3.850 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{A c+b B}{x}+B c \log (x) \]

[Out]

-(a*A)/(3*x^3) - (A*b + a*B)/(2*x^2) - (b*B + A*c)/x + B*c*Log[x]

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Rubi [A]  time = 0.0624709, antiderivative size = 41, 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 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{A c+b B}{x}+B c \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2))/x^4,x]

[Out]

-(a*A)/(3*x^3) - (A*b + a*B)/(2*x^2) - (b*B + A*c)/x + B*c*Log[x]

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Rubi in Sympy [A]  time = 9.59198, size = 36, normalized size = 0.88 \[ - \frac{A a}{3 x^{3}} + B c \log{\left (x \right )} - \frac{A c + B b}{x} - \frac{\frac{A b}{2} + \frac{B a}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0506744, size = 41, normalized size = 1. \[ B c \log (x)-\frac{a (2 A+3 B x)+3 x (A b+2 A c x+2 b B x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2))/x^4,x]

[Out]

-(a*(2*A + 3*B*x) + 3*x*(A*b + 2*b*B*x + 2*A*c*x))/(6*x^3) + B*c*Log[x]

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Maple [A]  time = 0.008, size = 42, normalized size = 1. \[ Bc\ln \left ( x \right ) -{\frac{aA}{3\,{x}^{3}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{Ba}{2\,{x}^{2}}}-{\frac{Ac}{x}}-{\frac{Bb}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)/x^4,x)

[Out]

B*c*ln(x)-1/3*a*A/x^3-1/2*A*b/x^2-1/2*B*a/x^2-1/x*A*c-B*b/x

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Maxima [A]  time = 0.694569, size = 51, normalized size = 1.24 \[ B c \log \left (x\right ) - \frac{6 \,{\left (B b + A c\right )} x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.309532, size = 55, normalized size = 1.34 \[ \frac{6 \, B c x^{3} \log \left (x\right ) - 6 \,{\left (B b + A c\right )} x^{2} - 2 \, A a - 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.89868, size = 41, normalized size = 1. \[ B c \log{\left (x \right )} - \frac{2 A a + x^{2} \left (6 A c + 6 B b\right ) + x \left (3 A b + 3 B a\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)/x**4,x)

[Out]

B*c*log(x) - (2*A*a + x**2*(6*A*c + 6*B*b) + x*(3*A*b + 3*B*a))/(6*x**3)

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GIAC/XCAS [A]  time = 0.268251, size = 53, normalized size = 1.29 \[ B c{\rm ln}\left ({\left | x \right |}\right ) - \frac{6 \,{\left (B b + A c\right )} x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

B*c*ln(abs(x)) - 1/6*(6*(B*b + A*c)*x^2 + 2*A*a + 3*(B*a + A*b)*x)/x^3