3.222 \(\int \frac{x^{12}}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=186 \[ -\frac{a^{12}}{9 b^{13} (a+b x)^9}+\frac{3 a^{11}}{2 b^{13} (a+b x)^8}-\frac{66 a^{10}}{7 b^{13} (a+b x)^7}+\frac{110 a^9}{3 b^{13} (a+b x)^6}-\frac{99 a^8}{b^{13} (a+b x)^5}+\frac{198 a^7}{b^{13} (a+b x)^4}-\frac{308 a^6}{b^{13} (a+b x)^3}+\frac{396 a^5}{b^{13} (a+b x)^2}-\frac{495 a^4}{b^{13} (a+b x)}+\frac{55 a^2 x}{b^{12}}-\frac{220 a^3 \log (a+b x)}{b^{13}}-\frac{5 a x^2}{b^{11}}+\frac{x^3}{3 b^{10}} \]

[Out]

(55*a^2*x)/b^12 - (5*a*x^2)/b^11 + x^3/(3*b^10) - a^12/(9*b^13*(a + b*x)^9) + (3*a^11)/(2*b^13*(a + b*x)^8) -
(66*a^10)/(7*b^13*(a + b*x)^7) + (110*a^9)/(3*b^13*(a + b*x)^6) - (99*a^8)/(b^13*(a + b*x)^5) + (198*a^7)/(b^1
3*(a + b*x)^4) - (308*a^6)/(b^13*(a + b*x)^3) + (396*a^5)/(b^13*(a + b*x)^2) - (495*a^4)/(b^13*(a + b*x)) - (2
20*a^3*Log[a + b*x])/b^13

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Rubi [A]  time = 0.176762, antiderivative size = 186, 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} \[ -\frac{a^{12}}{9 b^{13} (a+b x)^9}+\frac{3 a^{11}}{2 b^{13} (a+b x)^8}-\frac{66 a^{10}}{7 b^{13} (a+b x)^7}+\frac{110 a^9}{3 b^{13} (a+b x)^6}-\frac{99 a^8}{b^{13} (a+b x)^5}+\frac{198 a^7}{b^{13} (a+b x)^4}-\frac{308 a^6}{b^{13} (a+b x)^3}+\frac{396 a^5}{b^{13} (a+b x)^2}-\frac{495 a^4}{b^{13} (a+b x)}+\frac{55 a^2 x}{b^{12}}-\frac{220 a^3 \log (a+b x)}{b^{13}}-\frac{5 a x^2}{b^{11}}+\frac{x^3}{3 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^12/(a + b*x)^10,x]

[Out]

(55*a^2*x)/b^12 - (5*a*x^2)/b^11 + x^3/(3*b^10) - a^12/(9*b^13*(a + b*x)^9) + (3*a^11)/(2*b^13*(a + b*x)^8) -
(66*a^10)/(7*b^13*(a + b*x)^7) + (110*a^9)/(3*b^13*(a + b*x)^6) - (99*a^8)/(b^13*(a + b*x)^5) + (198*a^7)/(b^1
3*(a + b*x)^4) - (308*a^6)/(b^13*(a + b*x)^3) + (396*a^5)/(b^13*(a + b*x)^2) - (495*a^4)/(b^13*(a + b*x)) - (2
20*a^3*Log[a + b*x])/b^13

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^{12}}{(a+b x)^{10}} \, dx &=\int \left (\frac{55 a^2}{b^{12}}-\frac{10 a x}{b^{11}}+\frac{x^2}{b^{10}}+\frac{a^{12}}{b^{12} (a+b x)^{10}}-\frac{12 a^{11}}{b^{12} (a+b x)^9}+\frac{66 a^{10}}{b^{12} (a+b x)^8}-\frac{220 a^9}{b^{12} (a+b x)^7}+\frac{495 a^8}{b^{12} (a+b x)^6}-\frac{792 a^7}{b^{12} (a+b x)^5}+\frac{924 a^6}{b^{12} (a+b x)^4}-\frac{792 a^5}{b^{12} (a+b x)^3}+\frac{495 a^4}{b^{12} (a+b x)^2}-\frac{220 a^3}{b^{12} (a+b x)}\right ) \, dx\\ &=\frac{55 a^2 x}{b^{12}}-\frac{5 a x^2}{b^{11}}+\frac{x^3}{3 b^{10}}-\frac{a^{12}}{9 b^{13} (a+b x)^9}+\frac{3 a^{11}}{2 b^{13} (a+b x)^8}-\frac{66 a^{10}}{7 b^{13} (a+b x)^7}+\frac{110 a^9}{3 b^{13} (a+b x)^6}-\frac{99 a^8}{b^{13} (a+b x)^5}+\frac{198 a^7}{b^{13} (a+b x)^4}-\frac{308 a^6}{b^{13} (a+b x)^3}+\frac{396 a^5}{b^{13} (a+b x)^2}-\frac{495 a^4}{b^{13} (a+b x)}-\frac{220 a^3 \log (a+b x)}{b^{13}}\\ \end{align*}

Mathematica [A]  time = 0.115452, size = 161, normalized size = 0.87 \[ -\frac{1031616 a^{10} b^2 x^2+2074464 a^9 b^3 x^3+2529576 a^8 b^4 x^4+1831032 a^7 b^5 x^5+638568 a^6 b^6 x^6-58968 a^5 b^7 x^7-139482 a^4 b^8 x^8-43218 a^3 b^9 x^9-2772 a^2 b^{10} x^{10}+289089 a^{11} b x+27720 a^3 (a+b x)^9 \log (a+b x)+35201 a^{12}+252 a b^{11} x^{11}-42 b^{12} x^{12}}{126 b^{13} (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(a + b*x)^10,x]

[Out]

-(35201*a^12 + 289089*a^11*b*x + 1031616*a^10*b^2*x^2 + 2074464*a^9*b^3*x^3 + 2529576*a^8*b^4*x^4 + 1831032*a^
7*b^5*x^5 + 638568*a^6*b^6*x^6 - 58968*a^5*b^7*x^7 - 139482*a^4*b^8*x^8 - 43218*a^3*b^9*x^9 - 2772*a^2*b^10*x^
10 + 252*a*b^11*x^11 - 42*b^12*x^12 + 27720*a^3*(a + b*x)^9*Log[a + b*x])/(126*b^13*(a + b*x)^9)

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Maple [A]  time = 0.011, size = 177, normalized size = 1. \begin{align*} 55\,{\frac{{a}^{2}x}{{b}^{12}}}-5\,{\frac{a{x}^{2}}{{b}^{11}}}+{\frac{{x}^{3}}{3\,{b}^{10}}}-{\frac{{a}^{12}}{9\,{b}^{13} \left ( bx+a \right ) ^{9}}}+{\frac{3\,{a}^{11}}{2\,{b}^{13} \left ( bx+a \right ) ^{8}}}-{\frac{66\,{a}^{10}}{7\,{b}^{13} \left ( bx+a \right ) ^{7}}}+{\frac{110\,{a}^{9}}{3\,{b}^{13} \left ( bx+a \right ) ^{6}}}-99\,{\frac{{a}^{8}}{{b}^{13} \left ( bx+a \right ) ^{5}}}+198\,{\frac{{a}^{7}}{{b}^{13} \left ( bx+a \right ) ^{4}}}-308\,{\frac{{a}^{6}}{{b}^{13} \left ( bx+a \right ) ^{3}}}+396\,{\frac{{a}^{5}}{{b}^{13} \left ( bx+a \right ) ^{2}}}-495\,{\frac{{a}^{4}}{{b}^{13} \left ( bx+a \right ) }}-220\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12/(b*x+a)^10,x)

[Out]

55*a^2*x/b^12-5*a*x^2/b^11+1/3*x^3/b^10-1/9*a^12/b^13/(b*x+a)^9+3/2*a^11/b^13/(b*x+a)^8-66/7*a^10/b^13/(b*x+a)
^7+110/3*a^9/b^13/(b*x+a)^6-99*a^8/b^13/(b*x+a)^5+198*a^7/b^13/(b*x+a)^4-308*a^6/b^13/(b*x+a)^3+396*a^5/b^13/(
b*x+a)^2-495*a^4/b^13/(b*x+a)-220*a^3*ln(b*x+a)/b^13

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Maxima [A]  time = 1.19224, size = 316, normalized size = 1.7 \begin{align*} -\frac{62370 \, a^{4} b^{8} x^{8} + 449064 \, a^{5} b^{7} x^{7} + 1435896 \, a^{6} b^{6} x^{6} + 2652804 \, a^{7} b^{5} x^{5} + 3089394 \, a^{8} b^{4} x^{4} + 2318316 \, a^{9} b^{3} x^{3} + 1093356 \, a^{10} b^{2} x^{2} + 296019 \, a^{11} b x + 35201 \, a^{12}}{126 \,{\left (b^{22} x^{9} + 9 \, a b^{21} x^{8} + 36 \, a^{2} b^{20} x^{7} + 84 \, a^{3} b^{19} x^{6} + 126 \, a^{4} b^{18} x^{5} + 126 \, a^{5} b^{17} x^{4} + 84 \, a^{6} b^{16} x^{3} + 36 \, a^{7} b^{15} x^{2} + 9 \, a^{8} b^{14} x + a^{9} b^{13}\right )}} - \frac{220 \, a^{3} \log \left (b x + a\right )}{b^{13}} + \frac{b^{2} x^{3} - 15 \, a b x^{2} + 165 \, a^{2} x}{3 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/126*(62370*a^4*b^8*x^8 + 449064*a^5*b^7*x^7 + 1435896*a^6*b^6*x^6 + 2652804*a^7*b^5*x^5 + 3089394*a^8*b^4*x
^4 + 2318316*a^9*b^3*x^3 + 1093356*a^10*b^2*x^2 + 296019*a^11*b*x + 35201*a^12)/(b^22*x^9 + 9*a*b^21*x^8 + 36*
a^2*b^20*x^7 + 84*a^3*b^19*x^6 + 126*a^4*b^18*x^5 + 126*a^5*b^17*x^4 + 84*a^6*b^16*x^3 + 36*a^7*b^15*x^2 + 9*a
^8*b^14*x + a^9*b^13) - 220*a^3*log(b*x + a)/b^13 + 1/3*(b^2*x^3 - 15*a*b*x^2 + 165*a^2*x)/b^12

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Fricas [A]  time = 1.55658, size = 811, normalized size = 4.36 \begin{align*} \frac{42 \, b^{12} x^{12} - 252 \, a b^{11} x^{11} + 2772 \, a^{2} b^{10} x^{10} + 43218 \, a^{3} b^{9} x^{9} + 139482 \, a^{4} b^{8} x^{8} + 58968 \, a^{5} b^{7} x^{7} - 638568 \, a^{6} b^{6} x^{6} - 1831032 \, a^{7} b^{5} x^{5} - 2529576 \, a^{8} b^{4} x^{4} - 2074464 \, a^{9} b^{3} x^{3} - 1031616 \, a^{10} b^{2} x^{2} - 289089 \, a^{11} b x - 35201 \, a^{12} - 27720 \,{\left (a^{3} b^{9} x^{9} + 9 \, a^{4} b^{8} x^{8} + 36 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 126 \, a^{7} b^{5} x^{5} + 126 \, a^{8} b^{4} x^{4} + 84 \, a^{9} b^{3} x^{3} + 36 \, a^{10} b^{2} x^{2} + 9 \, a^{11} b x + a^{12}\right )} \log \left (b x + a\right )}{126 \,{\left (b^{22} x^{9} + 9 \, a b^{21} x^{8} + 36 \, a^{2} b^{20} x^{7} + 84 \, a^{3} b^{19} x^{6} + 126 \, a^{4} b^{18} x^{5} + 126 \, a^{5} b^{17} x^{4} + 84 \, a^{6} b^{16} x^{3} + 36 \, a^{7} b^{15} x^{2} + 9 \, a^{8} b^{14} x + a^{9} b^{13}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x+a)^10,x, algorithm="fricas")

[Out]

1/126*(42*b^12*x^12 - 252*a*b^11*x^11 + 2772*a^2*b^10*x^10 + 43218*a^3*b^9*x^9 + 139482*a^4*b^8*x^8 + 58968*a^
5*b^7*x^7 - 638568*a^6*b^6*x^6 - 1831032*a^7*b^5*x^5 - 2529576*a^8*b^4*x^4 - 2074464*a^9*b^3*x^3 - 1031616*a^1
0*b^2*x^2 - 289089*a^11*b*x - 35201*a^12 - 27720*(a^3*b^9*x^9 + 9*a^4*b^8*x^8 + 36*a^5*b^7*x^7 + 84*a^6*b^6*x^
6 + 126*a^7*b^5*x^5 + 126*a^8*b^4*x^4 + 84*a^9*b^3*x^3 + 36*a^10*b^2*x^2 + 9*a^11*b*x + a^12)*log(b*x + a))/(b
^22*x^9 + 9*a*b^21*x^8 + 36*a^2*b^20*x^7 + 84*a^3*b^19*x^6 + 126*a^4*b^18*x^5 + 126*a^5*b^17*x^4 + 84*a^6*b^16
*x^3 + 36*a^7*b^15*x^2 + 9*a^8*b^14*x + a^9*b^13)

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Sympy [A]  time = 2.24824, size = 248, normalized size = 1.33 \begin{align*} - \frac{220 a^{3} \log{\left (a + b x \right )}}{b^{13}} + \frac{55 a^{2} x}{b^{12}} - \frac{5 a x^{2}}{b^{11}} - \frac{35201 a^{12} + 296019 a^{11} b x + 1093356 a^{10} b^{2} x^{2} + 2318316 a^{9} b^{3} x^{3} + 3089394 a^{8} b^{4} x^{4} + 2652804 a^{7} b^{5} x^{5} + 1435896 a^{6} b^{6} x^{6} + 449064 a^{5} b^{7} x^{7} + 62370 a^{4} b^{8} x^{8}}{126 a^{9} b^{13} + 1134 a^{8} b^{14} x + 4536 a^{7} b^{15} x^{2} + 10584 a^{6} b^{16} x^{3} + 15876 a^{5} b^{17} x^{4} + 15876 a^{4} b^{18} x^{5} + 10584 a^{3} b^{19} x^{6} + 4536 a^{2} b^{20} x^{7} + 1134 a b^{21} x^{8} + 126 b^{22} x^{9}} + \frac{x^{3}}{3 b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(b*x+a)**10,x)

[Out]

-220*a**3*log(a + b*x)/b**13 + 55*a**2*x/b**12 - 5*a*x**2/b**11 - (35201*a**12 + 296019*a**11*b*x + 1093356*a*
*10*b**2*x**2 + 2318316*a**9*b**3*x**3 + 3089394*a**8*b**4*x**4 + 2652804*a**7*b**5*x**5 + 1435896*a**6*b**6*x
**6 + 449064*a**5*b**7*x**7 + 62370*a**4*b**8*x**8)/(126*a**9*b**13 + 1134*a**8*b**14*x + 4536*a**7*b**15*x**2
+ 10584*a**6*b**16*x**3 + 15876*a**5*b**17*x**4 + 15876*a**4*b**18*x**5 + 10584*a**3*b**19*x**6 + 4536*a**2*b
**20*x**7 + 1134*a*b**21*x**8 + 126*b**22*x**9) + x**3/(3*b**10)

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Giac [A]  time = 1.19218, size = 201, normalized size = 1.08 \begin{align*} -\frac{220 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{13}} - \frac{62370 \, a^{4} b^{8} x^{8} + 449064 \, a^{5} b^{7} x^{7} + 1435896 \, a^{6} b^{6} x^{6} + 2652804 \, a^{7} b^{5} x^{5} + 3089394 \, a^{8} b^{4} x^{4} + 2318316 \, a^{9} b^{3} x^{3} + 1093356 \, a^{10} b^{2} x^{2} + 296019 \, a^{11} b x + 35201 \, a^{12}}{126 \,{\left (b x + a\right )}^{9} b^{13}} + \frac{b^{20} x^{3} - 15 \, a b^{19} x^{2} + 165 \, a^{2} b^{18} x}{3 \, b^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x+a)^10,x, algorithm="giac")

[Out]

-220*a^3*log(abs(b*x + a))/b^13 - 1/126*(62370*a^4*b^8*x^8 + 449064*a^5*b^7*x^7 + 1435896*a^6*b^6*x^6 + 265280
4*a^7*b^5*x^5 + 3089394*a^8*b^4*x^4 + 2318316*a^9*b^3*x^3 + 1093356*a^10*b^2*x^2 + 296019*a^11*b*x + 35201*a^1
2)/((b*x + a)^9*b^13) + 1/3*(b^20*x^3 - 15*a*b^19*x^2 + 165*a^2*b^18*x)/b^30