3.151 \(\int \frac{(a+b x)^{10}}{x^{17}} \, dx\)

Optimal. Leaf size=116 \[ \frac{b^5 (a+b x)^{11}}{48048 a^6 x^{11}}-\frac{b^4 (a+b x)^{11}}{4368 a^5 x^{12}}+\frac{b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac{b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}-\frac{(a+b x)^{11}}{16 a x^{16}} \]

[Out]

-(a + b*x)^11/(16*a*x^16) + (b*(a + b*x)^11)/(48*a^2*x^15) - (b^2*(a + b*x)^11)/(168*a^3*x^14) + (b^3*(a + b*x
)^11)/(728*a^4*x^13) - (b^4*(a + b*x)^11)/(4368*a^5*x^12) + (b^5*(a + b*x)^11)/(48048*a^6*x^11)

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Rubi [A]  time = 0.0351052, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{b^5 (a+b x)^{11}}{48048 a^6 x^{11}}-\frac{b^4 (a+b x)^{11}}{4368 a^5 x^{12}}+\frac{b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac{b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}-\frac{(a+b x)^{11}}{16 a x^{16}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^17,x]

[Out]

-(a + b*x)^11/(16*a*x^16) + (b*(a + b*x)^11)/(48*a^2*x^15) - (b^2*(a + b*x)^11)/(168*a^3*x^14) + (b^3*(a + b*x
)^11)/(728*a^4*x^13) - (b^4*(a + b*x)^11)/(4368*a^5*x^12) + (b^5*(a + b*x)^11)/(48048*a^6*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^{17}} \, dx &=-\frac{(a+b x)^{11}}{16 a x^{16}}-\frac{(5 b) \int \frac{(a+b x)^{10}}{x^{16}} \, dx}{16 a}\\ &=-\frac{(a+b x)^{11}}{16 a x^{16}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}+\frac{b^2 \int \frac{(a+b x)^{10}}{x^{15}} \, dx}{12 a^2}\\ &=-\frac{(a+b x)^{11}}{16 a x^{16}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}-\frac{b^2 (a+b x)^{11}}{168 a^3 x^{14}}-\frac{b^3 \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{56 a^3}\\ &=-\frac{(a+b x)^{11}}{16 a x^{16}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}-\frac{b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac{b^3 (a+b x)^{11}}{728 a^4 x^{13}}+\frac{b^4 \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{364 a^4}\\ &=-\frac{(a+b x)^{11}}{16 a x^{16}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}-\frac{b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac{b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac{b^4 (a+b x)^{11}}{4368 a^5 x^{12}}-\frac{b^5 \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{4368 a^5}\\ &=-\frac{(a+b x)^{11}}{16 a x^{16}}+\frac{b (a+b x)^{11}}{48 a^2 x^{15}}-\frac{b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac{b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac{b^4 (a+b x)^{11}}{4368 a^5 x^{12}}+\frac{b^5 (a+b x)^{11}}{48048 a^6 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0049497, size = 132, normalized size = 1.14 \[ -\frac{45 a^8 b^2}{14 x^{14}}-\frac{120 a^7 b^3}{13 x^{13}}-\frac{35 a^6 b^4}{2 x^{12}}-\frac{252 a^5 b^5}{11 x^{11}}-\frac{21 a^4 b^6}{x^{10}}-\frac{40 a^3 b^7}{3 x^9}-\frac{45 a^2 b^8}{8 x^8}-\frac{2 a^9 b}{3 x^{15}}-\frac{a^{10}}{16 x^{16}}-\frac{10 a b^9}{7 x^7}-\frac{b^{10}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^17,x]

[Out]

-a^10/(16*x^16) - (2*a^9*b)/(3*x^15) - (45*a^8*b^2)/(14*x^14) - (120*a^7*b^3)/(13*x^13) - (35*a^6*b^4)/(2*x^12
) - (252*a^5*b^5)/(11*x^11) - (21*a^4*b^6)/x^10 - (40*a^3*b^7)/(3*x^9) - (45*a^2*b^8)/(8*x^8) - (10*a*b^9)/(7*
x^7) - b^10/(6*x^6)

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Maple [A]  time = 0.007, size = 113, normalized size = 1. \begin{align*} -21\,{\frac{{a}^{4}{b}^{6}}{{x}^{10}}}-{\frac{35\,{a}^{6}{b}^{4}}{2\,{x}^{12}}}-{\frac{2\,{a}^{9}b}{3\,{x}^{15}}}-{\frac{252\,{a}^{5}{b}^{5}}{11\,{x}^{11}}}-{\frac{{b}^{10}}{6\,{x}^{6}}}-{\frac{45\,{a}^{2}{b}^{8}}{8\,{x}^{8}}}-{\frac{120\,{a}^{7}{b}^{3}}{13\,{x}^{13}}}-{\frac{10\,a{b}^{9}}{7\,{x}^{7}}}-{\frac{{a}^{10}}{16\,{x}^{16}}}-{\frac{40\,{a}^{3}{b}^{7}}{3\,{x}^{9}}}-{\frac{45\,{a}^{8}{b}^{2}}{14\,{x}^{14}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^17,x)

[Out]

-21*a^4*b^6/x^10-35/2*a^6*b^4/x^12-2/3*a^9*b/x^15-252/11*a^5*b^5/x^11-1/6*b^10/x^6-45/8*a^2*b^8/x^8-120/13*a^7
*b^3/x^13-10/7*a*b^9/x^7-1/16*a^10/x^16-40/3*a^3*b^7/x^9-45/14*a^8*b^2/x^14

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Maxima [A]  time = 1.08816, size = 151, normalized size = 1.3 \begin{align*} -\frac{8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^17,x, algorithm="maxima")

[Out]

-1/48048*(8008*b^10*x^10 + 68640*a*b^9*x^9 + 270270*a^2*b^8*x^8 + 640640*a^3*b^7*x^7 + 1009008*a^4*b^6*x^6 + 1
100736*a^5*b^5*x^5 + 840840*a^6*b^4*x^4 + 443520*a^7*b^3*x^3 + 154440*a^8*b^2*x^2 + 32032*a^9*b*x + 3003*a^10)
/x^16

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Fricas [A]  time = 1.56903, size = 306, normalized size = 2.64 \begin{align*} -\frac{8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^17,x, algorithm="fricas")

[Out]

-1/48048*(8008*b^10*x^10 + 68640*a*b^9*x^9 + 270270*a^2*b^8*x^8 + 640640*a^3*b^7*x^7 + 1009008*a^4*b^6*x^6 + 1
100736*a^5*b^5*x^5 + 840840*a^6*b^4*x^4 + 443520*a^7*b^3*x^3 + 154440*a^8*b^2*x^2 + 32032*a^9*b*x + 3003*a^10)
/x^16

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Sympy [A]  time = 2.09986, size = 121, normalized size = 1.04 \begin{align*} - \frac{3003 a^{10} + 32032 a^{9} b x + 154440 a^{8} b^{2} x^{2} + 443520 a^{7} b^{3} x^{3} + 840840 a^{6} b^{4} x^{4} + 1100736 a^{5} b^{5} x^{5} + 1009008 a^{4} b^{6} x^{6} + 640640 a^{3} b^{7} x^{7} + 270270 a^{2} b^{8} x^{8} + 68640 a b^{9} x^{9} + 8008 b^{10} x^{10}}{48048 x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**17,x)

[Out]

-(3003*a**10 + 32032*a**9*b*x + 154440*a**8*b**2*x**2 + 443520*a**7*b**3*x**3 + 840840*a**6*b**4*x**4 + 110073
6*a**5*b**5*x**5 + 1009008*a**4*b**6*x**6 + 640640*a**3*b**7*x**7 + 270270*a**2*b**8*x**8 + 68640*a*b**9*x**9
+ 8008*b**10*x**10)/(48048*x**16)

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Giac [A]  time = 1.12558, size = 151, normalized size = 1.3 \begin{align*} -\frac{8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^17,x, algorithm="giac")

[Out]

-1/48048*(8008*b^10*x^10 + 68640*a*b^9*x^9 + 270270*a^2*b^8*x^8 + 640640*a^3*b^7*x^7 + 1009008*a^4*b^6*x^6 + 1
100736*a^5*b^5*x^5 + 840840*a^6*b^4*x^4 + 443520*a^7*b^3*x^3 + 154440*a^8*b^2*x^2 + 32032*a^9*b*x + 3003*a^10)
/x^16