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

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

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.84 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (a+b x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*(a + b*x)^4),x]

[Out]

IntegrateAlgebraic[1/(x*(a + b*x)^4), x]

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fricas [B]  time = 1.07, size = 124, normalized size = 2.18 \begin {gather*} \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 \relax (x)}{6 \, {\left (a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} + 3 \, a^{6} b x + a^{7}\right )}} \end {gather*}

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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giac [A]  time = 1.02, size = 54, normalized size = 0.95 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.01, size = 54, normalized size = 0.95 \begin {gather*} \frac {1}{3 \left (b x +a \right )^{3} a}+\frac {1}{2 \left (b x +a \right )^{2} a^{2}}+\frac {1}{\left (b x +a \right ) a^{3}}+\frac {\ln \relax (x )}{a^{4}}-\frac {\ln \left (b x +a \right )}{a^{4}} \end {gather*}

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 = 1.47, size = 73, normalized size = 1.28 \begin {gather*} \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 \relax (x)}{a^{4}} \end {gather*}

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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mupad [B]  time = 0.13, size = 60, normalized size = 1.05 \begin {gather*} \frac {\frac {\frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2}}{a}+\frac {1}{2\,a\,{\left (a+b\,x\right )}^2}}{a}+\frac {1}{3\,a\,{\left (a+b\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^4),x)

[Out]

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

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sympy [A]  time = 0.44, size = 70, normalized size = 1.23 \begin {gather*} \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 {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a^{4}} \end {gather*}

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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