3.180 \(\int \frac{c+d x}{x^3 (a+b x)} \, dx\)

Optimal. Leaf size=62 \[ \frac{b \log (x) (b c-a d)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}+\frac{b c-a d}{a^2 x}-\frac{c}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0997947, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{b \log (x) (b c-a d)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}+\frac{b c-a d}{a^2 x}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(x^3*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.6384, size = 53, normalized size = 0.85 \[ - \frac{c}{2 a x^{2}} - \frac{a d - b c}{a^{2} x} - \frac{b \left (a d - b c\right ) \log{\left (x \right )}}{a^{3}} + \frac{b \left (a d - b c\right ) \log{\left (a + b x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)/x**3/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0515368, size = 58, normalized size = 0.94 \[ \frac{-\frac{a (a c+2 a d x-2 b c x)}{x^2}+2 b \log (x) (b c-a d)+2 b (a d-b c) \log (a+b x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)/(x^3*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 75, normalized size = 1.2 \[ -{\frac{c}{2\,a{x}^{2}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}}-{\frac{b\ln \left ( x \right ) d}{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) c}{{a}^{3}}}+{\frac{b\ln \left ( bx+a \right ) d}{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) c}{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/x^3/(b*x+a),x)

[Out]

-1/2*c/a/x^2-1/a/x*d+1/a^2/x*b*c-1/a^2*b*ln(x)*d+1/a^3*b^2*ln(x)*c+1/a^2*b*ln(b*
x+a)*d-1/a^3*b^2*ln(b*x+a)*c

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Maxima [A]  time = 1.35835, size = 85, normalized size = 1.37 \[ -\frac{{\left (b^{2} c - a b d\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (b^{2} c - a b d\right )} \log \left (x\right )}{a^{3}} - \frac{a c - 2 \,{\left (b c - a d\right )} x}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)/((b*x + a)*x^3),x, algorithm="maxima")

[Out]

-(b^2*c - a*b*d)*log(b*x + a)/a^3 + (b^2*c - a*b*d)*log(x)/a^3 - 1/2*(a*c - 2*(b
*c - a*d)*x)/(a^2*x^2)

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Fricas [A]  time = 0.209225, size = 92, normalized size = 1.48 \[ -\frac{2 \,{\left (b^{2} c - a b d\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (b^{2} c - a b d\right )} x^{2} \log \left (x\right ) + a^{2} c - 2 \,{\left (a b c - a^{2} d\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)/((b*x + a)*x^3),x, algorithm="fricas")

[Out]

-1/2*(2*(b^2*c - a*b*d)*x^2*log(b*x + a) - 2*(b^2*c - a*b*d)*x^2*log(x) + a^2*c
- 2*(a*b*c - a^2*d)*x)/(a^3*x^2)

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Sympy [A]  time = 3.7108, size = 131, normalized size = 2.11 \[ - \frac{a c + x \left (2 a d - 2 b c\right )}{2 a^{2} x^{2}} - \frac{b \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b d - a b^{2} c - a b \left (a d - b c\right )}{2 a b^{2} d - 2 b^{3} c} \right )}}{a^{3}} + \frac{b \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b d - a b^{2} c + a b \left (a d - b c\right )}{2 a b^{2} d - 2 b^{3} c} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/x**3/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296036, size = 101, normalized size = 1.63 \[ \frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c - a b^{2} d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{a^{2} c - 2 \,{\left (a b c - a^{2} d\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)/((b*x + a)*x^3),x, algorithm="giac")

[Out]

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