3.223 \(\int \frac{x^{11}}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=177 \[ \frac{a^{11}}{9 b^{12} (a+b x)^9}-\frac{11 a^{10}}{8 b^{12} (a+b x)^8}+\frac{55 a^9}{7 b^{12} (a+b x)^7}-\frac{55 a^8}{2 b^{12} (a+b x)^6}+\frac{66 a^7}{b^{12} (a+b x)^5}-\frac{231 a^6}{2 b^{12} (a+b x)^4}+\frac{154 a^5}{b^{12} (a+b x)^3}-\frac{165 a^4}{b^{12} (a+b x)^2}+\frac{165 a^3}{b^{12} (a+b x)}+\frac{55 a^2 \log (a+b x)}{b^{12}}-\frac{10 a x}{b^{11}}+\frac{x^2}{2 b^{10}} \]

[Out]

(-10*a*x)/b^11 + x^2/(2*b^10) + a^11/(9*b^12*(a + b*x)^9) - (11*a^10)/(8*b^12*(a + b*x)^8) + (55*a^9)/(7*b^12*
(a + b*x)^7) - (55*a^8)/(2*b^12*(a + b*x)^6) + (66*a^7)/(b^12*(a + b*x)^5) - (231*a^6)/(2*b^12*(a + b*x)^4) +
(154*a^5)/(b^12*(a + b*x)^3) - (165*a^4)/(b^12*(a + b*x)^2) + (165*a^3)/(b^12*(a + b*x)) + (55*a^2*Log[a + b*x
])/b^12

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Rubi [A]  time = 0.142378, antiderivative size = 177, 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 = {43} \[ \frac{a^{11}}{9 b^{12} (a+b x)^9}-\frac{11 a^{10}}{8 b^{12} (a+b x)^8}+\frac{55 a^9}{7 b^{12} (a+b x)^7}-\frac{55 a^8}{2 b^{12} (a+b x)^6}+\frac{66 a^7}{b^{12} (a+b x)^5}-\frac{231 a^6}{2 b^{12} (a+b x)^4}+\frac{154 a^5}{b^{12} (a+b x)^3}-\frac{165 a^4}{b^{12} (a+b x)^2}+\frac{165 a^3}{b^{12} (a+b x)}+\frac{55 a^2 \log (a+b x)}{b^{12}}-\frac{10 a x}{b^{11}}+\frac{x^2}{2 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x)^10,x]

[Out]

(-10*a*x)/b^11 + x^2/(2*b^10) + a^11/(9*b^12*(a + b*x)^9) - (11*a^10)/(8*b^12*(a + b*x)^8) + (55*a^9)/(7*b^12*
(a + b*x)^7) - (55*a^8)/(2*b^12*(a + b*x)^6) + (66*a^7)/(b^12*(a + b*x)^5) - (231*a^6)/(2*b^12*(a + b*x)^4) +
(154*a^5)/(b^12*(a + b*x)^3) - (165*a^4)/(b^12*(a + b*x)^2) + (165*a^3)/(b^12*(a + b*x)) + (55*a^2*Log[a + b*x
])/b^12

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{11}}{(a+b x)^{10}} \, dx &=\int \left (-\frac{10 a}{b^{11}}+\frac{x}{b^{10}}-\frac{a^{11}}{b^{11} (a+b x)^{10}}+\frac{11 a^{10}}{b^{11} (a+b x)^9}-\frac{55 a^9}{b^{11} (a+b x)^8}+\frac{165 a^8}{b^{11} (a+b x)^7}-\frac{330 a^7}{b^{11} (a+b x)^6}+\frac{462 a^6}{b^{11} (a+b x)^5}-\frac{462 a^5}{b^{11} (a+b x)^4}+\frac{330 a^4}{b^{11} (a+b x)^3}-\frac{165 a^3}{b^{11} (a+b x)^2}+\frac{55 a^2}{b^{11} (a+b x)}\right ) \, dx\\ &=-\frac{10 a x}{b^{11}}+\frac{x^2}{2 b^{10}}+\frac{a^{11}}{9 b^{12} (a+b x)^9}-\frac{11 a^{10}}{8 b^{12} (a+b x)^8}+\frac{55 a^9}{7 b^{12} (a+b x)^7}-\frac{55 a^8}{2 b^{12} (a+b x)^6}+\frac{66 a^7}{b^{12} (a+b x)^5}-\frac{231 a^6}{2 b^{12} (a+b x)^4}+\frac{154 a^5}{b^{12} (a+b x)^3}-\frac{165 a^4}{b^{12} (a+b x)^2}+\frac{165 a^3}{b^{12} (a+b x)}+\frac{55 a^2 \log (a+b x)}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0468254, size = 150, normalized size = 0.85 \[ \frac{1281096 a^9 b^2 x^2+2656584 a^8 b^3 x^3+3402756 a^7 b^4 x^4+2704212 a^6 b^5 x^5+1220688 a^5 b^6 x^6+190512 a^4 b^7 x^7-77112 a^3 b^8 x^8-36288 a^2 b^9 x^9+351459 a^{10} b x+27720 a^2 (a+b x)^9 \log (a+b x)+42131 a^{11}-2772 a b^{10} x^{10}+252 b^{11} x^{11}}{504 b^{12} (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x)^10,x]

[Out]

(42131*a^11 + 351459*a^10*b*x + 1281096*a^9*b^2*x^2 + 2656584*a^8*b^3*x^3 + 3402756*a^7*b^4*x^4 + 2704212*a^6*
b^5*x^5 + 1220688*a^5*b^6*x^6 + 190512*a^4*b^7*x^7 - 77112*a^3*b^8*x^8 - 36288*a^2*b^9*x^9 - 2772*a*b^10*x^10
+ 252*b^11*x^11 + 27720*a^2*(a + b*x)^9*Log[a + b*x])/(504*b^12*(a + b*x)^9)

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Maple [A]  time = 0.011, size = 166, normalized size = 0.9 \begin{align*} -10\,{\frac{ax}{{b}^{11}}}+{\frac{{x}^{2}}{2\,{b}^{10}}}+{\frac{{a}^{11}}{9\,{b}^{12} \left ( bx+a \right ) ^{9}}}-{\frac{11\,{a}^{10}}{8\,{b}^{12} \left ( bx+a \right ) ^{8}}}+{\frac{55\,{a}^{9}}{7\,{b}^{12} \left ( bx+a \right ) ^{7}}}-{\frac{55\,{a}^{8}}{2\,{b}^{12} \left ( bx+a \right ) ^{6}}}+66\,{\frac{{a}^{7}}{{b}^{12} \left ( bx+a \right ) ^{5}}}-{\frac{231\,{a}^{6}}{2\,{b}^{12} \left ( bx+a \right ) ^{4}}}+154\,{\frac{{a}^{5}}{{b}^{12} \left ( bx+a \right ) ^{3}}}-165\,{\frac{{a}^{4}}{{b}^{12} \left ( bx+a \right ) ^{2}}}+165\,{\frac{{a}^{3}}{{b}^{12} \left ( bx+a \right ) }}+55\,{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x+a)^10,x)

[Out]

-10*a*x/b^11+1/2*x^2/b^10+1/9*a^11/b^12/(b*x+a)^9-11/8*a^10/b^12/(b*x+a)^8+55/7*a^9/b^12/(b*x+a)^7-55/2*a^8/b^
12/(b*x+a)^6+66*a^7/b^12/(b*x+a)^5-231/2*a^6/b^12/(b*x+a)^4+154*a^5/b^12/(b*x+a)^3-165*a^4/b^12/(b*x+a)^2+165*
a^3/b^12/(b*x+a)+55*a^2*ln(b*x+a)/b^12

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Maxima [A]  time = 1.12964, size = 301, normalized size = 1.7 \begin{align*} \frac{83160 \, a^{3} b^{8} x^{8} + 582120 \, a^{4} b^{7} x^{7} + 1823976 \, a^{5} b^{6} x^{6} + 3318084 \, a^{6} b^{5} x^{5} + 3817044 \, a^{7} b^{4} x^{4} + 2835756 \, a^{8} b^{3} x^{3} + 1326204 \, a^{9} b^{2} x^{2} + 356499 \, a^{10} b x + 42131 \, a^{11}}{504 \,{\left (b^{21} x^{9} + 9 \, a b^{20} x^{8} + 36 \, a^{2} b^{19} x^{7} + 84 \, a^{3} b^{18} x^{6} + 126 \, a^{4} b^{17} x^{5} + 126 \, a^{5} b^{16} x^{4} + 84 \, a^{6} b^{15} x^{3} + 36 \, a^{7} b^{14} x^{2} + 9 \, a^{8} b^{13} x + a^{9} b^{12}\right )}} + \frac{55 \, a^{2} \log \left (b x + a\right )}{b^{12}} + \frac{b x^{2} - 20 \, a x}{2 \, b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x+a)^10,x, algorithm="maxima")

[Out]

1/504*(83160*a^3*b^8*x^8 + 582120*a^4*b^7*x^7 + 1823976*a^5*b^6*x^6 + 3318084*a^6*b^5*x^5 + 3817044*a^7*b^4*x^
4 + 2835756*a^8*b^3*x^3 + 1326204*a^9*b^2*x^2 + 356499*a^10*b*x + 42131*a^11)/(b^21*x^9 + 9*a*b^20*x^8 + 36*a^
2*b^19*x^7 + 84*a^3*b^18*x^6 + 126*a^4*b^17*x^5 + 126*a^5*b^16*x^4 + 84*a^6*b^15*x^3 + 36*a^7*b^14*x^2 + 9*a^8
*b^13*x + a^9*b^12) + 55*a^2*log(b*x + a)/b^12 + 1/2*(b*x^2 - 20*a*x)/b^11

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Fricas [A]  time = 1.44952, size = 784, normalized size = 4.43 \begin{align*} \frac{252 \, b^{11} x^{11} - 2772 \, a b^{10} x^{10} - 36288 \, a^{2} b^{9} x^{9} - 77112 \, a^{3} b^{8} x^{8} + 190512 \, a^{4} b^{7} x^{7} + 1220688 \, a^{5} b^{6} x^{6} + 2704212 \, a^{6} b^{5} x^{5} + 3402756 \, a^{7} b^{4} x^{4} + 2656584 \, a^{8} b^{3} x^{3} + 1281096 \, a^{9} b^{2} x^{2} + 351459 \, a^{10} b x + 42131 \, a^{11} + 27720 \,{\left (a^{2} b^{9} x^{9} + 9 \, a^{3} b^{8} x^{8} + 36 \, a^{4} b^{7} x^{7} + 84 \, a^{5} b^{6} x^{6} + 126 \, a^{6} b^{5} x^{5} + 126 \, a^{7} b^{4} x^{4} + 84 \, a^{8} b^{3} x^{3} + 36 \, a^{9} b^{2} x^{2} + 9 \, a^{10} b x + a^{11}\right )} \log \left (b x + a\right )}{504 \,{\left (b^{21} x^{9} + 9 \, a b^{20} x^{8} + 36 \, a^{2} b^{19} x^{7} + 84 \, a^{3} b^{18} x^{6} + 126 \, a^{4} b^{17} x^{5} + 126 \, a^{5} b^{16} x^{4} + 84 \, a^{6} b^{15} x^{3} + 36 \, a^{7} b^{14} x^{2} + 9 \, a^{8} b^{13} x + a^{9} b^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x+a)^10,x, algorithm="fricas")

[Out]

1/504*(252*b^11*x^11 - 2772*a*b^10*x^10 - 36288*a^2*b^9*x^9 - 77112*a^3*b^8*x^8 + 190512*a^4*b^7*x^7 + 1220688
*a^5*b^6*x^6 + 2704212*a^6*b^5*x^5 + 3402756*a^7*b^4*x^4 + 2656584*a^8*b^3*x^3 + 1281096*a^9*b^2*x^2 + 351459*
a^10*b*x + 42131*a^11 + 27720*(a^2*b^9*x^9 + 9*a^3*b^8*x^8 + 36*a^4*b^7*x^7 + 84*a^5*b^6*x^6 + 126*a^6*b^5*x^5
+ 126*a^7*b^4*x^4 + 84*a^8*b^3*x^3 + 36*a^9*b^2*x^2 + 9*a^10*b*x + a^11)*log(b*x + a))/(b^21*x^9 + 9*a*b^20*x
^8 + 36*a^2*b^19*x^7 + 84*a^3*b^18*x^6 + 126*a^4*b^17*x^5 + 126*a^5*b^16*x^4 + 84*a^6*b^15*x^3 + 36*a^7*b^14*x
^2 + 9*a^8*b^13*x + a^9*b^12)

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Sympy [A]  time = 2.12916, size = 236, normalized size = 1.33 \begin{align*} \frac{55 a^{2} \log{\left (a + b x \right )}}{b^{12}} - \frac{10 a x}{b^{11}} + \frac{42131 a^{11} + 356499 a^{10} b x + 1326204 a^{9} b^{2} x^{2} + 2835756 a^{8} b^{3} x^{3} + 3817044 a^{7} b^{4} x^{4} + 3318084 a^{6} b^{5} x^{5} + 1823976 a^{5} b^{6} x^{6} + 582120 a^{4} b^{7} x^{7} + 83160 a^{3} b^{8} x^{8}}{504 a^{9} b^{12} + 4536 a^{8} b^{13} x + 18144 a^{7} b^{14} x^{2} + 42336 a^{6} b^{15} x^{3} + 63504 a^{5} b^{16} x^{4} + 63504 a^{4} b^{17} x^{5} + 42336 a^{3} b^{18} x^{6} + 18144 a^{2} b^{19} x^{7} + 4536 a b^{20} x^{8} + 504 b^{21} x^{9}} + \frac{x^{2}}{2 b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x+a)**10,x)

[Out]

55*a**2*log(a + b*x)/b**12 - 10*a*x/b**11 + (42131*a**11 + 356499*a**10*b*x + 1326204*a**9*b**2*x**2 + 2835756
*a**8*b**3*x**3 + 3817044*a**7*b**4*x**4 + 3318084*a**6*b**5*x**5 + 1823976*a**5*b**6*x**6 + 582120*a**4*b**7*
x**7 + 83160*a**3*b**8*x**8)/(504*a**9*b**12 + 4536*a**8*b**13*x + 18144*a**7*b**14*x**2 + 42336*a**6*b**15*x*
*3 + 63504*a**5*b**16*x**4 + 63504*a**4*b**17*x**5 + 42336*a**3*b**18*x**6 + 18144*a**2*b**19*x**7 + 4536*a*b*
*20*x**8 + 504*b**21*x**9) + x**2/(2*b**10)

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Giac [A]  time = 1.2484, size = 186, normalized size = 1.05 \begin{align*} \frac{55 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{12}} + \frac{b^{10} x^{2} - 20 \, a b^{9} x}{2 \, b^{20}} + \frac{83160 \, a^{3} b^{8} x^{8} + 582120 \, a^{4} b^{7} x^{7} + 1823976 \, a^{5} b^{6} x^{6} + 3318084 \, a^{6} b^{5} x^{5} + 3817044 \, a^{7} b^{4} x^{4} + 2835756 \, a^{8} b^{3} x^{3} + 1326204 \, a^{9} b^{2} x^{2} + 356499 \, a^{10} b x + 42131 \, a^{11}}{504 \,{\left (b x + a\right )}^{9} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x+a)^10,x, algorithm="giac")

[Out]

55*a^2*log(abs(b*x + a))/b^12 + 1/2*(b^10*x^2 - 20*a*b^9*x)/b^20 + 1/504*(83160*a^3*b^8*x^8 + 582120*a^4*b^7*x
^7 + 1823976*a^5*b^6*x^6 + 3318084*a^6*b^5*x^5 + 3817044*a^7*b^4*x^4 + 2835756*a^8*b^3*x^3 + 1326204*a^9*b^2*x
^2 + 356499*a^10*b*x + 42131*a^11)/((b*x + a)^9*b^12)