3.26 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^8} \, dx\)

Optimal. Leaf size=55 \[ -\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0885886, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.2038, size = 51, normalized size = 0.93 \[ - \frac{A b^{2}}{5 x^{5}} - \frac{B c^{2}}{2 x^{2}} - \frac{b \left (2 A c + B b\right )}{4 x^{4}} - \frac{c \left (A c + 2 B b\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.0273304, size = 53, normalized size = 0.96 \[ -\frac{2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )}{60 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^8,x]

[Out]

-(5*B*x*(3*b^2 + 8*b*c*x + 6*c^2*x^2) + 2*A*(6*b^2 + 15*b*c*x + 10*c^2*x^2))/(60
*x^5)

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Maple [A]  time = 0.008, size = 48, normalized size = 0.9 \[ -{\frac{{b}^{2}A}{5\,{x}^{5}}}-{\frac{b \left ( 2\,Ac+Bb \right ) }{4\,{x}^{4}}}-{\frac{c \left ( Ac+2\,Bb \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x^8,x)

[Out]

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

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Maxima [A]  time = 0.690403, size = 69, normalized size = 1.25 \[ -\frac{30 \, B c^{2} x^{3} + 12 \, A b^{2} + 20 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^8,x, algorithm="maxima")

[Out]

-1/60*(30*B*c^2*x^3 + 12*A*b^2 + 20*(2*B*b*c + A*c^2)*x^2 + 15*(B*b^2 + 2*A*b*c)
*x)/x^5

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Fricas [A]  time = 0.259315, size = 69, normalized size = 1.25 \[ -\frac{30 \, B c^{2} x^{3} + 12 \, A b^{2} + 20 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

-1/60*(30*B*c^2*x^3 + 12*A*b^2 + 20*(2*B*b*c + A*c^2)*x^2 + 15*(B*b^2 + 2*A*b*c)
*x)/x^5

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Sympy [A]  time = 4.00107, size = 54, normalized size = 0.98 \[ - \frac{12 A b^{2} + 30 B c^{2} x^{3} + x^{2} \left (20 A c^{2} + 40 B b c\right ) + x \left (30 A b c + 15 B b^{2}\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/x**8,x)

[Out]

-(12*A*b**2 + 30*B*c**2*x**3 + x**2*(20*A*c**2 + 40*B*b*c) + x*(30*A*b*c + 15*B*
b**2))/(60*x**5)

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GIAC/XCAS [A]  time = 0.266039, size = 69, normalized size = 1.25 \[ -\frac{30 \, B c^{2} x^{3} + 40 \, B b c x^{2} + 20 \, A c^{2} x^{2} + 15 \, B b^{2} x + 30 \, A b c x + 12 \, A b^{2}}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^8,x, algorithm="giac")

[Out]

-1/60*(30*B*c^2*x^3 + 40*B*b*c*x^2 + 20*A*c^2*x^2 + 15*B*b^2*x + 30*A*b*c*x + 12
*A*b^2)/x^5