3.191 \(\int \frac{1}{x^4 (a+b x)^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0477162, antiderivative size = 89, normalized size of antiderivative = 1., 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 = {44} \[ -\frac{4 b^3}{a^5 (a+b x)}-\frac{b^3}{2 a^4 (a+b x)^2}-\frac{6 b^2}{a^5 x}-\frac{10 b^3 \log (x)}{a^6}+\frac{10 b^3 \log (a+b x)}{a^6}+\frac{3 b}{2 a^4 x^2}-\frac{1}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.132316, size = 79, normalized size = 0.89 \[ -\frac{\frac{a \left (20 a^2 b^2 x^2-5 a^3 b x+2 a^4+90 a b^3 x^3+60 b^4 x^4\right )}{x^3 (a+b x)^2}-60 b^3 \log (a+b x)+60 b^3 \log (x)}{6 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^3),x]

[Out]

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

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Maple [A]  time = 0.013, size = 84, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{3\,b}{2\,{a}^{4}{x}^{2}}}-6\,{\frac{{b}^{2}}{{a}^{5}x}}-{\frac{{b}^{3}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}-4\,{\frac{{b}^{3}}{{a}^{5} \left ( bx+a \right ) }}-10\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{6}}}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.0925, size = 131, normalized size = 1.47 \begin{align*} -\frac{60 \, b^{4} x^{4} + 90 \, a b^{3} x^{3} + 20 \, a^{2} b^{2} x^{2} - 5 \, a^{3} b x + 2 \, a^{4}}{6 \,{\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} + \frac{10 \, b^{3} \log \left (b x + a\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.54313, size = 296, normalized size = 3.33 \begin{align*} -\frac{60 \, a b^{4} x^{4} + 90 \, a^{2} b^{3} x^{3} + 20 \, a^{3} b^{2} x^{2} - 5 \, a^{4} b x + 2 \, a^{5} - 60 \,{\left (b^{5} x^{5} + 2 \, a b^{4} x^{4} + a^{2} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + 2 \, a b^{4} x^{4} + a^{2} b^{3} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.814525, size = 92, normalized size = 1.03 \begin{align*} - \frac{2 a^{4} - 5 a^{3} b x + 20 a^{2} b^{2} x^{2} + 90 a b^{3} x^{3} + 60 b^{4} x^{4}}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac{10 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**3,x)

[Out]

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

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Giac [A]  time = 1.14114, size = 116, normalized size = 1.3 \begin{align*} \frac{10 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{6}} - \frac{10 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac{60 \, a b^{4} x^{4} + 90 \, a^{2} b^{3} x^{3} + 20 \, a^{3} b^{2} x^{2} - 5 \, a^{4} b x + 2 \, a^{5}}{6 \,{\left (b x + a\right )}^{2} a^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^3,x, algorithm="giac")

[Out]

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