3.25 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^7} \, dx\)

Optimal. Leaf size=45 \[ -\frac{(b+c x)^3 (4 b B-A c)}{12 b^2 x^3}-\frac{A (b+c x)^3}{4 b x^4} \]

[Out]

-(A*(b + c*x)^3)/(4*b*x^4) - ((4*b*B - A*c)*(b + c*x)^3)/(12*b^2*x^3)

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Rubi [A]  time = 0.0657799, 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 \[ -\frac{(b+c x)^3 (4 b B-A c)}{12 b^2 x^3}-\frac{A (b+c x)^3}{4 b x^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/x^7,x]

[Out]

-(A*(b + c*x)^3)/(4*b*x^4) - ((4*b*B - A*c)*(b + c*x)^3)/(12*b^2*x^3)

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Rubi in Sympy [A]  time = 11.1875, size = 48, normalized size = 1.07 \[ - \frac{A b^{2}}{4 x^{4}} - \frac{B c^{2}}{x} - \frac{b \left (2 A c + B b\right )}{3 x^{3}} - \frac{c \left (A c + 2 B b\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0276581, size = 50, normalized size = 1.11 \[ -\frac{A \left (3 b^2+8 b c x+6 c^2 x^2\right )+4 B x \left (b^2+3 b c x+3 c^2 x^2\right )}{12 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^7,x]

[Out]

-(4*B*x*(b^2 + 3*b*c*x + 3*c^2*x^2) + A*(3*b^2 + 8*b*c*x + 6*c^2*x^2))/(12*x^4)

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Maple [A]  time = 0.006, size = 48, normalized size = 1.1 \[ -{\frac{{b}^{2}A}{4\,{x}^{4}}}-{\frac{b \left ( 2\,Ac+Bb \right ) }{3\,{x}^{3}}}-{\frac{c \left ( Ac+2\,Bb \right ) }{2\,{x}^{2}}}-{\frac{B{c}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 0.694872, size = 69, normalized size = 1.53 \[ -\frac{12 \, B c^{2} x^{3} + 3 \, A b^{2} + 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 4 \,{\left (B b^{2} + 2 \, A b c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

-1/12*(12*B*c^2*x^3 + 3*A*b^2 + 6*(2*B*b*c + A*c^2)*x^2 + 4*(B*b^2 + 2*A*b*c)*x)
/x^4

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Fricas [A]  time = 0.258082, size = 69, normalized size = 1.53 \[ -\frac{12 \, B c^{2} x^{3} + 3 \, A b^{2} + 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 4 \,{\left (B b^{2} + 2 \, A b c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

-1/12*(12*B*c^2*x^3 + 3*A*b^2 + 6*(2*B*b*c + A*c^2)*x^2 + 4*(B*b^2 + 2*A*b*c)*x)
/x^4

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Sympy [A]  time = 3.37467, size = 54, normalized size = 1.2 \[ - \frac{3 A b^{2} + 12 B c^{2} x^{3} + x^{2} \left (6 A c^{2} + 12 B b c\right ) + x \left (8 A b c + 4 B b^{2}\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/x**7,x)

[Out]

-(3*A*b**2 + 12*B*c**2*x**3 + x**2*(6*A*c**2 + 12*B*b*c) + x*(8*A*b*c + 4*B*b**2
))/(12*x**4)

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GIAC/XCAS [A]  time = 0.267387, size = 69, normalized size = 1.53 \[ -\frac{12 \, B c^{2} x^{3} + 12 \, B b c x^{2} + 6 \, A c^{2} x^{2} + 4 \, B b^{2} x + 8 \, A b c x + 3 \, A b^{2}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^7,x, algorithm="giac")

[Out]

-1/12*(12*B*c^2*x^3 + 12*B*b*c*x^2 + 6*A*c^2*x^2 + 4*B*b^2*x + 8*A*b*c*x + 3*A*b
^2)/x^4