3.2.66 \(\int \frac {1}{x^4 (a+b x)} \, dx\) [166]

Optimal. Leaf size=56 \[ -\frac {1}{3 a x^3}+\frac {b}{2 a^2 x^2}-\frac {b^2}{a^3 x}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log (a+b x)}{a^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 56, 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^3 \log (x)}{a^4}+\frac {b^3 \log (a+b x)}{a^4}-\frac {b^2}{a^3 x}+\frac {b}{2 a^2 x^2}-\frac {1}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 56, normalized size = 1.00 \begin {gather*} -\frac {1}{3 a x^3}+\frac {b}{2 a^2 x^2}-\frac {b^2}{a^3 x}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log (a+b x)}{a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)),x]

[Out]

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

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Mathics [A]
time = 2.12, size = 53, normalized size = 0.95 \begin {gather*} \frac {a \left (-2 a^2+3 a b x-6 b^2 x^2\right )+6 b^3 x^3 \left (\text {Log}\left [\frac {a+b x}{b}\right ]-\text {Log}\left [x\right ]\right )}{6 a^4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^4*(a + b*x)),x]')

[Out]

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

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Maple [A]
time = 0.09, size = 53, normalized size = 0.95

method	result	size
default	\(-\frac {1}{3 a \,x^{3}}+\frac {b}{2 a^{2} x^{2}}-\frac {b^{2}}{a^{3} x}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (b x +a \right )}{a^{4}}\)	\(53\)
norman	\(\frac {-\frac {1}{3 a}+\frac {b x}{2 a^{2}}-\frac {b^{2} x^{2}}{a^{3}}}{x^{3}}+\frac {b^{3} \ln \left (b x +a \right )}{a^{4}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}\)	\(53\)
risch	\(\frac {-\frac {1}{3 a}+\frac {b x}{2 a^{2}}-\frac {b^{2} x^{2}}{a^{3}}}{x^{3}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (-b x -a \right )}{a^{4}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.24, size = 51, normalized size = 0.91 \begin {gather*} \frac {b^{3} \log \left (b x + a\right )}{a^{4}} - \frac {b^{3} \log \left (x\right )}{a^{4}} - \frac {6 \, b^{2} x^{2} - 3 \, a b x + 2 \, a^{2}}{6 \, a^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.32, size = 54, normalized size = 0.96 \begin {gather*} \frac {6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.11, size = 44, normalized size = 0.79 \begin {gather*} \frac {- 2 a^{2} + 3 a b x - 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac {b^{3} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 66, normalized size = 1.18 \begin {gather*} \frac {b^{4} \ln \left |x b+a\right |}{b a^{4}}-\frac {b^{3} \ln \left |x\right |}{a^{4}}+\frac {\frac {1}{6} \left (-6 b^{2} a x^{2}+3 b a^{2} x-2 a^{3}\right )}{a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.10, size = 48, normalized size = 0.86 \begin {gather*} \frac {2\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^4}-\frac {\frac {a^3}{3}-\frac {a^2\,b\,x}{2}+a\,b^2\,x^2}{a^4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x)),x)

[Out]

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

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