3.237 \(\int \frac{1}{x^3 (a+b x)^{10}} \, dx\)

Optimal. Leaf size=191 \[ \frac{55 b^2 \log (x)}{a^{12}}-\frac{55 b^2 \log (a+b x)}{a^{12}}+\frac{45 b^2}{a^{11} (a+b x)}+\frac{10 b}{a^{11} x}+\frac{18 b^2}{a^{10} (a+b x)^2}-\frac{1}{2 a^{10} x^2}+\frac{28 b^2}{3 a^9 (a+b x)^3}+\frac{21 b^2}{4 a^8 (a+b x)^4}+\frac{3 b^2}{a^7 (a+b x)^5}+\frac{5 b^2}{3 a^6 (a+b x)^6}+\frac{6 b^2}{7 a^5 (a+b x)^7}+\frac{3 b^2}{8 a^4 (a+b x)^8}+\frac{b^2}{9 a^3 (a+b x)^9} \]

[Out]

-1/(2*a^10*x^2) + (10*b)/(a^11*x) + b^2/(9*a^3*(a + b*x)^9) + (3*b^2)/(8*a^4*(a
+ b*x)^8) + (6*b^2)/(7*a^5*(a + b*x)^7) + (5*b^2)/(3*a^6*(a + b*x)^6) + (3*b^2)/
(a^7*(a + b*x)^5) + (21*b^2)/(4*a^8*(a + b*x)^4) + (28*b^2)/(3*a^9*(a + b*x)^3)
+ (18*b^2)/(a^10*(a + b*x)^2) + (45*b^2)/(a^11*(a + b*x)) + (55*b^2*Log[x])/a^12
- (55*b^2*Log[a + b*x])/a^12

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Rubi [A]  time = 0.323249, antiderivative size = 191, 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 \[ \frac{55 b^2 \log (x)}{a^{12}}-\frac{55 b^2 \log (a+b x)}{a^{12}}+\frac{45 b^2}{a^{11} (a+b x)}+\frac{10 b}{a^{11} x}+\frac{18 b^2}{a^{10} (a+b x)^2}-\frac{1}{2 a^{10} x^2}+\frac{28 b^2}{3 a^9 (a+b x)^3}+\frac{21 b^2}{4 a^8 (a+b x)^4}+\frac{3 b^2}{a^7 (a+b x)^5}+\frac{5 b^2}{3 a^6 (a+b x)^6}+\frac{6 b^2}{7 a^5 (a+b x)^7}+\frac{3 b^2}{8 a^4 (a+b x)^8}+\frac{b^2}{9 a^3 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^10),x]

[Out]

-1/(2*a^10*x^2) + (10*b)/(a^11*x) + b^2/(9*a^3*(a + b*x)^9) + (3*b^2)/(8*a^4*(a
+ b*x)^8) + (6*b^2)/(7*a^5*(a + b*x)^7) + (5*b^2)/(3*a^6*(a + b*x)^6) + (3*b^2)/
(a^7*(a + b*x)^5) + (21*b^2)/(4*a^8*(a + b*x)^4) + (28*b^2)/(3*a^9*(a + b*x)^3)
+ (18*b^2)/(a^10*(a + b*x)^2) + (45*b^2)/(a^11*(a + b*x)) + (55*b^2*Log[x])/a^12
- (55*b^2*Log[a + b*x])/a^12

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x+a)**10,x)

[Out]

Timed out

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Mathematica [A]  time = 0.181095, size = 145, normalized size = 0.76 \[ \frac{\frac{a \left (-252 a^{10}+2772 a^9 b x+78419 a^8 b^2 x^2+456291 a^7 b^3 x^3+1326204 a^6 b^4 x^4+2318316 a^5 b^5 x^5+2604294 a^4 b^6 x^6+1905750 a^3 b^7 x^7+882420 a^2 b^8 x^8+235620 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^2 (a+b x)^9}-27720 b^2 \log (a+b x)+27720 b^2 \log (x)}{504 a^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)^10),x]

[Out]

((a*(-252*a^10 + 2772*a^9*b*x + 78419*a^8*b^2*x^2 + 456291*a^7*b^3*x^3 + 1326204
*a^6*b^4*x^4 + 2318316*a^5*b^5*x^5 + 2604294*a^4*b^6*x^6 + 1905750*a^3*b^7*x^7 +
882420*a^2*b^8*x^8 + 235620*a*b^9*x^9 + 27720*b^10*x^10))/(x^2*(a + b*x)^9) + 2
7720*b^2*Log[x] - 27720*b^2*Log[a + b*x])/(504*a^12)

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Maple [A]  time = 0.022, size = 178, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{10}{x}^{2}}}+10\,{\frac{b}{{a}^{11}x}}+{\frac{{b}^{2}}{9\,{a}^{3} \left ( bx+a \right ) ^{9}}}+{\frac{3\,{b}^{2}}{8\,{a}^{4} \left ( bx+a \right ) ^{8}}}+{\frac{6\,{b}^{2}}{7\,{a}^{5} \left ( bx+a \right ) ^{7}}}+{\frac{5\,{b}^{2}}{3\,{a}^{6} \left ( bx+a \right ) ^{6}}}+3\,{\frac{{b}^{2}}{{a}^{7} \left ( bx+a \right ) ^{5}}}+{\frac{21\,{b}^{2}}{4\,{a}^{8} \left ( bx+a \right ) ^{4}}}+{\frac{28\,{b}^{2}}{3\,{a}^{9} \left ( bx+a \right ) ^{3}}}+18\,{\frac{{b}^{2}}{{a}^{10} \left ( bx+a \right ) ^{2}}}+45\,{\frac{{b}^{2}}{{a}^{11} \left ( bx+a \right ) }}+55\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{12}}}-55\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^10,x)

[Out]

-1/2/a^10/x^2+10*b/a^11/x+1/9*b^2/a^3/(b*x+a)^9+3/8*b^2/a^4/(b*x+a)^8+6/7*b^2/a^
5/(b*x+a)^7+5/3*b^2/a^6/(b*x+a)^6+3*b^2/a^7/(b*x+a)^5+21/4*b^2/a^8/(b*x+a)^4+28/
3*b^2/a^9/(b*x+a)^3+18*b^2/a^10/(b*x+a)^2+45*b^2/a^11/(b*x+a)+55*b^2*ln(x)/a^12-
55*b^2*ln(b*x+a)/a^12

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Maxima [A]  time = 1.37158, size = 324, normalized size = 1.7 \[ \frac{27720 \, b^{10} x^{10} + 235620 \, a b^{9} x^{9} + 882420 \, a^{2} b^{8} x^{8} + 1905750 \, a^{3} b^{7} x^{7} + 2604294 \, a^{4} b^{6} x^{6} + 2318316 \, a^{5} b^{5} x^{5} + 1326204 \, a^{6} b^{4} x^{4} + 456291 \, a^{7} b^{3} x^{3} + 78419 \, a^{8} b^{2} x^{2} + 2772 \, a^{9} b x - 252 \, a^{10}}{504 \,{\left (a^{11} b^{9} x^{11} + 9 \, a^{12} b^{8} x^{10} + 36 \, a^{13} b^{7} x^{9} + 84 \, a^{14} b^{6} x^{8} + 126 \, a^{15} b^{5} x^{7} + 126 \, a^{16} b^{4} x^{6} + 84 \, a^{17} b^{3} x^{5} + 36 \, a^{18} b^{2} x^{4} + 9 \, a^{19} b x^{3} + a^{20} x^{2}\right )}} - \frac{55 \, b^{2} \log \left (b x + a\right )}{a^{12}} + \frac{55 \, b^{2} \log \left (x\right )}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^10*x^3),x, algorithm="maxima")

[Out]

1/504*(27720*b^10*x^10 + 235620*a*b^9*x^9 + 882420*a^2*b^8*x^8 + 1905750*a^3*b^7
*x^7 + 2604294*a^4*b^6*x^6 + 2318316*a^5*b^5*x^5 + 1326204*a^6*b^4*x^4 + 456291*
a^7*b^3*x^3 + 78419*a^8*b^2*x^2 + 2772*a^9*b*x - 252*a^10)/(a^11*b^9*x^11 + 9*a^
12*b^8*x^10 + 36*a^13*b^7*x^9 + 84*a^14*b^6*x^8 + 126*a^15*b^5*x^7 + 126*a^16*b^
4*x^6 + 84*a^17*b^3*x^5 + 36*a^18*b^2*x^4 + 9*a^19*b*x^3 + a^20*x^2) - 55*b^2*lo
g(b*x + a)/a^12 + 55*b^2*log(x)/a^12

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Fricas [A]  time = 0.236545, size = 591, normalized size = 3.09 \[ \frac{27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11} - 27720 \,{\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 27720 \,{\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (x\right )}{504 \,{\left (a^{12} b^{9} x^{11} + 9 \, a^{13} b^{8} x^{10} + 36 \, a^{14} b^{7} x^{9} + 84 \, a^{15} b^{6} x^{8} + 126 \, a^{16} b^{5} x^{7} + 126 \, a^{17} b^{4} x^{6} + 84 \, a^{18} b^{3} x^{5} + 36 \, a^{19} b^{2} x^{4} + 9 \, a^{20} b x^{3} + a^{21} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^10*x^3),x, algorithm="fricas")

[Out]

1/504*(27720*a*b^10*x^10 + 235620*a^2*b^9*x^9 + 882420*a^3*b^8*x^8 + 1905750*a^4
*b^7*x^7 + 2604294*a^5*b^6*x^6 + 2318316*a^6*b^5*x^5 + 1326204*a^7*b^4*x^4 + 456
291*a^8*b^3*x^3 + 78419*a^9*b^2*x^2 + 2772*a^10*b*x - 252*a^11 - 27720*(b^11*x^1
1 + 9*a*b^10*x^10 + 36*a^2*b^9*x^9 + 84*a^3*b^8*x^8 + 126*a^4*b^7*x^7 + 126*a^5*
b^6*x^6 + 84*a^6*b^5*x^5 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^3 + a^9*b^2*x^2)*log(b*x
+ a) + 27720*(b^11*x^11 + 9*a*b^10*x^10 + 36*a^2*b^9*x^9 + 84*a^3*b^8*x^8 + 126
*a^4*b^7*x^7 + 126*a^5*b^6*x^6 + 84*a^6*b^5*x^5 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^3
+ a^9*b^2*x^2)*log(x))/(a^12*b^9*x^11 + 9*a^13*b^8*x^10 + 36*a^14*b^7*x^9 + 84*
a^15*b^6*x^8 + 126*a^16*b^5*x^7 + 126*a^17*b^4*x^6 + 84*a^18*b^3*x^5 + 36*a^19*b
^2*x^4 + 9*a^20*b*x^3 + a^21*x^2)

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Sympy [A]  time = 10.1191, size = 246, normalized size = 1.29 \[ \frac{- 252 a^{10} + 2772 a^{9} b x + 78419 a^{8} b^{2} x^{2} + 456291 a^{7} b^{3} x^{3} + 1326204 a^{6} b^{4} x^{4} + 2318316 a^{5} b^{5} x^{5} + 2604294 a^{4} b^{6} x^{6} + 1905750 a^{3} b^{7} x^{7} + 882420 a^{2} b^{8} x^{8} + 235620 a b^{9} x^{9} + 27720 b^{10} x^{10}}{504 a^{20} x^{2} + 4536 a^{19} b x^{3} + 18144 a^{18} b^{2} x^{4} + 42336 a^{17} b^{3} x^{5} + 63504 a^{16} b^{4} x^{6} + 63504 a^{15} b^{5} x^{7} + 42336 a^{14} b^{6} x^{8} + 18144 a^{13} b^{7} x^{9} + 4536 a^{12} b^{8} x^{10} + 504 a^{11} b^{9} x^{11}} + \frac{55 b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x+a)**10,x)

[Out]

(-252*a**10 + 2772*a**9*b*x + 78419*a**8*b**2*x**2 + 456291*a**7*b**3*x**3 + 132
6204*a**6*b**4*x**4 + 2318316*a**5*b**5*x**5 + 2604294*a**4*b**6*x**6 + 1905750*
a**3*b**7*x**7 + 882420*a**2*b**8*x**8 + 235620*a*b**9*x**9 + 27720*b**10*x**10)
/(504*a**20*x**2 + 4536*a**19*b*x**3 + 18144*a**18*b**2*x**4 + 42336*a**17*b**3*
x**5 + 63504*a**16*b**4*x**6 + 63504*a**15*b**5*x**7 + 42336*a**14*b**6*x**8 + 1
8144*a**13*b**7*x**9 + 4536*a**12*b**8*x**10 + 504*a**11*b**9*x**11) + 55*b**2*(
log(x) - log(a/b + x))/a**12

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GIAC/XCAS [A]  time = 0.215403, size = 205, normalized size = 1.07 \[ -\frac{55 \, b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{12}} + \frac{55 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{12}} + \frac{27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11}}{504 \,{\left (b x + a\right )}^{9} a^{12} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^10*x^3),x, algorithm="giac")

[Out]

-55*b^2*ln(abs(b*x + a))/a^12 + 55*b^2*ln(abs(x))/a^12 + 1/504*(27720*a*b^10*x^1
0 + 235620*a^2*b^9*x^9 + 882420*a^3*b^8*x^8 + 1905750*a^4*b^7*x^7 + 2604294*a^5*
b^6*x^6 + 2318316*a^6*b^5*x^5 + 1326204*a^7*b^4*x^4 + 456291*a^8*b^3*x^3 + 78419
*a^9*b^2*x^2 + 2772*a^10*b*x - 252*a^11)/((b*x + a)^9*a^12*x^2)