3.1.54 \(\int \frac {1}{x (a+b x)^4} \, dx\) [54]

Optimal. Leaf size=51 \[ \frac {\frac {11}{6 a}+\frac {5 b x}{2 a^2}+\frac {b^2 x^2}{a^3}}{(a+b x)^3}-\frac {\log \left (\frac {a+b x}{x}\right )}{a^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.12, 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 = {46} \begin {gather*} -\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} \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 46

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] && Lt
Q[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\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 {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.94 \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 (x)-6 \log (a+b 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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Maple [A]
time = 0.03, size = 54, normalized size = 1.06

method	result	size
risch	\(\frac {\frac {11}{6 a}+\frac {5 b x}{2 a^{2}}+\frac {b^{2} x^{2}}{a^{3}}}{\left (b x +a \right )^{3}}+\frac {\ln \left (-x \right )}{a^{4}}-\frac {\ln \left (b x +a \right )}{a^{4}}\)	\(52\)
default	\(-\frac {\ln \left (b x +a \right )}{a^{4}}+\frac {1}{a^{3} \left (b x +a \right )}+\frac {1}{2 a^{2} \left (b x +a \right )^{2}}+\frac {1}{3 a \left (b x +a \right )^{3}}+\frac {\ln \left (x \right )}{a^{4}}\)	\(54\)
norman	\(\frac {-\frac {3 b x}{a^{2}}-\frac {9 b^{2} x^{2}}{2 a^{3}}-\frac {11 b^{3} x^{3}}{6 a^{4}}}{\left (b x +a \right )^{3}}+\frac {\ln \left (x \right )}{a^{4}}-\frac {\ln \left (b x +a \right )}{a^{4}}\)	\(57\)
parallelrisch	\(\frac {6 \ln \left (x \right ) x^{3} b^{3}-6 \ln \left (b x +a \right ) x^{3} b^{3}+18 \ln \left (x \right ) x^{2} a \,b^{2}-18 \ln \left (b x +a \right ) x^{2} a \,b^{2}-11 b^{3} x^{3}+18 \ln \left (x \right ) x \,a^{2} b -18 \ln \left (b x +a \right ) x \,a^{2} b -27 a \,x^{2} b^{2}+6 \ln \left (x \right ) a^{3}-6 \ln \left (b x +a \right ) a^{3}-18 a^{2} x b}{6 a^{4} \left (b x +a \right )^{3}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 73, normalized size = 1.43 \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 \left (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, 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] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
time = 0.58, size = 124, normalized size = 2.43 \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 \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 {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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Sympy [A]
time = 0.17, size = 70, normalized size = 1.37 \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 {\left (x \right )} - \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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Giac [A]
time = 0.42, size = 54, normalized size = 1.06 \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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Mupad [B]
time = 0.09, size = 60, normalized size = 1.18 \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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