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

Optimal. Leaf size=33 \[ -\frac{A c+b B}{4 x^4}-\frac{A b}{5 x^5}-\frac{B c}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0395515, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{A c+b B}{4 x^4}-\frac{A b}{5 x^5}-\frac{B c}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.93461, size = 31, normalized size = 0.94 \[ - \frac{A b}{5 x^{5}} - \frac{B c}{3 x^{3}} - \frac{\frac{A c}{4} + \frac{B b}{4}}{x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0175523, size = 31, normalized size = 0.94 \[ -\frac{3 A (4 b+5 c x)+5 B x (3 b+4 c x)}{60 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(5*B*x*(3*b + 4*c*x) + 3*A*(4*b + 5*c*x))/(60*x^5)

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Maple [A]  time = 0.006, size = 28, normalized size = 0.9 \[ -{\frac{Ac+Bb}{4\,{x}^{4}}}-{\frac{Bc}{3\,{x}^{3}}}-{\frac{Ab}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x^7,x)

[Out]

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

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Maxima [A]  time = 0.696168, size = 36, normalized size = 1.09 \[ -\frac{20 \, B c x^{2} + 12 \, A b + 15 \,{\left (B b + A c\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28463, size = 36, normalized size = 1.09 \[ -\frac{20 \, B c x^{2} + 12 \, A b + 15 \,{\left (B b + A c\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.25023, size = 31, normalized size = 0.94 \[ - \frac{12 A b + 20 B c x^{2} + x \left (15 A c + 15 B b\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268401, size = 36, normalized size = 1.09 \[ -\frac{20 \, B c x^{2} + 15 \, B b x + 15 \, A c x + 12 \, A b}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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