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

Optimal. Leaf size=17 \[ -\frac{(a+b x)^{11}}{11 a x^{11}} \]

[Out]

-(a + b*x)^11/(11*a*x^11)

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Rubi [A]  time = 0.0016535, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {37} \[ -\frac{(a+b x)^{11}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^12,x]

[Out]

-(a + b*x)^11/(11*a*x^11)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{10}}{x^{12}} \, dx &=-\frac{(a+b x)^{11}}{11 a x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0112869, size = 114, normalized size = 6.71 \[ -\frac{5 a^8 b^2}{x^9}-\frac{15 a^7 b^3}{x^8}-\frac{30 a^6 b^4}{x^7}-\frac{42 a^5 b^5}{x^6}-\frac{42 a^4 b^6}{x^5}-\frac{30 a^3 b^7}{x^4}-\frac{15 a^2 b^8}{x^3}-\frac{a^9 b}{x^{10}}-\frac{a^{10}}{11 x^{11}}-\frac{5 a b^9}{x^2}-\frac{b^{10}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^12,x]

[Out]

-a^10/(11*x^11) - (a^9*b)/x^10 - (5*a^8*b^2)/x^9 - (15*a^7*b^3)/x^8 - (30*a^6*b^4)/x^7 - (42*a^5*b^5)/x^6 - (4
2*a^4*b^6)/x^5 - (30*a^3*b^7)/x^4 - (15*a^2*b^8)/x^3 - (5*a*b^9)/x^2 - b^10/x

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Maple [B]  time = 0.007, size = 113, normalized size = 6.7 \begin{align*} -{\frac{{a}^{9}b}{{x}^{10}}}-15\,{\frac{{a}^{2}{b}^{8}}{{x}^{3}}}-42\,{\frac{{a}^{4}{b}^{6}}{{x}^{5}}}-{\frac{{a}^{10}}{11\,{x}^{11}}}-30\,{\frac{{a}^{3}{b}^{7}}{{x}^{4}}}-42\,{\frac{{a}^{5}{b}^{5}}{{x}^{6}}}-15\,{\frac{{a}^{7}{b}^{3}}{{x}^{8}}}-5\,{\frac{a{b}^{9}}{{x}^{2}}}-30\,{\frac{{a}^{6}{b}^{4}}{{x}^{7}}}-{\frac{{b}^{10}}{x}}-5\,{\frac{{a}^{8}{b}^{2}}{{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^9*b/x^10-15*a^2*b^8/x^3-42*a^4*b^6/x^5-1/11*a^10/x^11-30*a^3*b^7/x^4-42*a^5*b^5/x^6-15*a^7*b^3/x^8-5*a*b^9/
x^2-30*a^6*b^4/x^7-b^10/x-5*a^8*b^2/x^9

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Maxima [B]  time = 0.992049, size = 149, normalized size = 8.76 \begin{align*} -\frac{11 \, b^{10} x^{10} + 55 \, a b^{9} x^{9} + 165 \, a^{2} b^{8} x^{8} + 330 \, a^{3} b^{7} x^{7} + 462 \, a^{4} b^{6} x^{6} + 462 \, a^{5} b^{5} x^{5} + 330 \, a^{6} b^{4} x^{4} + 165 \, a^{7} b^{3} x^{3} + 55 \, a^{8} b^{2} x^{2} + 11 \, a^{9} b x + a^{10}}{11 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11*(11*b^10*x^10 + 55*a*b^9*x^9 + 165*a^2*b^8*x^8 + 330*a^3*b^7*x^7 + 462*a^4*b^6*x^6 + 462*a^5*b^5*x^5 + 3
30*a^6*b^4*x^4 + 165*a^7*b^3*x^3 + 55*a^8*b^2*x^2 + 11*a^9*b*x + a^10)/x^11

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Fricas [B]  time = 1.53949, size = 252, normalized size = 14.82 \begin{align*} -\frac{11 \, b^{10} x^{10} + 55 \, a b^{9} x^{9} + 165 \, a^{2} b^{8} x^{8} + 330 \, a^{3} b^{7} x^{7} + 462 \, a^{4} b^{6} x^{6} + 462 \, a^{5} b^{5} x^{5} + 330 \, a^{6} b^{4} x^{4} + 165 \, a^{7} b^{3} x^{3} + 55 \, a^{8} b^{2} x^{2} + 11 \, a^{9} b x + a^{10}}{11 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11*(11*b^10*x^10 + 55*a*b^9*x^9 + 165*a^2*b^8*x^8 + 330*a^3*b^7*x^7 + 462*a^4*b^6*x^6 + 462*a^5*b^5*x^5 + 3
30*a^6*b^4*x^4 + 165*a^7*b^3*x^3 + 55*a^8*b^2*x^2 + 11*a^9*b*x + a^10)/x^11

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Sympy [B]  time = 1.31416, size = 119, normalized size = 7. \begin{align*} - \frac{a^{10} + 11 a^{9} b x + 55 a^{8} b^{2} x^{2} + 165 a^{7} b^{3} x^{3} + 330 a^{6} b^{4} x^{4} + 462 a^{5} b^{5} x^{5} + 462 a^{4} b^{6} x^{6} + 330 a^{3} b^{7} x^{7} + 165 a^{2} b^{8} x^{8} + 55 a b^{9} x^{9} + 11 b^{10} x^{10}}{11 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a**10 + 11*a**9*b*x + 55*a**8*b**2*x**2 + 165*a**7*b**3*x**3 + 330*a**6*b**4*x**4 + 462*a**5*b**5*x**5 + 462
*a**4*b**6*x**6 + 330*a**3*b**7*x**7 + 165*a**2*b**8*x**8 + 55*a*b**9*x**9 + 11*b**10*x**10)/(11*x**11)

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Giac [B]  time = 1.16533, size = 149, normalized size = 8.76 \begin{align*} -\frac{11 \, b^{10} x^{10} + 55 \, a b^{9} x^{9} + 165 \, a^{2} b^{8} x^{8} + 330 \, a^{3} b^{7} x^{7} + 462 \, a^{4} b^{6} x^{6} + 462 \, a^{5} b^{5} x^{5} + 330 \, a^{6} b^{4} x^{4} + 165 \, a^{7} b^{3} x^{3} + 55 \, a^{8} b^{2} x^{2} + 11 \, a^{9} b x + a^{10}}{11 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11*(11*b^10*x^10 + 55*a*b^9*x^9 + 165*a^2*b^8*x^8 + 330*a^3*b^7*x^7 + 462*a^4*b^6*x^6 + 462*a^5*b^5*x^5 + 3
30*a^6*b^4*x^4 + 165*a^7*b^3*x^3 + 55*a^8*b^2*x^2 + 11*a^9*b*x + a^10)/x^11