3.220 \(\int \frac{1}{x^3 (a+b x)^7} \, dx\)

Optimal. Leaf size=144 \[ \frac{28 b^2 \log (x)}{a^9}-\frac{28 b^2 \log (a+b x)}{a^9}+\frac{21 b^2}{a^8 (a+b x)}+\frac{7 b}{a^8 x}+\frac{15 b^2}{2 a^7 (a+b x)^2}-\frac{1}{2 a^7 x^2}+\frac{10 b^2}{3 a^6 (a+b x)^3}+\frac{3 b^2}{2 a^5 (a+b x)^4}+\frac{3 b^2}{5 a^4 (a+b x)^5}+\frac{b^2}{6 a^3 (a+b x)^6} \]

[Out]

-1/(2*a^7*x^2) + (7*b)/(a^8*x) + b^2/(6*a^3*(a + b*x)^6) + (3*b^2)/(5*a^4*(a + b
*x)^5) + (3*b^2)/(2*a^5*(a + b*x)^4) + (10*b^2)/(3*a^6*(a + b*x)^3) + (15*b^2)/(
2*a^7*(a + b*x)^2) + (21*b^2)/(a^8*(a + b*x)) + (28*b^2*Log[x])/a^9 - (28*b^2*Lo
g[a + b*x])/a^9

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Rubi [A]  time = 0.208567, antiderivative size = 144, 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 \[ \frac{28 b^2 \log (x)}{a^9}-\frac{28 b^2 \log (a+b x)}{a^9}+\frac{21 b^2}{a^8 (a+b x)}+\frac{7 b}{a^8 x}+\frac{15 b^2}{2 a^7 (a+b x)^2}-\frac{1}{2 a^7 x^2}+\frac{10 b^2}{3 a^6 (a+b x)^3}+\frac{3 b^2}{2 a^5 (a+b x)^4}+\frac{3 b^2}{5 a^4 (a+b x)^5}+\frac{b^2}{6 a^3 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^7),x]

[Out]

-1/(2*a^7*x^2) + (7*b)/(a^8*x) + b^2/(6*a^3*(a + b*x)^6) + (3*b^2)/(5*a^4*(a + b
*x)^5) + (3*b^2)/(2*a^5*(a + b*x)^4) + (10*b^2)/(3*a^6*(a + b*x)^3) + (15*b^2)/(
2*a^7*(a + b*x)^2) + (21*b^2)/(a^8*(a + b*x)) + (28*b^2*Log[x])/a^9 - (28*b^2*Lo
g[a + b*x])/a^9

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Rubi in Sympy [A]  time = 78.7256, size = 141, normalized size = 0.98 \[ \frac{b^{2}}{6 a^{3} \left (a + b x\right )^{6}} + \frac{3 b^{2}}{5 a^{4} \left (a + b x\right )^{5}} + \frac{3 b^{2}}{2 a^{5} \left (a + b x\right )^{4}} + \frac{10 b^{2}}{3 a^{6} \left (a + b x\right )^{3}} + \frac{15 b^{2}}{2 a^{7} \left (a + b x\right )^{2}} - \frac{1}{2 a^{7} x^{2}} + \frac{21 b^{2}}{a^{8} \left (a + b x\right )} + \frac{7 b}{a^{8} x} + \frac{28 b^{2} \log{\left (x \right )}}{a^{9}} - \frac{28 b^{2} \log{\left (a + b x \right )}}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x+a)**7,x)

[Out]

b**2/(6*a**3*(a + b*x)**6) + 3*b**2/(5*a**4*(a + b*x)**5) + 3*b**2/(2*a**5*(a +
b*x)**4) + 10*b**2/(3*a**6*(a + b*x)**3) + 15*b**2/(2*a**7*(a + b*x)**2) - 1/(2*
a**7*x**2) + 21*b**2/(a**8*(a + b*x)) + 7*b/(a**8*x) + 28*b**2*log(x)/a**9 - 28*
b**2*log(a + b*x)/a**9

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Mathematica [A]  time = 0.137494, size = 112, normalized size = 0.78 \[ \frac{\frac{a \left (-15 a^7+120 a^6 b x+2058 a^5 b^2 x^2+7308 a^4 b^3 x^3+11970 a^3 b^4 x^4+10360 a^2 b^5 x^5+4620 a b^6 x^6+840 b^7 x^7\right )}{x^2 (a+b x)^6}-840 b^2 \log (a+b x)+840 b^2 \log (x)}{30 a^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)^7),x]

[Out]

((a*(-15*a^7 + 120*a^6*b*x + 2058*a^5*b^2*x^2 + 7308*a^4*b^3*x^3 + 11970*a^3*b^4
*x^4 + 10360*a^2*b^5*x^5 + 4620*a*b^6*x^6 + 840*b^7*x^7))/(x^2*(a + b*x)^6) + 84
0*b^2*Log[x] - 840*b^2*Log[a + b*x])/(30*a^9)

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Maple [A]  time = 0.019, size = 133, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{7}{x}^{2}}}+7\,{\frac{b}{{a}^{8}x}}+{\frac{{b}^{2}}{6\,{a}^{3} \left ( bx+a \right ) ^{6}}}+{\frac{3\,{b}^{2}}{5\,{a}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{3\,{b}^{2}}{2\,{a}^{5} \left ( bx+a \right ) ^{4}}}+{\frac{10\,{b}^{2}}{3\,{a}^{6} \left ( bx+a \right ) ^{3}}}+{\frac{15\,{b}^{2}}{2\,{a}^{7} \left ( bx+a \right ) ^{2}}}+21\,{\frac{{b}^{2}}{{a}^{8} \left ( bx+a \right ) }}+28\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{9}}}-28\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^7,x)

[Out]

-1/2/a^7/x^2+7*b/a^8/x+1/6*b^2/a^3/(b*x+a)^6+3/5*b^2/a^4/(b*x+a)^5+3/2*b^2/a^5/(
b*x+a)^4+10/3*b^2/a^6/(b*x+a)^3+15/2*b^2/a^7/(b*x+a)^2+21*b^2/a^8/(b*x+a)+28*b^2
*ln(x)/a^9-28*b^2*ln(b*x+a)/a^9

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Maxima [A]  time = 1.34842, size = 235, normalized size = 1.63 \[ \frac{840 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 10360 \, a^{2} b^{5} x^{5} + 11970 \, a^{3} b^{4} x^{4} + 7308 \, a^{4} b^{3} x^{3} + 2058 \, a^{5} b^{2} x^{2} + 120 \, a^{6} b x - 15 \, a^{7}}{30 \,{\left (a^{8} b^{6} x^{8} + 6 \, a^{9} b^{5} x^{7} + 15 \, a^{10} b^{4} x^{6} + 20 \, a^{11} b^{3} x^{5} + 15 \, a^{12} b^{2} x^{4} + 6 \, a^{13} b x^{3} + a^{14} x^{2}\right )}} - \frac{28 \, b^{2} \log \left (b x + a\right )}{a^{9}} + \frac{28 \, b^{2} \log \left (x\right )}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^7*x^3),x, algorithm="maxima")

[Out]

1/30*(840*b^7*x^7 + 4620*a*b^6*x^6 + 10360*a^2*b^5*x^5 + 11970*a^3*b^4*x^4 + 730
8*a^4*b^3*x^3 + 2058*a^5*b^2*x^2 + 120*a^6*b*x - 15*a^7)/(a^8*b^6*x^8 + 6*a^9*b^
5*x^7 + 15*a^10*b^4*x^6 + 20*a^11*b^3*x^5 + 15*a^12*b^2*x^4 + 6*a^13*b*x^3 + a^1
4*x^2) - 28*b^2*log(b*x + a)/a^9 + 28*b^2*log(x)/a^9

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Fricas [A]  time = 0.221672, size = 413, normalized size = 2.87 \[ \frac{840 \, a b^{7} x^{7} + 4620 \, a^{2} b^{6} x^{6} + 10360 \, a^{3} b^{5} x^{5} + 11970 \, a^{4} b^{4} x^{4} + 7308 \, a^{5} b^{3} x^{3} + 2058 \, a^{6} b^{2} x^{2} + 120 \, a^{7} b x - 15 \, a^{8} - 840 \,{\left (b^{8} x^{8} + 6 \, a b^{7} x^{7} + 15 \, a^{2} b^{6} x^{6} + 20 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 6 \, a^{5} b^{3} x^{3} + a^{6} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 840 \,{\left (b^{8} x^{8} + 6 \, a b^{7} x^{7} + 15 \, a^{2} b^{6} x^{6} + 20 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 6 \, a^{5} b^{3} x^{3} + a^{6} b^{2} x^{2}\right )} \log \left (x\right )}{30 \,{\left (a^{9} b^{6} x^{8} + 6 \, a^{10} b^{5} x^{7} + 15 \, a^{11} b^{4} x^{6} + 20 \, a^{12} b^{3} x^{5} + 15 \, a^{13} b^{2} x^{4} + 6 \, a^{14} b x^{3} + a^{15} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^7*x^3),x, algorithm="fricas")

[Out]

1/30*(840*a*b^7*x^7 + 4620*a^2*b^6*x^6 + 10360*a^3*b^5*x^5 + 11970*a^4*b^4*x^4 +
7308*a^5*b^3*x^3 + 2058*a^6*b^2*x^2 + 120*a^7*b*x - 15*a^8 - 840*(b^8*x^8 + 6*a
*b^7*x^7 + 15*a^2*b^6*x^6 + 20*a^3*b^5*x^5 + 15*a^4*b^4*x^4 + 6*a^5*b^3*x^3 + a^
6*b^2*x^2)*log(b*x + a) + 840*(b^8*x^8 + 6*a*b^7*x^7 + 15*a^2*b^6*x^6 + 20*a^3*b
^5*x^5 + 15*a^4*b^4*x^4 + 6*a^5*b^3*x^3 + a^6*b^2*x^2)*log(x))/(a^9*b^6*x^8 + 6*
a^10*b^5*x^7 + 15*a^11*b^4*x^6 + 20*a^12*b^3*x^5 + 15*a^13*b^2*x^4 + 6*a^14*b*x^
3 + a^15*x^2)

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Sympy [A]  time = 4.41825, size = 175, normalized size = 1.22 \[ \frac{- 15 a^{7} + 120 a^{6} b x + 2058 a^{5} b^{2} x^{2} + 7308 a^{4} b^{3} x^{3} + 11970 a^{3} b^{4} x^{4} + 10360 a^{2} b^{5} x^{5} + 4620 a b^{6} x^{6} + 840 b^{7} x^{7}}{30 a^{14} x^{2} + 180 a^{13} b x^{3} + 450 a^{12} b^{2} x^{4} + 600 a^{11} b^{3} x^{5} + 450 a^{10} b^{4} x^{6} + 180 a^{9} b^{5} x^{7} + 30 a^{8} b^{6} x^{8}} + \frac{28 b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x+a)**7,x)

[Out]

(-15*a**7 + 120*a**6*b*x + 2058*a**5*b**2*x**2 + 7308*a**4*b**3*x**3 + 11970*a**
3*b**4*x**4 + 10360*a**2*b**5*x**5 + 4620*a*b**6*x**6 + 840*b**7*x**7)/(30*a**14
*x**2 + 180*a**13*b*x**3 + 450*a**12*b**2*x**4 + 600*a**11*b**3*x**5 + 450*a**10
*b**4*x**6 + 180*a**9*b**5*x**7 + 30*a**8*b**6*x**8) + 28*b**2*(log(x) - log(a/b
+ x))/a**9

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GIAC/XCAS [A]  time = 0.233528, size = 161, normalized size = 1.12 \[ -\frac{28 \, b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{9}} + \frac{28 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, a b^{7} x^{7} + 4620 \, a^{2} b^{6} x^{6} + 10360 \, a^{3} b^{5} x^{5} + 11970 \, a^{4} b^{4} x^{4} + 7308 \, a^{5} b^{3} x^{3} + 2058 \, a^{6} b^{2} x^{2} + 120 \, a^{7} b x - 15 \, a^{8}}{30 \,{\left (b x + a\right )}^{6} a^{9} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^7*x^3),x, algorithm="giac")

[Out]

-28*b^2*ln(abs(b*x + a))/a^9 + 28*b^2*ln(abs(x))/a^9 + 1/30*(840*a*b^7*x^7 + 462
0*a^2*b^6*x^6 + 10360*a^3*b^5*x^5 + 11970*a^4*b^4*x^4 + 7308*a^5*b^3*x^3 + 2058*
a^6*b^2*x^2 + 120*a^7*b*x - 15*a^8)/((b*x + a)^6*a^9*x^2)