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

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

[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 = 0.10823, 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 \[ -\frac{\log (a+b x)}{a^7}+\frac{\log (x)}{a^7}+\frac{1}{a^6 (a+b x)}+\frac{1}{2 a^5 (a+b x)^2}+\frac{1}{3 a^4 (a+b x)^3}+\frac{1}{4 a^3 (a+b x)^4}+\frac{1}{5 a^2 (a+b x)^5}+\frac{1}{6 a (a+b x)^6} \]

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

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Rubi in Sympy [A]  time = 20.1217, size = 92, normalized size = 0.93 \[ \frac{1}{6 a \left (a + b x\right )^{6}} + \frac{1}{5 a^{2} \left (a + b x\right )^{5}} + \frac{1}{4 a^{3} \left (a + b x\right )^{4}} + \frac{1}{3 a^{4} \left (a + b x\right )^{3}} + \frac{1}{2 a^{5} \left (a + b x\right )^{2}} + \frac{1}{a^{6} \left (a + b x\right )} + \frac{\log{\left (x \right )}}{a^{7}} - \frac{\log{\left (a + b x \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)**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

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Mathematica [A]  time = 0.0929195, size = 81, normalized size = 0.82 \[ \frac{\frac{a \left (147 a^5+522 a^4 b x+855 a^3 b^2 x^2+740 a^2 b^3 x^3+330 a b^4 x^4+60 b^5 x^5\right )}{(a+b x)^6}-60 \log (a+b x)+60 \log (x)}{60 a^7} \]

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 = 0.016, size = 90, normalized size = 0.9 \[{\frac{1}{6\,a \left ( bx+a \right ) ^{6}}}+{\frac{1}{5\,{a}^{2} \left ( bx+a \right ) ^{5}}}+{\frac{1}{4\,{a}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{1}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{1}{2\,{a}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{1}{{a}^{6} \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{7}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{7}}} \]

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 = 1.34929, size = 188, normalized size = 1.9 \[ \frac{60 \, b^{5} x^{5} + 330 \, a b^{4} x^{4} + 740 \, a^{2} b^{3} x^{3} + 855 \, a^{3} b^{2} x^{2} + 522 \, a^{4} b x + 147 \, a^{5}}{60 \,{\left (a^{6} b^{6} x^{6} + 6 \, a^{7} b^{5} x^{5} + 15 \, a^{8} b^{4} x^{4} + 20 \, a^{9} b^{3} x^{3} + 15 \, a^{10} b^{2} x^{2} + 6 \, a^{11} b x + a^{12}\right )}} - \frac{\log \left (b x + a\right )}{a^{7}} + \frac{\log \left (x\right )}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^7*x),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 + 1
5*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 [A]  time = 0.222602, size = 346, normalized size = 3.49 \[ \frac{60 \, a b^{5} x^{5} + 330 \, a^{2} b^{4} x^{4} + 740 \, a^{3} b^{3} x^{3} + 855 \, a^{4} b^{2} x^{2} + 522 \, a^{5} b x + 147 \, a^{6} - 60 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (x\right )}{60 \,{\left (a^{7} b^{6} x^{6} + 6 \, a^{8} b^{5} x^{5} + 15 \, a^{9} b^{4} x^{4} + 20 \, a^{10} b^{3} x^{3} + 15 \, a^{11} b^{2} x^{2} + 6 \, a^{12} b x + a^{13}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^7*x),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 = 3.18036, size = 141, normalized size = 1.42 \[ \frac{147 a^{5} + 522 a^{4} b x + 855 a^{3} b^{2} x^{2} + 740 a^{2} b^{3} x^{3} + 330 a b^{4} x^{4} + 60 b^{5} x^{5}}{60 a^{12} + 360 a^{11} b x + 900 a^{10} b^{2} x^{2} + 1200 a^{9} b^{3} x^{3} + 900 a^{8} b^{4} x^{4} + 360 a^{7} b^{5} x^{5} + 60 a^{6} b^{6} x^{6}} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a^{7}} \]

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/XCAS [A]  time = 0.228818, size = 117, normalized size = 1.18 \[ -\frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{7}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{5} + 330 \, a^{2} b^{4} x^{4} + 740 \, a^{3} b^{3} x^{3} + 855 \, a^{4} b^{2} x^{2} + 522 \, a^{5} b x + 147 \, a^{6}}{60 \,{\left (b x + a\right )}^{6} a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(b*x + a))/a^7 + ln(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)