3.2.78 \(\int \frac {1}{x^4 (a+b x)^2} \, dx\) [178]

Optimal. Leaf size=69 \[ -\frac {1}{3 a^2 x^3}+\frac {b}{a^3 x^2}-\frac {3 b^2}{a^4 x}-\frac {b^3}{a^4 (a+b x)}-\frac {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log (a+b x)}{a^5} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 69, 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 {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log (a+b x)}{a^5}-\frac {b^3}{a^4 (a+b x)}-\frac {3 b^2}{a^4 x}+\frac {b}{a^3 x^2}-\frac {1}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.96 \begin {gather*} -\frac {\frac {a \left (a^3-2 a^2 b x+6 a b^2 x^2+12 b^3 x^3\right )}{x^3 (a+b x)}+12 b^3 \log (x)-12 b^3 \log (a+b x)}{3 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^2),x]

[Out]

-1/3*((a*(a^3 - 2*a^2*b*x + 6*a*b^2*x^2 + 12*b^3*x^3))/(x^3*(a + b*x)) + 12*b^3*Log[x] - 12*b^3*Log[a + b*x])/
a^5

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Mathics [A]
time = 2.35, size = 76, normalized size = 1.10 \begin {gather*} \frac {a \left (-a^3+2 a^2 b x-6 a b^2 x^2-12 b^3 x^3\right )-12 b^3 x^3 \left (a+b x\right ) \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )}{3 a^5 x^3 \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^4*(a + b*x)^2),x]')

[Out]

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

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Maple [A]
time = 0.08, size = 68, normalized size = 0.99

method	result	size
default	\(-\frac {1}{3 a^{2} x^{3}}+\frac {b}{a^{3} x^{2}}-\frac {3 b^{2}}{a^{4} x}-\frac {b^{3}}{a^{4} \left (b x +a \right )}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+\frac {4 b^{3} \ln \left (b x +a \right )}{a^{5}}\)	\(68\)
norman	\(\frac {\frac {4 b^{4} x^{4}}{a^{5}}-\frac {1}{3 a}+\frac {2 b x}{3 a^{2}}-\frac {2 b^{2} x^{2}}{a^{3}}}{x^{3} \left (b x +a \right )}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+\frac {4 b^{3} \ln \left (b x +a \right )}{a^{5}}\)	\(72\)
risch	\(\frac {-\frac {4 b^{3} x^{3}}{a^{4}}-\frac {2 b^{2} x^{2}}{a^{3}}+\frac {2 b x}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (b x +a \right )}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+\frac {4 b^{3} \ln \left (-b x -a \right )}{a^{5}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 73, normalized size = 1.06 \begin {gather*} -\frac {12 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}}{3 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac {4 \, b^{3} \log \left (b x + a\right )}{a^{5}} - \frac {4 \, b^{3} \log \left (x\right )}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 95, normalized size = 1.38 \begin {gather*} -\frac {12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4} - 12 \, {\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (x\right )}{3 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.17, size = 66, normalized size = 0.96 \begin {gather*} \frac {- a^{3} + 2 a^{2} b x - 6 a b^{2} x^{2} - 12 b^{3} x^{3}}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} + \frac {4 b^{3} \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**4/(b*x+a)**2,x)

[Out]

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

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Giac [A]
time = 0.00, size = 84, normalized size = 1.22 \begin {gather*} \frac {4 b^{4} \ln \left |x b+a\right |}{b a^{5}}-\frac {4 b^{3} \ln \left |x\right |}{a^{5}}+\frac {\frac {1}{3} \left (-12 b^{3} a x^{3}-6 b^{2} a^{2} x^{2}+2 b a^{3} x-a^{4}\right )}{a^{5} x^{3} \left (b x+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^2,x)

[Out]

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

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Mupad [B]
time = 0.08, size = 69, normalized size = 1.00 \begin {gather*} \frac {8\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5}-\frac {\frac {1}{3\,a}+\frac {2\,b^2\,x^2}{a^3}+\frac {4\,b^3\,x^3}{a^4}-\frac {2\,b\,x}{3\,a^2}}{b\,x^4+a\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x)^2),x)

[Out]

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

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