3.183 \(\int \frac{A+B x}{x^5 (a+b x)} \, dx\)

Optimal. Leaf size=106 \[ \frac{b^2 (A b-a B)}{a^4 x}+\frac{b^3 \log (x) (A b-a B)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{4 a x^4} \]

[Out]

-A/(4*a*x^4) + (A*b - a*B)/(3*a^2*x^3) - (b*(A*b - a*B))/(2*a^3*x^2) + (b^2*(A*b - a*B))/(a^4*x) + (b^3*(A*b -
a*B)*Log[x])/a^5 - (b^3*(A*b - a*B)*Log[a + b*x])/a^5

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Rubi [A]  time = 0.0595751, antiderivative size = 106, 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, Rules used = {77} \[ \frac{b^2 (A b-a B)}{a^4 x}+\frac{b^3 \log (x) (A b-a B)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^5*(a + b*x)),x]

[Out]

-A/(4*a*x^4) + (A*b - a*B)/(3*a^2*x^3) - (b*(A*b - a*B))/(2*a^3*x^2) + (b^2*(A*b - a*B))/(a^4*x) + (b^3*(A*b -
a*B)*Log[x])/a^5 - (b^3*(A*b - a*B)*Log[a + b*x])/a^5

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{x^5 (a+b x)} \, dx &=\int \left (\frac{A}{a x^5}+\frac{-A b+a B}{a^2 x^4}-\frac{b (-A b+a B)}{a^3 x^3}+\frac{b^2 (-A b+a B)}{a^4 x^2}-\frac{b^3 (-A b+a B)}{a^5 x}+\frac{b^4 (-A b+a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{A}{4 a x^4}+\frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{b^2 (A b-a B)}{a^4 x}+\frac{b^3 (A b-a B) \log (x)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.062978, size = 100, normalized size = 0.94 \[ \frac{\frac{a \left (2 a^2 b x (2 A+3 B x)+a^3 (-(3 A+4 B x))-6 a b^2 x^2 (A+2 B x)+12 A b^3 x^3\right )}{x^4}+12 b^3 \log (x) (A b-a B)-12 b^3 (A b-a B) \log (a+b x)}{12 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^5*(a + b*x)),x]

[Out]

((a*(12*A*b^3*x^3 - 6*a*b^2*x^2*(A + 2*B*x) + 2*a^2*b*x*(2*A + 3*B*x) - a^3*(3*A + 4*B*x)))/x^4 + 12*b^3*(A*b
- a*B)*Log[x] - 12*b^3*(A*b - a*B)*Log[a + b*x])/(12*a^5)

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Maple [A]  time = 0.008, size = 125, normalized size = 1.2 \begin{align*} -{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{3\,{a}^{2}{x}^{3}}}-{\frac{B}{3\,a{x}^{3}}}+{\frac{{b}^{4}\ln \left ( x \right ) A}{{a}^{5}}}-{\frac{{b}^{3}B\ln \left ( x \right ) }{{a}^{4}}}-{\frac{{b}^{2}A}{2\,{a}^{3}{x}^{2}}}+{\frac{bB}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{3}A}{{a}^{4}x}}-{\frac{{b}^{2}B}{{a}^{3}x}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) A}{{a}^{5}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) B}{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^5/(b*x+a),x)

[Out]

-1/4*A/a/x^4+1/3/a^2/x^3*A*b-1/3/a/x^3*B+1/a^5*b^4*ln(x)*A-1/a^4*b^3*ln(x)*B-1/2/a^3*b^2/x^2*A+1/2/a^2*b/x^2*B
+1/a^4*b^3/x*A-1/a^3*b^2/x*B-1/a^5*b^4*ln(b*x+a)*A+1/a^4*b^3*ln(b*x+a)*B

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Maxima [A]  time = 1.00312, size = 151, normalized size = 1.42 \begin{align*} \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (x\right )}{a^{5}} - \frac{3 \, A a^{3} + 12 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 4 \,{\left (B a^{3} - A a^{2} b\right )} x}{12 \, a^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a),x, algorithm="maxima")

[Out]

(B*a*b^3 - A*b^4)*log(b*x + a)/a^5 - (B*a*b^3 - A*b^4)*log(x)/a^5 - 1/12*(3*A*a^3 + 12*(B*a*b^2 - A*b^3)*x^3 -
6*(B*a^2*b - A*a*b^2)*x^2 + 4*(B*a^3 - A*a^2*b)*x)/(a^4*x^4)

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Fricas [A]  time = 1.95608, size = 250, normalized size = 2.36 \begin{align*} \frac{12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (x\right ) - 3 \, A a^{4} - 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a),x, algorithm="fricas")

[Out]

1/12*(12*(B*a*b^3 - A*b^4)*x^4*log(b*x + a) - 12*(B*a*b^3 - A*b^4)*x^4*log(x) - 3*A*a^4 - 12*(B*a^2*b^2 - A*a*
b^3)*x^3 + 6*(B*a^3*b - A*a^2*b^2)*x^2 - 4*(B*a^4 - A*a^3*b)*x)/(a^5*x^4)

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Sympy [B]  time = 0.939351, size = 189, normalized size = 1.78 \begin{align*} - \frac{3 A a^{3} + x^{3} \left (- 12 A b^{3} + 12 B a b^{2}\right ) + x^{2} \left (6 A a b^{2} - 6 B a^{2} b\right ) + x \left (- 4 A a^{2} b + 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**5/(b*x+a),x)

[Out]

-(3*A*a**3 + x**3*(-12*A*b**3 + 12*B*a*b**2) + x**2*(6*A*a*b**2 - 6*B*a**2*b) + x*(-4*A*a**2*b + 4*B*a**3))/(1
2*a**4*x**4) - b**3*(-A*b + B*a)*log(x + (-A*a*b**4 + B*a**2*b**3 - a*b**3*(-A*b + B*a))/(-2*A*b**5 + 2*B*a*b*
*4))/a**5 + b**3*(-A*b + B*a)*log(x + (-A*a*b**4 + B*a**2*b**3 + a*b**3*(-A*b + B*a))/(-2*A*b**5 + 2*B*a*b**4)
)/a**5

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Giac [A]  time = 1.184, size = 165, normalized size = 1.56 \begin{align*} -\frac{{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{{\left (B a b^{4} - A b^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{3 \, A a^{4} + 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a),x, algorithm="giac")

[Out]

-(B*a*b^3 - A*b^4)*log(abs(x))/a^5 + (B*a*b^4 - A*b^5)*log(abs(b*x + a))/(a^5*b) - 1/12*(3*A*a^4 + 12*(B*a^2*b
^2 - A*a*b^3)*x^3 - 6*(B*a^3*b - A*a^2*b^2)*x^2 + 4*(B*a^4 - A*a^3*b)*x)/(a^5*x^4)