 (3 b B+A c)}{3 x^3}-\frac{B c^3}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0201825, size = 74, normalized size = 0.99 \[ -\frac{A \left (36 b^2 c x+10 b^3+45 b c^2 x^2+20 c^3 x^3\right )+3 B x \left (15 b^2 c x+4 b^3+20 b c^2 x^2+10 c^3 x^3\right )}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^10,x]

[Out]

-(3*B*x*(4*b^3 + 15*b^2*c*x + 20*b*c^2*x^2 + 10*c^3*x^3) + A*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 + 20*c^3*x^3)
)/(60*x^6)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.9 \begin{align*} -{\frac{A{b}^{3}}{6\,{x}^{6}}}-{\frac{{b}^{2} \left ( 3\,Ac+bB \right ) }{5\,{x}^{5}}}-{\frac{3\,bc \left ( Ac+bB \right ) }{4\,{x}^{4}}}-{\frac{{c}^{2} \left ( Ac+3\,bB \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{3}}{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)^3/x^10,x)

[Out]

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

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Maxima [A]  time = 1.05769, size = 99, normalized size = 1.32 \begin{align*} -\frac{30 \, B c^{3} x^{4} + 10 \, A b^{3} + 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 12 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^10,x, algorithm="maxima")

[Out]

-1/60*(30*B*c^3*x^4 + 10*A*b^3 + 20*(3*B*b*c^2 + A*c^3)*x^3 + 45*(B*b^2*c + A*b*c^2)*x^2 + 12*(B*b^3 + 3*A*b^2
*c)*x)/x^6

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Fricas [A]  time = 1.72531, size = 165, normalized size = 2.2 \begin{align*} -\frac{30 \, B c^{3} x^{4} + 10 \, A b^{3} + 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 12 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^10,x, algorithm="fricas")

[Out]

-1/60*(30*B*c^3*x^4 + 10*A*b^3 + 20*(3*B*b*c^2 + A*c^3)*x^3 + 45*(B*b^2*c + A*b*c^2)*x^2 + 12*(B*b^3 + 3*A*b^2
*c)*x)/x^6

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Sympy [A]  time = 2.55909, size = 78, normalized size = 1.04 \begin{align*} - \frac{10 A b^{3} + 30 B c^{3} x^{4} + x^{3} \left (20 A c^{3} + 60 B b c^{2}\right ) + x^{2} \left (45 A b c^{2} + 45 B b^{2} c\right ) + x \left (36 A b^{2} c + 12 B b^{3}\right )}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**10,x)

[Out]

-(10*A*b**3 + 30*B*c**3*x**4 + x**3*(20*A*c**3 + 60*B*b*c**2) + x**2*(45*A*b*c**2 + 45*B*b**2*c) + x*(36*A*b**
2*c + 12*B*b**3))/(60*x**6)

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Giac [A]  time = 1.12239, size = 101, normalized size = 1.35 \begin{align*} -\frac{30 \, B c^{3} x^{4} + 60 \, B b c^{2} x^{3} + 20 \, A c^{3} x^{3} + 45 \, B b^{2} c x^{2} + 45 \, A b c^{2} x^{2} + 12 \, B b^{3} x + 36 \, A b^{2} c x + 10 \, A b^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^10,x, algorithm="giac")

[Out]

-1/60*(30*B*c^3*x^4 + 60*B*b*c^2*x^3 + 20*A*c^3*x^3 + 45*B*b^2*c*x^2 + 45*A*b*c^2*x^2 + 12*B*b^3*x + 36*A*b^2*
c*x + 10*A*b^3)/x^6