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

Optimal. Leaf size=47 \[ -\frac{a B+A b}{5 x^5}-\frac{a A}{6 x^6}-\frac{A c+b B}{4 x^4}-\frac{B c}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0668822, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{a B+A b}{5 x^5}-\frac{a A}{6 x^6}-\frac{A c+b B}{4 x^4}-\frac{B c}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a*A)/(6*x^6) - (A*b + 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 = 9.75336, size = 44, normalized size = 0.94 \[ - \frac{A a}{6 x^{6}} - \frac{B c}{3 x^{3}} - \frac{\frac{A c}{4} + \frac{B b}{4}}{x^{4}} - \frac{\frac{A b}{5} + \frac{B a}{5}}{x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0310304, size = 45, normalized size = 0.96 \[ -\frac{2 a (5 A+6 B x)+x (3 A (4 b+5 c x)+5 B x (3 b+4 c x))}{60 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(2*a*(5*A + 6*B*x) + x*(5*B*x*(3*b + 4*c*x) + 3*A*(4*b + 5*c*x)))/(60*x^6)

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Maple [A]  time = 0.008, size = 40, normalized size = 0.9 \[ -{\frac{aA}{6\,{x}^{6}}}-{\frac{Ac+Bb}{4\,{x}^{4}}}-{\frac{Bc}{3\,{x}^{3}}}-{\frac{Ab+Ba}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*a*A/x^6-1/4*(A*c+B*b)/x^4-1/3*B*c/x^3-1/5*(A*b+B*a)/x^5

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Maxima [A]  time = 0.685763, size = 53, normalized size = 1.13 \[ -\frac{20 \, B c x^{3} + 15 \,{\left (B b + A c\right )} x^{2} + 10 \, A a + 12 \,{\left (B a + A b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*c*x^3 + 15*(B*b + A*c)*x^2 + 10*A*a + 12*(B*a + A*b)*x)/x^6

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Fricas [A]  time = 0.265091, size = 53, normalized size = 1.13 \[ -\frac{20 \, B c x^{3} + 15 \,{\left (B b + A c\right )} x^{2} + 10 \, A a + 12 \,{\left (B a + A b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*c*x^3 + 15*(B*b + A*c)*x^2 + 10*A*a + 12*(B*a + A*b)*x)/x^6

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Sympy [A]  time = 10.225, size = 44, normalized size = 0.94 \[ - \frac{10 A a + 20 B c x^{3} + x^{2} \left (15 A c + 15 B b\right ) + x \left (12 A b + 12 B a\right )}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(10*A*a + 20*B*c*x**3 + x**2*(15*A*c + 15*B*b) + x*(12*A*b + 12*B*a))/(60*x**6)

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GIAC/XCAS [A]  time = 0.266992, size = 55, normalized size = 1.17 \[ -\frac{20 \, B c x^{3} + 15 \, B b x^{2} + 15 \, A c x^{2} + 12 \, B a x + 12 \, A b x + 10 \, A a}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*c*x^3 + 15*B*b*x^2 + 15*A*c*x^2 + 12*B*a*x + 12*A*b*x + 10*A*a)/x^6