3.2.67 \(\int \frac {1}{x^5 (a+b x)} \, dx\) [167]

Optimal. Leaf size=68 \[ -\frac {1}{4 a x^4}+\frac {b}{3 a^2 x^3}-\frac {b^2}{2 a^3 x^2}+\frac {b^3}{a^4 x}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 68, 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 = {46} \begin {gather*} \frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5}+\frac {b^3}{a^4 x}-\frac {b^2}{2 a^3 x^2}+\frac {b}{3 a^2 x^3}-\frac {1}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)),x]

[Out]

-1/4*1/(a*x^4) + b/(3*a^2*x^3) - b^2/(2*a^3*x^2) + b^3/(a^4*x) + (b^4*Log[x])/a^5 - (b^4*Log[a + b*x])/a^5

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 {align*} \int \frac {1}{x^5 (a+b x)} \, dx &=\int \left (\frac {1}{a x^5}-\frac {b}{a^2 x^4}+\frac {b^2}{a^3 x^3}-\frac {b^3}{a^4 x^2}+\frac {b^4}{a^5 x}-\frac {b^5}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac {1}{4 a x^4}+\frac {b}{3 a^2 x^3}-\frac {b^2}{2 a^3 x^2}+\frac {b^3}{a^4 x}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 68, normalized size = 1.00 \begin {gather*} -\frac {1}{4 a x^4}+\frac {b}{3 a^2 x^3}-\frac {b^2}{2 a^3 x^2}+\frac {b^3}{a^4 x}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)),x]

[Out]

-1/4*1/(a*x^4) + b/(3*a^2*x^3) - b^2/(2*a^3*x^2) + b^3/(a^4*x) + (b^4*Log[x])/a^5 - (b^4*Log[a + b*x])/a^5

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Mathics [A]
time = 2.20, size = 64, normalized size = 0.94 \begin {gather*} \frac {a \left (-3 a^3+4 a^2 b x-6 a b^2 x^2+12 b^3 x^3\right )+12 b^4 x^4 \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )}{12 a^5 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^5*(a + b*x)),x]')

[Out]

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

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Maple [A]
time = 0.09, size = 63, normalized size = 0.93

method	result	size
default	\(-\frac {1}{4 a \,x^{4}}+\frac {b}{3 a^{2} x^{3}}-\frac {b^{2}}{2 a^{3} x^{2}}+\frac {b^{3}}{a^{4} x}+\frac {b^{4} \ln \left (x \right )}{a^{5}}-\frac {b^{4} \ln \left (b x +a \right )}{a^{5}}\)	\(63\)
norman	\(\frac {\frac {b^{3} x^{3}}{a^{4}}-\frac {1}{4 a}+\frac {b x}{3 a^{2}}-\frac {b^{2} x^{2}}{2 a^{3}}}{x^{4}}+\frac {b^{4} \ln \left (x \right )}{a^{5}}-\frac {b^{4} \ln \left (b x +a \right )}{a^{5}}\)	\(63\)
risch	\(\frac {\frac {b^{3} x^{3}}{a^{4}}-\frac {1}{4 a}+\frac {b x}{3 a^{2}}-\frac {b^{2} x^{2}}{2 a^{3}}}{x^{4}}-\frac {b^{4} \ln \left (b x +a \right )}{a^{5}}+\frac {b^{4} \ln \left (-x \right )}{a^{5}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.24, size = 62, normalized size = 0.91 \begin {gather*} -\frac {b^{4} \log \left (b x + a\right )}{a^{5}} + \frac {b^{4} \log \left (x\right )}{a^{5}} + \frac {12 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 4 \, a^{2} b x - 3 \, a^{3}}{12 \, a^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 65, normalized size = 0.96 \begin {gather*} -\frac {12 \, b^{4} x^{4} \log \left (b x + a\right ) - 12 \, b^{4} x^{4} \log \left (x\right ) - 12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 3 \, a^{4}}{12 \, a^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.13, size = 56, normalized size = 0.82 \begin {gather*} \frac {- 3 a^{3} + 4 a^{2} b x - 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{12 a^{4} x^{4}} + \frac {b^{4} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x+a),x)

[Out]

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

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Giac [A]
time = 0.00, size = 77, normalized size = 1.13 \begin {gather*} -\frac {b^{5} \ln \left |x b+a\right |}{b a^{5}}+\frac {b^{4} \ln \left |x\right |}{a^{5}}+\frac {\frac {1}{12} \left (12 b^{3} a x^{3}-6 b^{2} a^{2} x^{2}+4 b a^{3} x-3 a^{4}\right )}{a^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a),x)

[Out]

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

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Mupad [B]
time = 0.07, size = 60, normalized size = 0.88 \begin {gather*} -\frac {\frac {a^4}{4}-\frac {a^3\,b\,x}{3}+\frac {a^2\,b^2\,x^2}{2}-a\,b^3\,x^3}{a^5\,x^4}-\frac {2\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x)),x)

[Out]

- (a^4/4 - a*b^3*x^3 + (a^2*b^2*x^2)/2 - (a^3*b*x)/3)/(a^5*x^4) - (2*b^4*atanh((2*b*x)/a + 1))/a^5

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