\(\int \frac {1}{x (a+b x)^7} \, dx\) [218]

Optimal result

Integrand size = 11, antiderivative size = 99 \[ \int \frac {1}{x (a+b x)^7} \, dx=\frac {1}{6 a (a+b x)^6}+\frac {1}{5 a^2 (a+b x)^5}+\frac {1}{4 a^3 (a+b x)^4}+\frac {1}{3 a^4 (a+b x)^3}+\frac {1}{2 a^5 (a+b x)^2}+\frac {1}{a^6 (a+b x)}+\frac {\log (x)}{a^7}-\frac {\log (a+b x)}{a^7} \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x (a+b x)^7} \, dx=-\frac {\log (a+b x)}{a^7}+\frac {\log (x)}{a^7}+\frac {1}{a^6 (a+b x)}+\frac {1}{2 a^5 (a+b x)^2}+\frac {1}{3 a^4 (a+b x)^3}+\frac {1}{4 a^3 (a+b x)^4}+\frac {1}{5 a^2 (a+b x)^5}+\frac {1}{6 a (a+b x)^6} \]

[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 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{align*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x (a+b x)^7} \, dx=\frac {\frac {a \left (147 a^5+522 a^4 b x+855 a^3 b^2 x^2+740 a^2 b^3 x^3+330 a b^4 x^4+60 b^5 x^5\right )}{(a+b x)^6}+60 \log (x)-60 \log (a+b x)}{60 a^7} \]

[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)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {\frac {b^{5} x^{5}}{a^{6}}+\frac {11 b^{4} x^{4}}{2 a^{5}}+\frac {37 b^{3} x^{3}}{3 a^{4}}+\frac {57 b^{2} x^{2}}{4 a^{3}}+\frac {87 b x}{10 a^{2}}+\frac {49}{20 a}}{\left (b x +a \right )^{6}}+\frac {\ln \left (-x \right )}{a^{7}}-\frac {\ln \left (b x +a \right )}{a^{7}}\)	\(85\)
default	\(\frac {1}{6 a \left (b x +a \right )^{6}}+\frac {1}{5 a^{2} \left (b x +a \right )^{5}}+\frac {1}{4 a^{3} \left (b x +a \right )^{4}}+\frac {1}{3 a^{4} \left (b x +a \right )^{3}}+\frac {1}{2 a^{5} \left (b x +a \right )^{2}}+\frac {1}{a^{6} \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{7}}-\frac {\ln \left (b x +a \right )}{a^{7}}\)	\(90\)
norman	\(\frac {-\frac {6 b x}{a^{2}}-\frac {45 b^{2} x^{2}}{2 a^{3}}-\frac {110 b^{3} x^{3}}{3 a^{4}}-\frac {125 b^{4} x^{4}}{4 a^{5}}-\frac {137 b^{5} x^{5}}{10 a^{6}}-\frac {49 b^{6} x^{6}}{20 a^{7}}}{\left (b x +a \right )^{6}}+\frac {\ln \left (x \right )}{a^{7}}-\frac {\ln \left (b x +a \right )}{a^{7}}\)	\(90\)
parallelrisch	\(\frac {60 \ln \left (x \right ) a^{6}-60 \ln \left (b x +a \right ) a^{6}+900 \ln \left (x \right ) x^{2} a^{4} b^{2}+360 \ln \left (x \right ) x \,a^{5} b +1200 \ln \left (x \right ) x^{3} a^{3} b^{3}+360 \ln \left (x \right ) x^{5} a \,b^{5}-360 \ln \left (b x +a \right ) x^{5} a \,b^{5}+900 \ln \left (x \right ) x^{4} a^{2} b^{4}-900 \ln \left (b x +a \right ) x^{4} a^{2} b^{4}-147 b^{6} x^{6}-1200 \ln \left (b x +a \right ) x^{3} a^{3} b^{3}-900 \ln \left (b x +a \right ) x^{2} a^{4} b^{2}-360 \ln \left (b x +a \right ) x \,a^{5} b -360 a^{5} x b -60 \ln \left (b x +a \right ) x^{6} b^{6}+60 \ln \left (x \right ) x^{6} b^{6}-1875 a^{2} x^{4} b^{4}-2200 a^{3} x^{3} b^{3}-1350 a^{4} x^{2} b^{2}-822 a \,x^{5} b^{5}}{60 a^{7} \left (b x +a \right )^{6}}\)	\(251\)

[In]

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

[Out]

(b^5/a^6*x^5+11/2*b^4/a^5*x^4+37/3*b^3/a^4*x^3+57/4*b^2/a^3*x^2+87/10*b/a^2*x+49/20/a)/(b*x+a)^6+1/a^7*ln(-x)-
ln(b*x+a)/a^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (89) = 178\).

Time = 0.23 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x (a+b x)^7} \, dx=\frac {60 \, a b^{5} x^{5} + 330 \, a^{2} b^{4} x^{4} + 740 \, a^{3} b^{3} x^{3} + 855 \, a^{4} b^{2} x^{2} + 522 \, a^{5} b x + 147 \, a^{6} - 60 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (x\right )}{60 \, {\left (a^{7} b^{6} x^{6} + 6 \, a^{8} b^{5} x^{5} + 15 \, a^{9} b^{4} x^{4} + 20 \, a^{10} b^{3} x^{3} + 15 \, a^{11} b^{2} x^{2} + 6 \, a^{12} b x + a^{13}\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x (a+b x)^7} \, dx=\frac {147 a^{5} + 522 a^{4} b x + 855 a^{3} b^{2} x^{2} + 740 a^{2} b^{3} x^{3} + 330 a b^{4} x^{4} + 60 b^{5} x^{5}}{60 a^{12} + 360 a^{11} b x + 900 a^{10} b^{2} x^{2} + 1200 a^{9} b^{3} x^{3} + 900 a^{8} b^{4} x^{4} + 360 a^{7} b^{5} x^{5} + 60 a^{6} b^{6} x^{6}} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a^{7}} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x (a+b x)^7} \, dx=\frac {60 \, b^{5} x^{5} + 330 \, a b^{4} x^{4} + 740 \, a^{2} b^{3} x^{3} + 855 \, a^{3} b^{2} x^{2} + 522 \, a^{4} b x + 147 \, a^{5}}{60 \, {\left (a^{6} b^{6} x^{6} + 6 \, a^{7} b^{5} x^{5} + 15 \, a^{8} b^{4} x^{4} + 20 \, a^{9} b^{3} x^{3} + 15 \, a^{10} b^{2} x^{2} + 6 \, a^{11} b x + a^{12}\right )}} - \frac {\log \left (b x + a\right )}{a^{7}} + \frac {\log \left (x\right )}{a^{7}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x (a+b x)^7} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{7}} + \frac {\log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{5} + 330 \, a^{2} b^{4} x^{4} + 740 \, a^{3} b^{3} x^{3} + 855 \, a^{4} b^{2} x^{2} + 522 \, a^{5} b x + 147 \, a^{6}}{60 \, {\left (b x + a\right )}^{6} a^{7}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x (a+b x)^7} \, dx=-\frac {\ln \left (\frac {a+b\,x}{x}\right )-\frac {15\,b^2\,x^2}{2\,{\left (a+b\,x\right )}^2}+\frac {20\,b^3\,x^3}{3\,{\left (a+b\,x\right )}^3}-\frac {15\,b^4\,x^4}{4\,{\left (a+b\,x\right )}^4}+\frac {6\,b^5\,x^5}{5\,{\left (a+b\,x\right )}^5}-\frac {b^6\,x^6}{6\,{\left (a+b\,x\right )}^6}+\frac {6\,b\,x}{a+b\,x}}{a^7} \]

[In]

int(1/(x*(a + b*x)^7),x)

[Out]

-(log((a + b*x)/x) - (15*b^2*x^2)/(2*(a + b*x)^2) + (20*b^3*x^3)/(3*(a + b*x)^3) - (15*b^4*x^4)/(4*(a + b*x)^4
) + (6*b^5*x^5)/(5*(a + b*x)^5) - (b^6*x^6)/(6*(a + b*x)^6) + (6*b*x)/(a + b*x))/a^7