3.147 \(\int \frac{(a+b x)^{10}}{x^{13}} \, dx\)

Optimal. Leaf size=36 \[ \frac{b (a+b x)^{11}}{132 a^2 x^{11}}-\frac{(a+b x)^{11}}{12 a x^{12}} \]

[Out]

-(a + b*x)^11/(12*a*x^12) + (b*(a + b*x)^11)/(132*a^2*x^11)

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Rubi [A]  time = 0.0048916, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{b (a+b x)^{11}}{132 a^2 x^{11}}-\frac{(a+b x)^{11}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^13,x]

[Out]

-(a + b*x)^11/(12*a*x^12) + (b*(a + b*x)^11)/(132*a^2*x^11)

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^{13}} \, dx &=-\frac{(a+b x)^{11}}{12 a x^{12}}-\frac{b \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{12 a}\\ &=-\frac{(a+b x)^{11}}{12 a x^{12}}+\frac{b (a+b x)^{11}}{132 a^2 x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0059036, size = 128, normalized size = 3.56 \[ -\frac{9 a^8 b^2}{2 x^{10}}-\frac{40 a^7 b^3}{3 x^9}-\frac{105 a^6 b^4}{4 x^8}-\frac{36 a^5 b^5}{x^7}-\frac{35 a^4 b^6}{x^6}-\frac{24 a^3 b^7}{x^5}-\frac{45 a^2 b^8}{4 x^4}-\frac{10 a^9 b}{11 x^{11}}-\frac{a^{10}}{12 x^{12}}-\frac{10 a b^9}{3 x^3}-\frac{b^{10}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^13,x]

[Out]

-a^10/(12*x^12) - (10*a^9*b)/(11*x^11) - (9*a^8*b^2)/(2*x^10) - (40*a^7*b^3)/(3*x^9) - (105*a^6*b^4)/(4*x^8) -
(36*a^5*b^5)/x^7 - (35*a^4*b^6)/x^6 - (24*a^3*b^7)/x^5 - (45*a^2*b^8)/(4*x^4) - (10*a*b^9)/(3*x^3) - b^10/(2*
x^2)

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Maple [B]  time = 0.007, size = 113, normalized size = 3.1 \begin{align*} -{\frac{9\,{a}^{8}{b}^{2}}{2\,{x}^{10}}}-{\frac{{a}^{10}}{12\,{x}^{12}}}-{\frac{10\,a{b}^{9}}{3\,{x}^{3}}}-24\,{\frac{{a}^{3}{b}^{7}}{{x}^{5}}}-{\frac{10\,{a}^{9}b}{11\,{x}^{11}}}-{\frac{45\,{a}^{2}{b}^{8}}{4\,{x}^{4}}}-35\,{\frac{{a}^{4}{b}^{6}}{{x}^{6}}}-{\frac{105\,{a}^{6}{b}^{4}}{4\,{x}^{8}}}-{\frac{{b}^{10}}{2\,{x}^{2}}}-36\,{\frac{{a}^{5}{b}^{5}}{{x}^{7}}}-{\frac{40\,{a}^{7}{b}^{3}}{3\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^13,x)

[Out]

-9/2*a^8*b^2/x^10-1/12*a^10/x^12-10/3*a*b^9/x^3-24*a^3*b^7/x^5-10/11*a^9*b/x^11-45/4*a^2*b^8/x^4-35*a^4*b^6/x^
6-105/4*a^6*b^4/x^8-1/2*b^10/x^2-36*a^5*b^5/x^7-40/3*a^7*b^3/x^9

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Maxima [B]  time = 1.02893, size = 151, normalized size = 4.19 \begin{align*} -\frac{66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^13,x, algorithm="maxima")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 4620*a^4*b^6*x^6 + 4752*a^5*b^5*x
^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

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Fricas [B]  time = 1.56472, size = 270, normalized size = 7.5 \begin{align*} -\frac{66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^13,x, algorithm="fricas")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 4620*a^4*b^6*x^6 + 4752*a^5*b^5*x
^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

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Sympy [B]  time = 1.34703, size = 121, normalized size = 3.36 \begin{align*} - \frac{11 a^{10} + 120 a^{9} b x + 594 a^{8} b^{2} x^{2} + 1760 a^{7} b^{3} x^{3} + 3465 a^{6} b^{4} x^{4} + 4752 a^{5} b^{5} x^{5} + 4620 a^{4} b^{6} x^{6} + 3168 a^{3} b^{7} x^{7} + 1485 a^{2} b^{8} x^{8} + 440 a b^{9} x^{9} + 66 b^{10} x^{10}}{132 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**13,x)

[Out]

-(11*a**10 + 120*a**9*b*x + 594*a**8*b**2*x**2 + 1760*a**7*b**3*x**3 + 3465*a**6*b**4*x**4 + 4752*a**5*b**5*x*
*5 + 4620*a**4*b**6*x**6 + 3168*a**3*b**7*x**7 + 1485*a**2*b**8*x**8 + 440*a*b**9*x**9 + 66*b**10*x**10)/(132*
x**12)

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Giac [B]  time = 1.24711, size = 151, normalized size = 4.19 \begin{align*} -\frac{66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^13,x, algorithm="giac")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 4620*a^4*b^6*x^6 + 4752*a^5*b^5*x
^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12