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

Optimal. Leaf size=99 \[ \frac{1}{a^6 (a+b x)}+\frac{1}{2 a^5 (a+b x)^2}+\frac{1}{3 a^4 (a+b x)^3}+\frac{1}{4 a^3 (a+b x)^4}+\frac{1}{5 a^2 (a+b x)^5}-\frac{\log (a+b x)}{a^7}+\frac{\log (x)}{a^7}+\frac{1}{6 a (a+b x)^6} \]

[Out]

1/(6*a*(a + b*x)^6) + 1/(5*a^2*(a + b*x)^5) + 1/(4*a^3*(a + b*x)^4) + 1/(3*a^4*(a + b*x)^3) + 1/(2*a^5*(a + b*
x)^2) + 1/(a^6*(a + b*x)) + Log[x]/a^7 - Log[a + b*x]/a^7

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Rubi [A]  time = 0.0488924, antiderivative size = 99, 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} \[ \frac{1}{a^6 (a+b x)}+\frac{1}{2 a^5 (a+b x)^2}+\frac{1}{3 a^4 (a+b x)^3}+\frac{1}{4 a^3 (a+b x)^4}+\frac{1}{5 a^2 (a+b x)^5}-\frac{\log (a+b x)}{a^7}+\frac{\log (x)}{a^7}+\frac{1}{6 a (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^7),x]

[Out]

1/(6*a*(a + b*x)^6) + 1/(5*a^2*(a + b*x)^5) + 1/(4*a^3*(a + b*x)^4) + 1/(3*a^4*(a + b*x)^3) + 1/(2*a^5*(a + b*
x)^2) + 1/(a^6*(a + b*x)) + Log[x]/a^7 - Log[a + b*x]/a^7

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

Mathematica [A]  time = 0.104877, size = 81, normalized size = 0.82 \[ \frac{\frac{a \left (855 a^3 b^2 x^2+740 a^2 b^3 x^3+522 a^4 b x+147 a^5+330 a b^4 x^4+60 b^5 x^5\right )}{(a+b x)^6}-60 \log (a+b x)+60 \log (x)}{60 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)^7),x]

[Out]

((a*(147*a^5 + 522*a^4*b*x + 855*a^3*b^2*x^2 + 740*a^2*b^3*x^3 + 330*a*b^4*x^4 + 60*b^5*x^5))/(a + b*x)^6 + 60
*Log[x] - 60*Log[a + b*x])/(60*a^7)

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Maple [A]  time = 0.01, size = 90, normalized size = 0.9 \begin{align*}{\frac{1}{6\,a \left ( bx+a \right ) ^{6}}}+{\frac{1}{5\,{a}^{2} \left ( bx+a \right ) ^{5}}}+{\frac{1}{4\,{a}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{1}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{1}{2\,{a}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{1}{{a}^{6} \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{7}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/a/(b*x+a)^6+1/5/a^2/(b*x+a)^5+1/4/a^3/(b*x+a)^4+1/3/a^4/(b*x+a)^3+1/2/a^5/(b*x+a)^2+1/a^6/(b*x+a)+ln(x)/a^
7-ln(b*x+a)/a^7

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Maxima [A]  time = 1.12927, size = 188, normalized size = 1.9 \begin{align*} \frac{60 \, b^{5} x^{5} + 330 \, a b^{4} x^{4} + 740 \, a^{2} b^{3} x^{3} + 855 \, a^{3} b^{2} x^{2} + 522 \, a^{4} b x + 147 \, a^{5}}{60 \,{\left (a^{6} b^{6} x^{6} + 6 \, a^{7} b^{5} x^{5} + 15 \, a^{8} b^{4} x^{4} + 20 \, a^{9} b^{3} x^{3} + 15 \, a^{10} b^{2} x^{2} + 6 \, a^{11} b x + a^{12}\right )}} - \frac{\log \left (b x + a\right )}{a^{7}} + \frac{\log \left (x\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b^5*x^5 + 330*a*b^4*x^4 + 740*a^2*b^3*x^3 + 855*a^3*b^2*x^2 + 522*a^4*b*x + 147*a^5)/(a^6*b^6*x^6 + 6
*a^7*b^5*x^5 + 15*a^8*b^4*x^4 + 20*a^9*b^3*x^3 + 15*a^10*b^2*x^2 + 6*a^11*b*x + a^12) - log(b*x + a)/a^7 + log
(x)/a^7

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Fricas [B]  time = 1.63107, size = 564, normalized size = 5.7 \begin{align*} \frac{60 \, a b^{5} x^{5} + 330 \, a^{2} b^{4} x^{4} + 740 \, a^{3} b^{3} x^{3} + 855 \, a^{4} b^{2} x^{2} + 522 \, a^{5} b x + 147 \, a^{6} - 60 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (x\right )}{60 \,{\left (a^{7} b^{6} x^{6} + 6 \, a^{8} b^{5} x^{5} + 15 \, a^{9} b^{4} x^{4} + 20 \, a^{10} b^{3} x^{3} + 15 \, a^{11} b^{2} x^{2} + 6 \, a^{12} b x + a^{13}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*a*b^5*x^5 + 330*a^2*b^4*x^4 + 740*a^3*b^3*x^3 + 855*a^4*b^2*x^2 + 522*a^5*b*x + 147*a^6 - 60*(b^6*x^6
+ 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*log(b*x + a) + 60*(b^6*x^
6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*log(x))/(a^7*b^6*x^6 + 6
*a^8*b^5*x^5 + 15*a^9*b^4*x^4 + 20*a^10*b^3*x^3 + 15*a^11*b^2*x^2 + 6*a^12*b*x + a^13)

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Sympy [A]  time = 1.279, size = 141, normalized size = 1.42 \begin{align*} \frac{147 a^{5} + 522 a^{4} b x + 855 a^{3} b^{2} x^{2} + 740 a^{2} b^{3} x^{3} + 330 a b^{4} x^{4} + 60 b^{5} x^{5}}{60 a^{12} + 360 a^{11} b x + 900 a^{10} b^{2} x^{2} + 1200 a^{9} b^{3} x^{3} + 900 a^{8} b^{4} x^{4} + 360 a^{7} b^{5} x^{5} + 60 a^{6} b^{6} x^{6}} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(147*a**5 + 522*a**4*b*x + 855*a**3*b**2*x**2 + 740*a**2*b**3*x**3 + 330*a*b**4*x**4 + 60*b**5*x**5)/(60*a**12
+ 360*a**11*b*x + 900*a**10*b**2*x**2 + 1200*a**9*b**3*x**3 + 900*a**8*b**4*x**4 + 360*a**7*b**5*x**5 + 60*a*
*6*b**6*x**6) + (log(x) - log(a/b + x))/a**7

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Giac [A]  time = 1.22369, size = 117, normalized size = 1.18 \begin{align*} -\frac{\log \left ({\left | b x + a \right |}\right )}{a^{7}} + \frac{\log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{5} + 330 \, a^{2} b^{4} x^{4} + 740 \, a^{3} b^{3} x^{3} + 855 \, a^{4} b^{2} x^{2} + 522 \, a^{5} b x + 147 \, a^{6}}{60 \,{\left (b x + a\right )}^{6} a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(b*x + a))/a^7 + log(abs(x))/a^7 + 1/60*(60*a*b^5*x^5 + 330*a^2*b^4*x^4 + 740*a^3*b^3*x^3 + 855*a^4*b^
2*x^2 + 522*a^5*b*x + 147*a^6)/((b*x + a)^6*a^7)