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

Optimal. Leaf size=45 \[ -\frac{(b+c x)^4 (5 b B-A c)}{20 b^2 x^4}-\frac{A (b+c x)^4}{5 b x^5} \]

[Out]

-(A*(b + c*x)^4)/(5*b*x^5) - ((5*b*B - A*c)*(b + c*x)^4)/(20*b^2*x^4)

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Rubi [A]  time = 0.0677126, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{(b+c x)^4 (5 b B-A c)}{20 b^2 x^4}-\frac{A (b+c x)^4}{5 b x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(A*(b + c*x)^4)/(5*b*x^5) - ((5*b*B - A*c)*(b + c*x)^4)/(20*b^2*x^4)

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Rubi in Sympy [A]  time = 15.238, size = 65, normalized size = 1.44 \[ - \frac{A b^{3}}{5 x^{5}} - \frac{B c^{3}}{x} - \frac{b^{2} \left (3 A c + B b\right )}{4 x^{4}} - \frac{b c \left (A c + B b\right )}{x^{3}} - \frac{c^{2} \left (A c + 3 B b\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0353892, size = 72, normalized size = 1.6 \[ -\frac{A \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )}{20 x^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 66, normalized size = 1.5 \[ -{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{4\,{x}^{4}}}-{\frac{bc \left ( Ac+Bb \right ) }{{x}^{3}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{2\,{x}^{2}}}-{\frac{A{b}^{3}}{5\,{x}^{5}}}-{\frac{B{c}^{3}}{x}} \]

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]

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

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Maxima [A]  time = 0.710597, size = 99, normalized size = 2.2 \[ -\frac{20 \, B c^{3} x^{4} + 4 \, A b^{3} + 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 20 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/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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Fricas [A]  time = 0.268187, size = 99, normalized size = 2.2 \[ -\frac{20 \, B c^{3} x^{4} + 4 \, A b^{3} + 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 20 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/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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Sympy [A]  time = 5.72578, size = 78, normalized size = 1.73 \[ - \frac{4 A b^{3} + 20 B c^{3} x^{4} + x^{3} \left (10 A c^{3} + 30 B b c^{2}\right ) + x^{2} \left (20 A b c^{2} + 20 B b^{2} c\right ) + x \left (15 A b^{2} c + 5 B b^{3}\right )}{20 x^{5}} \]

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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GIAC/XCAS [A]  time = 0.268744, size = 101, normalized size = 2.24 \[ -\frac{20 \, B c^{3} x^{4} + 30 \, B b c^{2} x^{3} + 10 \, A c^{3} x^{3} + 20 \, B b^{2} c x^{2} + 20 \, A b c^{2} x^{2} + 5 \, B b^{3} x + 15 \, A b^{2} c x + 4 \, A b^{3}}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/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