3.202 \(\int \frac{1}{x (a+b x)^4} \, dx\)

Optimal. Leaf size=57 \[ \frac{1}{a^3 (a+b x)}+\frac{1}{2 a^2 (a+b x)^2}-\frac{\log (a+b x)}{a^4}+\frac{\log (x)}{a^4}+\frac{1}{3 a (a+b x)^3} \]

[Out]

1/(3*a*(a + b*x)^3) + 1/(2*a^2*(a + b*x)^2) + 1/(a^3*(a + b*x)) + Log[x]/a^4 - Log[a + b*x]/a^4

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Rubi [A]  time = 0.0283706, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {44} \[ \frac{1}{a^3 (a+b x)}+\frac{1}{2 a^2 (a+b x)^2}-\frac{\log (a+b x)}{a^4}+\frac{\log (x)}{a^4}+\frac{1}{3 a (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^4),x]

[Out]

1/(3*a*(a + b*x)^3) + 1/(2*a^2*(a + b*x)^2) + 1/(a^3*(a + b*x)) + Log[x]/a^4 - Log[a + b*x]/a^4

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x (a+b x)^4} \, dx &=\int \left (\frac{1}{a^4 x}-\frac{b}{a (a+b x)^4}-\frac{b}{a^2 (a+b x)^3}-\frac{b}{a^3 (a+b x)^2}-\frac{b}{a^4 (a+b x)}\right ) \, dx\\ &=\frac{1}{3 a (a+b x)^3}+\frac{1}{2 a^2 (a+b x)^2}+\frac{1}{a^3 (a+b x)}+\frac{\log (x)}{a^4}-\frac{\log (a+b x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0566188, size = 48, normalized size = 0.84 \[ \frac{\frac{a \left (11 a^2+15 a b x+6 b^2 x^2\right )}{(a+b x)^3}-6 \log (a+b x)+6 \log (x)}{6 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)^4),x]

[Out]

((a*(11*a^2 + 15*a*b*x + 6*b^2*x^2))/(a + b*x)^3 + 6*Log[x] - 6*Log[a + b*x])/(6*a^4)

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Maple [A]  time = 0.006, size = 54, normalized size = 1. \begin{align*}{\frac{1}{3\,a \left ( bx+a \right ) ^{3}}}+{\frac{1}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{1}{{a}^{3} \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{4}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.0721, size = 99, normalized size = 1.74 \begin{align*} \frac{6 \, b^{2} x^{2} + 15 \, a b x + 11 \, a^{2}}{6 \,{\left (a^{3} b^{3} x^{3} + 3 \, a^{4} b^{2} x^{2} + 3 \, a^{5} b x + a^{6}\right )}} - \frac{\log \left (b x + a\right )}{a^{4}} + \frac{\log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^4,x, algorithm="maxima")

[Out]

1/6*(6*b^2*x^2 + 15*a*b*x + 11*a^2)/(a^3*b^3*x^3 + 3*a^4*b^2*x^2 + 3*a^5*b*x + a^6) - log(b*x + a)/a^4 + log(x
)/a^4

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Fricas [B]  time = 1.57789, size = 271, normalized size = 4.75 \begin{align*} \frac{6 \, a b^{2} x^{2} + 15 \, a^{2} b x + 11 \, a^{3} - 6 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} + 3 \, a^{6} b x + a^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/6*(6*a*b^2*x^2 + 15*a^2*b*x + 11*a^3 - 6*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*log(b*x + a) + 6*(b^3*x^3
+ 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*log(x))/(a^4*b^3*x^3 + 3*a^5*b^2*x^2 + 3*a^6*b*x + a^7)

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Sympy [A]  time = 0.769741, size = 70, normalized size = 1.23 \begin{align*} \frac{11 a^{2} + 15 a b x + 6 b^{2} x^{2}}{6 a^{6} + 18 a^{5} b x + 18 a^{4} b^{2} x^{2} + 6 a^{3} b^{3} x^{3}} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)**4,x)

[Out]

(11*a**2 + 15*a*b*x + 6*b**2*x**2)/(6*a**6 + 18*a**5*b*x + 18*a**4*b**2*x**2 + 6*a**3*b**3*x**3) + (log(x) - l
og(a/b + x))/a**4

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Giac [A]  time = 1.16938, size = 73, normalized size = 1.28 \begin{align*} -\frac{\log \left ({\left | b x + a \right |}\right )}{a^{4}} + \frac{\log \left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, a b^{2} x^{2} + 15 \, a^{2} b x + 11 \, a^{3}}{6 \,{\left (b x + a\right )}^{3} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^4,x, algorithm="giac")

[Out]

-log(abs(b*x + a))/a^4 + log(abs(x))/a^4 + 1/6*(6*a*b^2*x^2 + 15*a^2*b*x + 11*a^3)/((b*x + a)^3*a^4)