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

Optimal. Leaf size=93 \[ \frac{6 b^2}{a^5 (a+b x)}+\frac{3 b^2}{2 a^4 (a+b x)^2}+\frac{b^2}{3 a^3 (a+b x)^3}+\frac{10 b^2 \log (x)}{a^6}-\frac{10 b^2 \log (a+b x)}{a^6}+\frac{4 b}{a^5 x}-\frac{1}{2 a^4 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0481338, antiderivative size = 93, 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} \[ \frac{6 b^2}{a^5 (a+b x)}+\frac{3 b^2}{2 a^4 (a+b x)^2}+\frac{b^2}{3 a^3 (a+b x)^3}+\frac{10 b^2 \log (x)}{a^6}-\frac{10 b^2 \log (a+b x)}{a^6}+\frac{4 b}{a^5 x}-\frac{1}{2 a^4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)^4),x]

[Out]

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

Mathematica [A]  time = 0.10632, size = 79, normalized size = 0.85 \[ \frac{\frac{a \left (110 a^2 b^2 x^2+15 a^3 b x-3 a^4+150 a b^3 x^3+60 b^4 x^4\right )}{x^2 (a+b x)^3}-60 b^2 \log (a+b x)+60 b^2 \log (x)}{6 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x)^4),x]

[Out]

((a*(-3*a^4 + 15*a^3*b*x + 110*a^2*b^2*x^2 + 150*a*b^3*x^3 + 60*b^4*x^4))/(x^2*(a + b*x)^3) + 60*b^2*Log[x] -
60*b^2*Log[a + b*x])/(6*a^6)

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Maple [A]  time = 0.01, size = 88, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{4}{x}^{2}}}+4\,{\frac{b}{{a}^{5}x}}+{\frac{{b}^{2}}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{3\,{b}^{2}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{b}^{2}}{{a}^{5} \left ( bx+a \right ) }}+10\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{6}}}-10\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04163, size = 146, normalized size = 1.57 \begin{align*} \frac{60 \, b^{4} x^{4} + 150 \, a b^{3} x^{3} + 110 \, a^{2} b^{2} x^{2} + 15 \, a^{3} b x - 3 \, a^{4}}{6 \,{\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}} - \frac{10 \, b^{2} \log \left (b x + a\right )}{a^{6}} + \frac{10 \, b^{2} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(60*b^4*x^4 + 150*a*b^3*x^3 + 110*a^2*b^2*x^2 + 15*a^3*b*x - 3*a^4)/(a^5*b^3*x^5 + 3*a^6*b^2*x^4 + 3*a^7*b
*x^3 + a^8*x^2) - 10*b^2*log(b*x + a)/a^6 + 10*b^2*log(x)/a^6

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Fricas [B]  time = 1.61244, size = 363, normalized size = 3.9 \begin{align*} \frac{60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5} - 60 \,{\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{3} x^{5} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{3} + a^{9} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(60*a*b^4*x^4 + 150*a^2*b^3*x^3 + 110*a^3*b^2*x^2 + 15*a^4*b*x - 3*a^5 - 60*(b^5*x^5 + 3*a*b^4*x^4 + 3*a^2
*b^3*x^3 + a^3*b^2*x^2)*log(b*x + a) + 60*(b^5*x^5 + 3*a*b^4*x^4 + 3*a^2*b^3*x^3 + a^3*b^2*x^2)*log(x))/(a^6*b
^3*x^5 + 3*a^7*b^2*x^4 + 3*a^8*b*x^3 + a^9*x^2)

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Sympy [A]  time = 0.8636, size = 104, normalized size = 1.12 \begin{align*} \frac{- 3 a^{4} + 15 a^{3} b x + 110 a^{2} b^{2} x^{2} + 150 a b^{3} x^{3} + 60 b^{4} x^{4}}{6 a^{8} x^{2} + 18 a^{7} b x^{3} + 18 a^{6} b^{2} x^{4} + 6 a^{5} b^{3} x^{5}} + \frac{10 b^{2} \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**3/(b*x+a)**4,x)

[Out]

(-3*a**4 + 15*a**3*b*x + 110*a**2*b**2*x**2 + 150*a*b**3*x**3 + 60*b**4*x**4)/(6*a**8*x**2 + 18*a**7*b*x**3 +
18*a**6*b**2*x**4 + 6*a**5*b**3*x**5) + 10*b**2*(log(x) - log(a/b + x))/a**6

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Giac [A]  time = 1.19504, size = 116, normalized size = 1.25 \begin{align*} -\frac{10 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac{10 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5}}{6 \,{\left (b x + a\right )}^{3} a^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*b^2*log(abs(b*x + a))/a^6 + 10*b^2*log(abs(x))/a^6 + 1/6*(60*a*b^4*x^4 + 150*a^2*b^3*x^3 + 110*a^3*b^2*x^2
+ 15*a^4*b*x - 3*a^5)/((b*x + a)^3*a^6*x^2)