3.148 \(\int \frac{(a+b x)^{10}}{x^{14}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b^2 (a+b x)^{11}}{858 a^3 x^{11}}+\frac{b (a+b x)^{11}}{78 a^2 x^{12}}-\frac{(a+b x)^{11}}{13 a x^{13}} \]

[Out]

-(a + b*x)^11/(13*a*x^13) + (b*(a + b*x)^11)/(78*a^2*x^12) - (b^2*(a + b*x)^11)/(858*a^3*x^11)

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Rubi [A]  time = 0.0099815, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^2 (a+b x)^{11}}{858 a^3 x^{11}}+\frac{b (a+b x)^{11}}{78 a^2 x^{12}}-\frac{(a+b x)^{11}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^14,x]

[Out]

-(a + b*x)^11/(13*a*x^13) + (b*(a + b*x)^11)/(78*a^2*x^12) - (b^2*(a + b*x)^11)/(858*a^3*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^{14}} \, dx &=-\frac{(a+b x)^{11}}{13 a x^{13}}-\frac{(2 b) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{13 a}\\ &=-\frac{(a+b x)^{11}}{13 a x^{13}}+\frac{b (a+b x)^{11}}{78 a^2 x^{12}}+\frac{b^2 \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{78 a^2}\\ &=-\frac{(a+b x)^{11}}{13 a x^{13}}+\frac{b (a+b x)^{11}}{78 a^2 x^{12}}-\frac{b^2 (a+b x)^{11}}{858 a^3 x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0158769, size = 126, normalized size = 2.25 \[ -\frac{45 a^8 b^2}{11 x^{11}}-\frac{12 a^7 b^3}{x^{10}}-\frac{70 a^6 b^4}{3 x^9}-\frac{63 a^5 b^5}{2 x^8}-\frac{30 a^4 b^6}{x^7}-\frac{20 a^3 b^7}{x^6}-\frac{9 a^2 b^8}{x^5}-\frac{5 a^9 b}{6 x^{12}}-\frac{a^{10}}{13 x^{13}}-\frac{5 a b^9}{2 x^4}-\frac{b^{10}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^14,x]

[Out]

-a^10/(13*x^13) - (5*a^9*b)/(6*x^12) - (45*a^8*b^2)/(11*x^11) - (12*a^7*b^3)/x^10 - (70*a^6*b^4)/(3*x^9) - (63
*a^5*b^5)/(2*x^8) - (30*a^4*b^6)/x^7 - (20*a^3*b^7)/x^6 - (9*a^2*b^8)/x^5 - (5*a*b^9)/(2*x^4) - b^10/(3*x^3)

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Maple [B]  time = 0.007, size = 113, normalized size = 2. \begin{align*} -12\,{\frac{{a}^{7}{b}^{3}}{{x}^{10}}}-{\frac{5\,{a}^{9}b}{6\,{x}^{12}}}-{\frac{{b}^{10}}{3\,{x}^{3}}}-9\,{\frac{{a}^{2}{b}^{8}}{{x}^{5}}}-{\frac{45\,{a}^{8}{b}^{2}}{11\,{x}^{11}}}-{\frac{5\,a{b}^{9}}{2\,{x}^{4}}}-20\,{\frac{{a}^{3}{b}^{7}}{{x}^{6}}}-{\frac{63\,{a}^{5}{b}^{5}}{2\,{x}^{8}}}-{\frac{{a}^{10}}{13\,{x}^{13}}}-30\,{\frac{{a}^{4}{b}^{6}}{{x}^{7}}}-{\frac{70\,{a}^{6}{b}^{4}}{3\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^14,x)

[Out]

-12*a^7*b^3/x^10-5/6*a^9*b/x^12-1/3*b^10/x^3-9*a^2*b^8/x^5-45/11*a^8*b^2/x^11-5/2*a*b^9/x^4-20*a^3*b^7/x^6-63/
2*a^5*b^5/x^8-1/13*a^10/x^13-30*a^4*b^6/x^7-70/3*a^6*b^4/x^9

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Maxima [B]  time = 1.00961, size = 151, normalized size = 2.7 \begin{align*} -\frac{286 \, b^{10} x^{10} + 2145 \, a b^{9} x^{9} + 7722 \, a^{2} b^{8} x^{8} + 17160 \, a^{3} b^{7} x^{7} + 25740 \, a^{4} b^{6} x^{6} + 27027 \, a^{5} b^{5} x^{5} + 20020 \, a^{6} b^{4} x^{4} + 10296 \, a^{7} b^{3} x^{3} + 3510 \, a^{8} b^{2} x^{2} + 715 \, a^{9} b x + 66 \, a^{10}}{858 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^14,x, algorithm="maxima")

[Out]

-1/858*(286*b^10*x^10 + 2145*a*b^9*x^9 + 7722*a^2*b^8*x^8 + 17160*a^3*b^7*x^7 + 25740*a^4*b^6*x^6 + 27027*a^5*
b^5*x^5 + 20020*a^6*b^4*x^4 + 10296*a^7*b^3*x^3 + 3510*a^8*b^2*x^2 + 715*a^9*b*x + 66*a^10)/x^13

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Fricas [B]  time = 1.49559, size = 281, normalized size = 5.02 \begin{align*} -\frac{286 \, b^{10} x^{10} + 2145 \, a b^{9} x^{9} + 7722 \, a^{2} b^{8} x^{8} + 17160 \, a^{3} b^{7} x^{7} + 25740 \, a^{4} b^{6} x^{6} + 27027 \, a^{5} b^{5} x^{5} + 20020 \, a^{6} b^{4} x^{4} + 10296 \, a^{7} b^{3} x^{3} + 3510 \, a^{8} b^{2} x^{2} + 715 \, a^{9} b x + 66 \, a^{10}}{858 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^14,x, algorithm="fricas")

[Out]

-1/858*(286*b^10*x^10 + 2145*a*b^9*x^9 + 7722*a^2*b^8*x^8 + 17160*a^3*b^7*x^7 + 25740*a^4*b^6*x^6 + 27027*a^5*
b^5*x^5 + 20020*a^6*b^4*x^4 + 10296*a^7*b^3*x^3 + 3510*a^8*b^2*x^2 + 715*a^9*b*x + 66*a^10)/x^13

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Sympy [B]  time = 1.81789, size = 121, normalized size = 2.16 \begin{align*} - \frac{66 a^{10} + 715 a^{9} b x + 3510 a^{8} b^{2} x^{2} + 10296 a^{7} b^{3} x^{3} + 20020 a^{6} b^{4} x^{4} + 27027 a^{5} b^{5} x^{5} + 25740 a^{4} b^{6} x^{6} + 17160 a^{3} b^{7} x^{7} + 7722 a^{2} b^{8} x^{8} + 2145 a b^{9} x^{9} + 286 b^{10} x^{10}}{858 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**14,x)

[Out]

-(66*a**10 + 715*a**9*b*x + 3510*a**8*b**2*x**2 + 10296*a**7*b**3*x**3 + 20020*a**6*b**4*x**4 + 27027*a**5*b**
5*x**5 + 25740*a**4*b**6*x**6 + 17160*a**3*b**7*x**7 + 7722*a**2*b**8*x**8 + 2145*a*b**9*x**9 + 286*b**10*x**1
0)/(858*x**13)

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Giac [B]  time = 1.1832, size = 151, normalized size = 2.7 \begin{align*} -\frac{286 \, b^{10} x^{10} + 2145 \, a b^{9} x^{9} + 7722 \, a^{2} b^{8} x^{8} + 17160 \, a^{3} b^{7} x^{7} + 25740 \, a^{4} b^{6} x^{6} + 27027 \, a^{5} b^{5} x^{5} + 20020 \, a^{6} b^{4} x^{4} + 10296 \, a^{7} b^{3} x^{3} + 3510 \, a^{8} b^{2} x^{2} + 715 \, a^{9} b x + 66 \, a^{10}}{858 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^14,x, algorithm="giac")

[Out]

-1/858*(286*b^10*x^10 + 2145*a*b^9*x^9 + 7722*a^2*b^8*x^8 + 17160*a^3*b^7*x^7 + 25740*a^4*b^6*x^6 + 27027*a^5*
b^5*x^5 + 20020*a^6*b^4*x^4 + 10296*a^7*b^3*x^3 + 3510*a^8*b^2*x^2 + 715*a^9*b*x + 66*a^10)/x^13