3.1.55 \(\int \frac {1}{x^2 (a+b x)^4} \, dx\) [55]

Optimal. Leaf size=62 \[ \frac {-\frac {22 b}{3 a^2}-\frac {1}{a x}-\frac {10 b^2 x}{a^3}-\frac {4 b^3 x^2}{a^4}}{(a+b x)^3}+\frac {4 b \log \left (\frac {a+b x}{x}\right )}{a^5} \]

[Out]

-(1/a/x+22/3*b/a^2+10*b^2*x/a^3+4*b^3*x^2/a^4)/(b*x+a)^3+4*b/a^5*ln((b*x+a)/x)

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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.13, 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 \log (x)}{a^5}+\frac {4 b \log (a+b x)}{a^5}-\frac {3 b}{a^4 (a+b x)}-\frac {1}{a^4 x}-\frac {b}{a^3 (a+b x)^2}-\frac {b}{3 a^2 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^4),x]

[Out]

-(1/(a^4*x)) - b/(3*a^2*(a + b*x)^3) - b/(a^3*(a + b*x)^2) - (3*b)/(a^4*(a + b*x)) - (4*b*Log[x])/a^5 + (4*b*L
og[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 {gather*} \begin {aligned} \text {Integral} &=\int \left (\frac {1}{a^4 x^2}-\frac {4 b}{a^5 x}+\frac {b^2}{a^2 (a+b x)^4}+\frac {2 b^2}{a^3 (a+b x)^3}+\frac {3 b^2}{a^4 (a+b x)^2}+\frac {4 b^2}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a^4 x}-\frac {b}{3 a^2 (a+b x)^3}-\frac {b}{a^3 (a+b x)^2}-\frac {3 b}{a^4 (a+b x)}-\frac {4 b \log (x)}{a^5}+\frac {4 b \log (a+b x)}{a^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 64, normalized size = 1.03 \begin {gather*} -\frac {\frac {a \left (3 a^3+22 a^2 b x+30 a b^2 x^2+12 b^3 x^3\right )}{x (a+b x)^3}+12 b \log (x)-12 b \log (a+b x)}{3 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^4),x]

[Out]

-1/3*((a*(3*a^3 + 22*a^2*b*x + 30*a*b^2*x^2 + 12*b^3*x^3))/(x*(a + b*x)^3) + 12*b*Log[x] - 12*b*Log[a + b*x])/
a^5

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Maple [A]
time = 0.04, size = 69, normalized size = 1.11

method	result	size
default	\(-\frac {b}{3 a^{2} \left (b x +a \right )^{3}}+\frac {4 b \ln \left (b x +a \right )}{a^{5}}-\frac {3 b}{a^{4} \left (b x +a \right )}-\frac {b}{a^{3} \left (b x +a \right )^{2}}-\frac {1}{a^{4} x}-\frac {4 b \ln \left (x \right )}{a^{5}}\)	\(69\)
risch	\(\frac {-\frac {4 b^{3} x^{3}}{a^{4}}-\frac {10 b^{2} x^{2}}{a^{3}}-\frac {22 b x}{3 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{3}}+\frac {4 b \ln \left (-b x -a \right )}{a^{5}}-\frac {4 b \ln \left (x \right )}{a^{5}}\)	\(71\)
norman	\(\frac {-\frac {1}{a}+\frac {12 b^{2} x^{2}}{a^{3}}+\frac {18 b^{3} x^{3}}{a^{4}}+\frac {22 b^{4} x^{4}}{3 a^{5}}}{x \left (b x +a \right )^{3}}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {4 b \ln \left (b x +a \right )}{a^{5}}\)	\(72\)
parallelrisch	\(-\frac {12 \ln \left (x \right ) x^{4} b^{4}-12 \ln \left (b x +a \right ) x^{4} b^{4}+36 \ln \left (x \right ) x^{3} a \,b^{3}-36 \ln \left (b x +a \right ) x^{3} a \,b^{3}-22 b^{4} x^{4}+36 \ln \left (x \right ) x^{2} a^{2} b^{2}-36 \ln \left (b x +a \right ) x^{2} a^{2} b^{2}-54 a \,b^{3} x^{3}+12 \ln \left (x \right ) x \,a^{3} b -12 \ln \left (b x +a \right ) x \,a^{3} b -36 b^{2} a^{2} x^{2}+3 a^{4}}{3 a^{5} x \left (b x +a \right )^{3}}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 91, normalized size = 1.47 \begin {gather*} -\frac {12 \, b^{3} x^{3} + 30 \, a b^{2} x^{2} + 22 \, a^{2} b x + 3 \, a^{3}}{3 \, {\left (a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{3} + 3 \, a^{6} b x^{2} + a^{7} x\right )}} + \frac {4 \, b \log \left (b x + a\right )}{a^{5}} - \frac {4 \, b \log \left (x\right )}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*b^3*x^3 + 30*a*b^2*x^2 + 22*a^2*b*x + 3*a^3)/(a^4*b^3*x^4 + 3*a^5*b^2*x^3 + 3*a^6*b*x^2 + a^7*x) + 4*
b*log(b*x + a)/a^5 - 4*b*log(x)/a^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (61) = 122\).
time = 0.60, size = 153, normalized size = 2.47 \begin {gather*} -\frac {12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4} - 12 \, {\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (x\right )}{3 \, {\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*a*b^3*x^3 + 30*a^2*b^2*x^2 + 22*a^3*b*x + 3*a^4 - 12*(b^4*x^4 + 3*a*b^3*x^3 + 3*a^2*b^2*x^2 + a^3*b*x
)*log(b*x + a) + 12*(b^4*x^4 + 3*a*b^3*x^3 + 3*a^2*b^2*x^2 + a^3*b*x)*log(x))/(a^5*b^3*x^4 + 3*a^6*b^2*x^3 + 3
*a^7*b*x^2 + a^8*x)

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Sympy [A]
time = 0.20, size = 90, normalized size = 1.45 \begin {gather*} \frac {- 3 a^{3} - 22 a^{2} b x - 30 a b^{2} x^{2} - 12 b^{3} x^{3}}{3 a^{7} x + 9 a^{6} b x^{2} + 9 a^{5} b^{2} x^{3} + 3 a^{4} b^{3} x^{4}} + \frac {4 b \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**2/(b*x+a)**4,x)

[Out]

(-3*a**3 - 22*a**2*b*x - 30*a*b**2*x**2 - 12*b**3*x**3)/(3*a**7*x + 9*a**6*b*x**2 + 9*a**5*b**2*x**3 + 3*a**4*
b**3*x**4) + 4*b*(-log(x) + log(a/b + x))/a**5

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Giac [A]
time = 0.53, size = 71, normalized size = 1.15 \begin {gather*} \frac {4 \, b \log \left ({\left | b x + a \right |}\right )}{a^{5}} - \frac {4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4}}{3 \, {\left (b x + a\right )}^{3} a^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*log(abs(b*x + a))/a^5 - 4*b*log(abs(x))/a^5 - 1/3*(12*a*b^3*x^3 + 30*a^2*b^2*x^2 + 22*a^3*b*x + 3*a^4)/((b
*x + a)^3*a^5*x)

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Mupad [B]
time = 0.07, size = 85, normalized size = 1.37 \begin {gather*} \frac {8\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5}-\frac {\frac {1}{a}+\frac {10\,b^2\,x^2}{a^3}+\frac {4\,b^3\,x^3}{a^4}+\frac {22\,b\,x}{3\,a^2}}{a^3\,x+3\,a^2\,b\,x^2+3\,a\,b^2\,x^3+b^3\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x)^4),x)

[Out]

(8*b*atanh((2*b*x)/a + 1))/a^5 - (1/a + (10*b^2*x^2)/a^3 + (4*b^3*x^3)/a^4 + (22*b*x)/(3*a^2))/(a^3*x + b^3*x^
4 + 3*a^2*b*x^2 + 3*a*b^2*x^3)

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