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

Optimal. Leaf size=141 \[ -\frac{66 a^{10} b^2}{7 x^7}-\frac{110 a^9 b^3}{3 x^6}-\frac{99 a^8 b^4}{x^5}-\frac{198 a^7 b^5}{x^4}-\frac{308 a^6 b^6}{x^3}-\frac{396 a^5 b^7}{x^2}-\frac{495 a^4 b^8}{x}+66 a^2 b^{10} x+220 a^3 b^9 \log (x)-\frac{3 a^{11} b}{2 x^8}-\frac{a^{12}}{9 x^9}+6 a b^{11} x^2+\frac{b^{12} x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.078976, antiderivative size = 141, 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} \[ -\frac{66 a^{10} b^2}{7 x^7}-\frac{110 a^9 b^3}{3 x^6}-\frac{99 a^8 b^4}{x^5}-\frac{198 a^7 b^5}{x^4}-\frac{308 a^6 b^6}{x^3}-\frac{396 a^5 b^7}{x^2}-\frac{495 a^4 b^8}{x}+66 a^2 b^{10} x+220 a^3 b^9 \log (x)-\frac{3 a^{11} b}{2 x^8}-\frac{a^{12}}{9 x^9}+6 a b^{11} x^2+\frac{b^{12} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^12/x^10,x]

[Out]

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

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

Mathematica [A]  time = 0.0187305, size = 141, normalized size = 1. \[ -\frac{66 a^{10} b^2}{7 x^7}-\frac{110 a^9 b^3}{3 x^6}-\frac{99 a^8 b^4}{x^5}-\frac{198 a^7 b^5}{x^4}-\frac{308 a^6 b^6}{x^3}-\frac{396 a^5 b^7}{x^2}-\frac{495 a^4 b^8}{x}+66 a^2 b^{10} x+220 a^3 b^9 \log (x)-\frac{3 a^{11} b}{2 x^8}-\frac{a^{12}}{9 x^9}+6 a b^{11} x^2+\frac{b^{12} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^12/x^10,x]

[Out]

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

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Maple [A]  time = 0.007, size = 132, normalized size = 0.9 \begin{align*} -{\frac{{a}^{12}}{9\,{x}^{9}}}-{\frac{3\,{a}^{11}b}{2\,{x}^{8}}}-{\frac{66\,{a}^{10}{b}^{2}}{7\,{x}^{7}}}-{\frac{110\,{a}^{9}{b}^{3}}{3\,{x}^{6}}}-99\,{\frac{{a}^{8}{b}^{4}}{{x}^{5}}}-198\,{\frac{{a}^{7}{b}^{5}}{{x}^{4}}}-308\,{\frac{{a}^{6}{b}^{6}}{{x}^{3}}}-396\,{\frac{{a}^{5}{b}^{7}}{{x}^{2}}}-495\,{\frac{{a}^{4}{b}^{8}}{x}}+66\,{a}^{2}{b}^{10}x+6\,a{b}^{11}{x}^{2}+{\frac{{b}^{12}{x}^{3}}{3}}+220\,{a}^{3}{b}^{9}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09162, size = 178, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{12} x^{3} + 6 \, a b^{11} x^{2} + 66 \, a^{2} b^{10} x + 220 \, a^{3} b^{9} \log \left (x\right ) - \frac{62370 \, a^{4} b^{8} x^{8} + 49896 \, a^{5} b^{7} x^{7} + 38808 \, a^{6} b^{6} x^{6} + 24948 \, a^{7} b^{5} x^{5} + 12474 \, a^{8} b^{4} x^{4} + 4620 \, a^{9} b^{3} x^{3} + 1188 \, a^{10} b^{2} x^{2} + 189 \, a^{11} b x + 14 \, a^{12}}{126 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^12*x^3 + 6*a*b^11*x^2 + 66*a^2*b^10*x + 220*a^3*b^9*log(x) - 1/126*(62370*a^4*b^8*x^8 + 49896*a^5*b^7*x^
7 + 38808*a^6*b^6*x^6 + 24948*a^7*b^5*x^5 + 12474*a^8*b^4*x^4 + 4620*a^9*b^3*x^3 + 1188*a^10*b^2*x^2 + 189*a^1
1*b*x + 14*a^12)/x^9

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Fricas [A]  time = 1.55274, size = 346, normalized size = 2.45 \begin{align*} \frac{42 \, b^{12} x^{12} + 756 \, a b^{11} x^{11} + 8316 \, a^{2} b^{10} x^{10} + 27720 \, a^{3} b^{9} x^{9} \log \left (x\right ) - 62370 \, a^{4} b^{8} x^{8} - 49896 \, a^{5} b^{7} x^{7} - 38808 \, a^{6} b^{6} x^{6} - 24948 \, a^{7} b^{5} x^{5} - 12474 \, a^{8} b^{4} x^{4} - 4620 \, a^{9} b^{3} x^{3} - 1188 \, a^{10} b^{2} x^{2} - 189 \, a^{11} b x - 14 \, a^{12}}{126 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/126*(42*b^12*x^12 + 756*a*b^11*x^11 + 8316*a^2*b^10*x^10 + 27720*a^3*b^9*x^9*log(x) - 62370*a^4*b^8*x^8 - 49
896*a^5*b^7*x^7 - 38808*a^6*b^6*x^6 - 24948*a^7*b^5*x^5 - 12474*a^8*b^4*x^4 - 4620*a^9*b^3*x^3 - 1188*a^10*b^2
*x^2 - 189*a^11*b*x - 14*a^12)/x^9

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Sympy [A]  time = 1.36934, size = 141, normalized size = 1. \begin{align*} 220 a^{3} b^{9} \log{\left (x \right )} + 66 a^{2} b^{10} x + 6 a b^{11} x^{2} + \frac{b^{12} x^{3}}{3} - \frac{14 a^{12} + 189 a^{11} b x + 1188 a^{10} b^{2} x^{2} + 4620 a^{9} b^{3} x^{3} + 12474 a^{8} b^{4} x^{4} + 24948 a^{7} b^{5} x^{5} + 38808 a^{6} b^{6} x^{6} + 49896 a^{5} b^{7} x^{7} + 62370 a^{4} b^{8} x^{8}}{126 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

220*a**3*b**9*log(x) + 66*a**2*b**10*x + 6*a*b**11*x**2 + b**12*x**3/3 - (14*a**12 + 189*a**11*b*x + 1188*a**1
0*b**2*x**2 + 4620*a**9*b**3*x**3 + 12474*a**8*b**4*x**4 + 24948*a**7*b**5*x**5 + 38808*a**6*b**6*x**6 + 49896
*a**5*b**7*x**7 + 62370*a**4*b**8*x**8)/(126*x**9)

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Giac [A]  time = 1.19574, size = 180, normalized size = 1.28 \begin{align*} \frac{1}{3} \, b^{12} x^{3} + 6 \, a b^{11} x^{2} + 66 \, a^{2} b^{10} x + 220 \, a^{3} b^{9} \log \left ({\left | x \right |}\right ) - \frac{62370 \, a^{4} b^{8} x^{8} + 49896 \, a^{5} b^{7} x^{7} + 38808 \, a^{6} b^{6} x^{6} + 24948 \, a^{7} b^{5} x^{5} + 12474 \, a^{8} b^{4} x^{4} + 4620 \, a^{9} b^{3} x^{3} + 1188 \, a^{10} b^{2} x^{2} + 189 \, a^{11} b x + 14 \, a^{12}}{126 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^12*x^3 + 6*a*b^11*x^2 + 66*a^2*b^10*x + 220*a^3*b^9*log(abs(x)) - 1/126*(62370*a^4*b^8*x^8 + 49896*a^5*b
^7*x^7 + 38808*a^6*b^6*x^6 + 24948*a^7*b^5*x^5 + 12474*a^8*b^4*x^4 + 4620*a^9*b^3*x^3 + 1188*a^10*b^2*x^2 + 18
9*a^11*b*x + 14*a^12)/x^9