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

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

[Out]

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

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*}

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Mathematica [A]  time = 0.02, size = 74, normalized size = 0.99 \[ -\frac {A \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )+3 B x \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )}{60 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(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))/x^6

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fricas [A]  time = 1.04, size = 73, normalized size = 0.97 \[ -\frac {30 \, B c^{3} x^{4} + 10 \, A b^{3} + 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 12 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{60 \, x^{6}} \]

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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giac [A]  time = 0.18, size = 75, normalized size = 1.00 \[ -\frac {30 \, B c^{3} x^{4} + 60 \, B b c^{2} x^{3} + 20 \, A c^{3} x^{3} + 45 \, B b^{2} c x^{2} + 45 \, A b c^{2} x^{2} + 12 \, B b^{3} x + 36 \, A b^{2} c x + 10 \, A b^{3}}{60 \, x^{6}} \]

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

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

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

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maxima [A]  time = 0.81, size = 73, normalized size = 0.97 \[ -\frac {30 \, B c^{3} x^{4} + 10 \, A b^{3} + 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 12 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x}{60 \, x^{6}} \]

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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mupad [B]  time = 0.04, size = 73, normalized size = 0.97 \[ -\frac {x^2\,\left (\frac {3\,B\,b^2\,c}{4}+\frac {3\,A\,b\,c^2}{4}\right )+x\,\left (\frac {B\,b^3}{5}+\frac {3\,A\,c\,b^2}{5}\right )+\frac {A\,b^3}{6}+x^3\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+\frac {B\,c^3\,x^4}{2}}{x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.63, size = 82, normalized size = 1.09 \[ \frac {- 10 A b^{3} - 30 B c^{3} x^{4} + x^{3} \left (- 20 A c^{3} - 60 B b c^{2}\right ) + x^{2} \left (- 45 A b c^{2} - 45 B b^{2} c\right ) + x \left (- 36 A b^{2} c - 12 B b^{3}\right )}{60 x^{6}} \]

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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