3.1644 \(\int \frac{1}{(a+\frac{b}{x})^3 x^7} \, dx\)

Optimal. Leaf size=89 \[ -\frac{4 a^3}{b^5 (a x+b)}-\frac{a^3}{2 b^4 (a x+b)^2}-\frac{6 a^2}{b^5 x}-\frac{10 a^3 \log (x)}{b^6}+\frac{10 a^3 \log (a x+b)}{b^6}+\frac{3 a}{2 b^4 x^2}-\frac{1}{3 b^3 x^3} \]

[Out]

-1/(3*b^3*x^3) + (3*a)/(2*b^4*x^2) - (6*a^2)/(b^5*x) - a^3/(2*b^4*(b + a*x)^2) - (4*a^3)/(b^5*(b + a*x)) - (10
*a^3*Log[x])/b^6 + (10*a^3*Log[b + a*x])/b^6

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Rubi [A]  time = 0.0486551, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 44} \[ -\frac{4 a^3}{b^5 (a x+b)}-\frac{a^3}{2 b^4 (a x+b)^2}-\frac{6 a^2}{b^5 x}-\frac{10 a^3 \log (x)}{b^6}+\frac{10 a^3 \log (a x+b)}{b^6}+\frac{3 a}{2 b^4 x^2}-\frac{1}{3 b^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^7),x]

[Out]

-1/(3*b^3*x^3) + (3*a)/(2*b^4*x^2) - (6*a^2)/(b^5*x) - a^3/(2*b^4*(b + a*x)^2) - (4*a^3)/(b^5*(b + a*x)) - (10
*a^3*Log[x])/b^6 + (10*a^3*Log[b + a*x])/b^6

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^7} \, dx &=\int \frac{1}{x^4 (b+a x)^3} \, dx\\ &=\int \left (\frac{1}{b^3 x^4}-\frac{3 a}{b^4 x^3}+\frac{6 a^2}{b^5 x^2}-\frac{10 a^3}{b^6 x}+\frac{a^4}{b^4 (b+a x)^3}+\frac{4 a^4}{b^5 (b+a x)^2}+\frac{10 a^4}{b^6 (b+a x)}\right ) \, dx\\ &=-\frac{1}{3 b^3 x^3}+\frac{3 a}{2 b^4 x^2}-\frac{6 a^2}{b^5 x}-\frac{a^3}{2 b^4 (b+a x)^2}-\frac{4 a^3}{b^5 (b+a x)}-\frac{10 a^3 \log (x)}{b^6}+\frac{10 a^3 \log (b+a x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0656797, size = 79, normalized size = 0.89 \[ -\frac{\frac{b \left (20 a^2 b^2 x^2+90 a^3 b x^3+60 a^4 x^4-5 a b^3 x+2 b^4\right )}{x^3 (a x+b)^2}-60 a^3 \log (a x+b)+60 a^3 \log (x)}{6 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^7),x]

[Out]

-((b*(2*b^4 - 5*a*b^3*x + 20*a^2*b^2*x^2 + 90*a^3*b*x^3 + 60*a^4*x^4))/(x^3*(b + a*x)^2) + 60*a^3*Log[x] - 60*
a^3*Log[b + a*x])/(6*b^6)

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Maple [A]  time = 0.011, size = 84, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{b}^{3}{x}^{3}}}+{\frac{3\,a}{2\,{b}^{4}{x}^{2}}}-6\,{\frac{{a}^{2}}{{b}^{5}x}}-{\frac{{a}^{3}}{2\,{b}^{4} \left ( ax+b \right ) ^{2}}}-4\,{\frac{{a}^{3}}{{b}^{5} \left ( ax+b \right ) }}-10\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{6}}}+10\,{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^3/x^7,x)

[Out]

-1/3/b^3/x^3+3/2*a/b^4/x^2-6*a^2/b^5/x-1/2*a^3/b^4/(a*x+b)^2-4*a^3/b^5/(a*x+b)-10*a^3*ln(x)/b^6+10*a^3*ln(a*x+
b)/b^6

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Maxima [A]  time = 1.03254, size = 131, normalized size = 1.47 \begin{align*} -\frac{60 \, a^{4} x^{4} + 90 \, a^{3} b x^{3} + 20 \, a^{2} b^{2} x^{2} - 5 \, a b^{3} x + 2 \, b^{4}}{6 \,{\left (a^{2} b^{5} x^{5} + 2 \, a b^{6} x^{4} + b^{7} x^{3}\right )}} + \frac{10 \, a^{3} \log \left (a x + b\right )}{b^{6}} - \frac{10 \, a^{3} \log \left (x\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^7,x, algorithm="maxima")

[Out]

-1/6*(60*a^4*x^4 + 90*a^3*b*x^3 + 20*a^2*b^2*x^2 - 5*a*b^3*x + 2*b^4)/(a^2*b^5*x^5 + 2*a*b^6*x^4 + b^7*x^3) +
10*a^3*log(a*x + b)/b^6 - 10*a^3*log(x)/b^6

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Fricas [A]  time = 1.69291, size = 296, normalized size = 3.33 \begin{align*} -\frac{60 \, a^{4} b x^{4} + 90 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 5 \, a b^{4} x + 2 \, b^{5} - 60 \,{\left (a^{5} x^{5} + 2 \, a^{4} b x^{4} + a^{3} b^{2} x^{3}\right )} \log \left (a x + b\right ) + 60 \,{\left (a^{5} x^{5} + 2 \, a^{4} b x^{4} + a^{3} b^{2} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + b^{8} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^7,x, algorithm="fricas")

[Out]

-1/6*(60*a^4*b*x^4 + 90*a^3*b^2*x^3 + 20*a^2*b^3*x^2 - 5*a*b^4*x + 2*b^5 - 60*(a^5*x^5 + 2*a^4*b*x^4 + a^3*b^2
*x^3)*log(a*x + b) + 60*(a^5*x^5 + 2*a^4*b*x^4 + a^3*b^2*x^3)*log(x))/(a^2*b^6*x^5 + 2*a*b^7*x^4 + b^8*x^3)

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Sympy [A]  time = 0.566538, size = 92, normalized size = 1.03 \begin{align*} \frac{10 a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{6}} - \frac{60 a^{4} x^{4} + 90 a^{3} b x^{3} + 20 a^{2} b^{2} x^{2} - 5 a b^{3} x + 2 b^{4}}{6 a^{2} b^{5} x^{5} + 12 a b^{6} x^{4} + 6 b^{7} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**3/x**7,x)

[Out]

10*a**3*(-log(x) + log(x + b/a))/b**6 - (60*a**4*x**4 + 90*a**3*b*x**3 + 20*a**2*b**2*x**2 - 5*a*b**3*x + 2*b*
*4)/(6*a**2*b**5*x**5 + 12*a*b**6*x**4 + 6*b**7*x**3)

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Giac [A]  time = 1.10008, size = 116, normalized size = 1.3 \begin{align*} \frac{10 \, a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{6}} - \frac{10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{4} + 90 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 5 \, a b^{4} x + 2 \, b^{5}}{6 \,{\left (a x + b\right )}^{2} b^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^7,x, algorithm="giac")

[Out]

10*a^3*log(abs(a*x + b))/b^6 - 10*a^3*log(abs(x))/b^6 - 1/6*(60*a^4*b*x^4 + 90*a^3*b^2*x^3 + 20*a^2*b^3*x^2 -
5*a*b^4*x + 2*b^5)/((a*x + b)^2*b^6*x^3)