3.151 \(\int \frac{A+B x}{x^4 (a+b x)} \, dx\)

Optimal. Leaf size=86 \[ -\frac{b^2 \log (x) (A b-a B)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}-\frac{b (A b-a B)}{a^3 x}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{3 a x^3} \]

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(2*a^2*x^2) - (b*(A*b - a*B))/(a^3*x) - (b^2*(A*b - a
*B)*Log[x])/a^4 + (b^2*(A*b - a*B)*Log[a + b*x])/a^4

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Rubi [A]  time = 0.12521, antiderivative size = 86, 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 \[ -\frac{b^2 \log (x) (A b-a B)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}-\frac{b (A b-a B)}{a^3 x}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(2*a^2*x^2) - (b*(A*b - a*B))/(a^3*x) - (b^2*(A*b - a
*B)*Log[x])/a^4 + (b^2*(A*b - a*B)*Log[a + b*x])/a^4

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Rubi in Sympy [A]  time = 25.9118, size = 73, normalized size = 0.85 \[ - \frac{A}{3 a x^{3}} + \frac{A b - B a}{2 a^{2} x^{2}} - \frac{b \left (A b - B a\right )}{a^{3} x} - \frac{b^{2} \left (A b - B a\right ) \log{\left (x \right )}}{a^{4}} + \frac{b^{2} \left (A b - B a\right ) \log{\left (a + b x \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0849843, size = 81, normalized size = 0.94 \[ \frac{\frac{a \left (a^2 (-(2 A+3 B x))+3 a b x (A+2 B x)-6 A b^2 x^2\right )}{x^3}+6 b^2 \log (x) (a B-A b)+6 b^2 (A b-a B) \log (a+b x)}{6 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((a*(-6*A*b^2*x^2 + 3*a*b*x*(A + 2*B*x) - a^2*(2*A + 3*B*x)))/x^3 + 6*b^2*(-(A*b
) + a*B)*Log[x] + 6*b^2*(A*b - a*B)*Log[a + b*x])/(6*a^4)

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Maple [A]  time = 0.013, size = 101, normalized size = 1.2 \[ -{\frac{A}{3\,a{x}^{3}}}+{\frac{Ab}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{2\,a{x}^{2}}}-{\frac{{b}^{2}A}{{a}^{3}x}}+{\frac{bB}{{a}^{2}x}}-{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{4}}}+{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{4}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.36167, size = 120, normalized size = 1.4 \[ -\frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{a^{4}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, A a^{2} - 6 \,{\left (B a b - A b^{2}\right )} x^{2} + 3 \,{\left (B a^{2} - A a b\right )} x}{6 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216383, size = 127, normalized size = 1.48 \[ -\frac{6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (x\right ) + 2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.09543, size = 165, normalized size = 1.92 \[ \frac{- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*A*a**2 + x**2*(-6*A*b**2 + 6*B*a*b) + x*(3*A*a*b - 3*B*a**2))/(6*a**3*x**3)
+ b**2*(-A*b + B*a)*log(x + (-A*a*b**3 + B*a**2*b**2 - a*b**2*(-A*b + B*a))/(-2*
A*b**4 + 2*B*a*b**3))/a**4 - b**2*(-A*b + B*a)*log(x + (-A*a*b**3 + B*a**2*b**2
+ a*b**2*(-A*b + B*a))/(-2*A*b**4 + 2*B*a*b**3))/a**4

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GIAC/XCAS [A]  time = 0.278971, size = 134, normalized size = 1.56 \[ \frac{{\left (B a b^{2} - A b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b^{3} - A b^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*a*b^2 - A*b^3)*ln(abs(x))/a^4 - (B*a*b^3 - A*b^4)*ln(abs(b*x + a))/(a^4*b) -
1/6*(2*A*a^3 - 6*(B*a^2*b - A*a*b^2)*x^2 + 3*(B*a^3 - A*a^2*b)*x)/(a^4*x^3)