3.2.76 \(\int \frac {1}{x^2 (a+b x)^2} \, dx\) [176]

Optimal. Leaf size=42 \[ -\frac {1}{a^2 x}-\frac {b}{a^2 (a+b x)}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 42, 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 {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}-\frac {b}{a^2 (a+b x)}-\frac {1}{a^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^2),x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.83 \begin {gather*} -\frac {a \left (\frac {1}{x}+\frac {b}{a+b x}\right )+2 b \log (x)-2 b \log (a+b x)}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^2),x]

[Out]

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

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Mathics [A]
time = 2.10, size = 49, normalized size = 1.17 \begin {gather*} \frac {a \left (-a-2 b x\right )-2 b x \left (a+b x\right ) \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )}{a^3 x \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^2*(a + b*x)^2),x]')

[Out]

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

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Maple [A]
time = 0.09, size = 43, normalized size = 1.02

method	result	size
default	\(-\frac {1}{a^{2} x}-\frac {b}{a^{2} \left (b x +a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (b x +a \right )}{a^{3}}\)	\(43\)
risch	\(\frac {-\frac {2 b x}{a^{2}}-\frac {1}{a}}{x \left (b x +a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (-b x -a \right )}{a^{3}}\)	\(49\)
norman	\(\frac {\frac {2 b^{2} x^{2}}{a^{3}}-\frac {1}{a}}{x \left (b x +a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (b x +a \right )}{a^{3}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 45, normalized size = 1.07 \begin {gather*} -\frac {2 \, b x + a}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \log \left (x\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.31, size = 63, normalized size = 1.50 \begin {gather*} -\frac {2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (x\right )}{a^{3} b x^{2} + a^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.13, size = 37, normalized size = 0.88 \begin {gather*} \frac {- a - 2 b x}{a^{3} x + a^{2} b x^{2}} + \frac {2 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**2,x)

[Out]

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

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Giac [A]
time = 0.00, size = 52, normalized size = 1.24 \begin {gather*} -\frac {2 b \ln \left |x\right |}{a^{3}}+\frac {2 b^{2} \ln \left |x b+a\right |}{b a^{3}}-\frac {2 x b+a}{a^{2} \left (x^{2} b+x a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^2,x)

[Out]

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

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Mupad [B]
time = 0.12, size = 45, normalized size = 1.07 \begin {gather*} \frac {2\,b\,\ln \left (\frac {a+b\,x}{x}\right )}{a^3}-\frac {1}{a\,x\,\left (a+b\,x\right )}-\frac {2\,b}{a^2\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x)^2),x)

[Out]

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

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