3.154 \(\int \frac{(a+b x)^{10}}{x^{20}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{45 a^8 b^2}{17 x^{17}}-\frac{15 a^7 b^3}{2 x^{16}}-\frac{14 a^6 b^4}{x^{15}}-\frac{18 a^5 b^5}{x^{14}}-\frac{210 a^4 b^6}{13 x^{13}}-\frac{10 a^3 b^7}{x^{12}}-\frac{45 a^2 b^8}{11 x^{11}}-\frac{5 a^9 b}{9 x^{18}}-\frac{a^{10}}{19 x^{19}}-\frac{a b^9}{x^{10}}-\frac{b^{10}}{9 x^9} \]

[Out]

-a^10/(19*x^19) - (5*a^9*b)/(9*x^18) - (45*a^8*b^2)/(17*x^17) - (15*a^7*b^3)/(2*x^16) - (14*a^6*b^4)/x^15 - (1
8*a^5*b^5)/x^14 - (210*a^4*b^6)/(13*x^13) - (10*a^3*b^7)/x^12 - (45*a^2*b^8)/(11*x^11) - (a*b^9)/x^10 - b^10/(
9*x^9)

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Rubi [A]  time = 0.0493592, antiderivative size = 126, 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} \[ -\frac{45 a^8 b^2}{17 x^{17}}-\frac{15 a^7 b^3}{2 x^{16}}-\frac{14 a^6 b^4}{x^{15}}-\frac{18 a^5 b^5}{x^{14}}-\frac{210 a^4 b^6}{13 x^{13}}-\frac{10 a^3 b^7}{x^{12}}-\frac{45 a^2 b^8}{11 x^{11}}-\frac{5 a^9 b}{9 x^{18}}-\frac{a^{10}}{19 x^{19}}-\frac{a b^9}{x^{10}}-\frac{b^{10}}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^20,x]

[Out]

-a^10/(19*x^19) - (5*a^9*b)/(9*x^18) - (45*a^8*b^2)/(17*x^17) - (15*a^7*b^3)/(2*x^16) - (14*a^6*b^4)/x^15 - (1
8*a^5*b^5)/x^14 - (210*a^4*b^6)/(13*x^13) - (10*a^3*b^7)/x^12 - (45*a^2*b^8)/(11*x^11) - (a*b^9)/x^10 - b^10/(
9*x^9)

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)^{10}}{x^{20}} \, dx &=\int \left (\frac{a^{10}}{x^{20}}+\frac{10 a^9 b}{x^{19}}+\frac{45 a^8 b^2}{x^{18}}+\frac{120 a^7 b^3}{x^{17}}+\frac{210 a^6 b^4}{x^{16}}+\frac{252 a^5 b^5}{x^{15}}+\frac{210 a^4 b^6}{x^{14}}+\frac{120 a^3 b^7}{x^{13}}+\frac{45 a^2 b^8}{x^{12}}+\frac{10 a b^9}{x^{11}}+\frac{b^{10}}{x^{10}}\right ) \, dx\\ &=-\frac{a^{10}}{19 x^{19}}-\frac{5 a^9 b}{9 x^{18}}-\frac{45 a^8 b^2}{17 x^{17}}-\frac{15 a^7 b^3}{2 x^{16}}-\frac{14 a^6 b^4}{x^{15}}-\frac{18 a^5 b^5}{x^{14}}-\frac{210 a^4 b^6}{13 x^{13}}-\frac{10 a^3 b^7}{x^{12}}-\frac{45 a^2 b^8}{11 x^{11}}-\frac{a b^9}{x^{10}}-\frac{b^{10}}{9 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0072441, size = 126, normalized size = 1. \[ -\frac{45 a^8 b^2}{17 x^{17}}-\frac{15 a^7 b^3}{2 x^{16}}-\frac{14 a^6 b^4}{x^{15}}-\frac{18 a^5 b^5}{x^{14}}-\frac{210 a^4 b^6}{13 x^{13}}-\frac{10 a^3 b^7}{x^{12}}-\frac{45 a^2 b^8}{11 x^{11}}-\frac{5 a^9 b}{9 x^{18}}-\frac{a^{10}}{19 x^{19}}-\frac{a b^9}{x^{10}}-\frac{b^{10}}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^20,x]

[Out]

-a^10/(19*x^19) - (5*a^9*b)/(9*x^18) - (45*a^8*b^2)/(17*x^17) - (15*a^7*b^3)/(2*x^16) - (14*a^6*b^4)/x^15 - (1
8*a^5*b^5)/x^14 - (210*a^4*b^6)/(13*x^13) - (10*a^3*b^7)/x^12 - (45*a^2*b^8)/(11*x^11) - (a*b^9)/x^10 - b^10/(
9*x^9)

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Maple [A]  time = 0.007, size = 113, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{19\,{x}^{19}}}-{\frac{5\,{a}^{9}b}{9\,{x}^{18}}}-{\frac{45\,{a}^{8}{b}^{2}}{17\,{x}^{17}}}-{\frac{15\,{a}^{7}{b}^{3}}{2\,{x}^{16}}}-14\,{\frac{{a}^{6}{b}^{4}}{{x}^{15}}}-18\,{\frac{{a}^{5}{b}^{5}}{{x}^{14}}}-{\frac{210\,{a}^{4}{b}^{6}}{13\,{x}^{13}}}-10\,{\frac{{a}^{3}{b}^{7}}{{x}^{12}}}-{\frac{45\,{a}^{2}{b}^{8}}{11\,{x}^{11}}}-{\frac{a{b}^{9}}{{x}^{10}}}-{\frac{{b}^{10}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^20,x)

[Out]

-1/19*a^10/x^19-5/9*a^9*b/x^18-45/17*a^8*b^2/x^17-15/2*a^7*b^3/x^16-14*a^6*b^4/x^15-18*a^5*b^5/x^14-210/13*a^4
*b^6/x^13-10*a^3*b^7/x^12-45/11*a^2*b^8/x^11-a*b^9/x^10-1/9*b^10/x^9

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Maxima [A]  time = 1.0694, size = 151, normalized size = 1.2 \begin{align*} -\frac{92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^20,x, algorithm="maxima")

[Out]

-1/831402*(92378*b^10*x^10 + 831402*a*b^9*x^9 + 3401190*a^2*b^8*x^8 + 8314020*a^3*b^7*x^7 + 13430340*a^4*b^6*x
^6 + 14965236*a^5*b^5*x^5 + 11639628*a^6*b^4*x^4 + 6235515*a^7*b^3*x^3 + 2200770*a^8*b^2*x^2 + 461890*a^9*b*x
+ 43758*a^10)/x^19

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Fricas [A]  time = 1.50204, size = 324, normalized size = 2.57 \begin{align*} -\frac{92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^20,x, algorithm="fricas")

[Out]

-1/831402*(92378*b^10*x^10 + 831402*a*b^9*x^9 + 3401190*a^2*b^8*x^8 + 8314020*a^3*b^7*x^7 + 13430340*a^4*b^6*x
^6 + 14965236*a^5*b^5*x^5 + 11639628*a^6*b^4*x^4 + 6235515*a^7*b^3*x^3 + 2200770*a^8*b^2*x^2 + 461890*a^9*b*x
+ 43758*a^10)/x^19

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Sympy [A]  time = 2.03177, size = 121, normalized size = 0.96 \begin{align*} - \frac{43758 a^{10} + 461890 a^{9} b x + 2200770 a^{8} b^{2} x^{2} + 6235515 a^{7} b^{3} x^{3} + 11639628 a^{6} b^{4} x^{4} + 14965236 a^{5} b^{5} x^{5} + 13430340 a^{4} b^{6} x^{6} + 8314020 a^{3} b^{7} x^{7} + 3401190 a^{2} b^{8} x^{8} + 831402 a b^{9} x^{9} + 92378 b^{10} x^{10}}{831402 x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**20,x)

[Out]

-(43758*a**10 + 461890*a**9*b*x + 2200770*a**8*b**2*x**2 + 6235515*a**7*b**3*x**3 + 11639628*a**6*b**4*x**4 +
14965236*a**5*b**5*x**5 + 13430340*a**4*b**6*x**6 + 8314020*a**3*b**7*x**7 + 3401190*a**2*b**8*x**8 + 831402*a
*b**9*x**9 + 92378*b**10*x**10)/(831402*x**19)

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Giac [A]  time = 1.19186, size = 151, normalized size = 1.2 \begin{align*} -\frac{92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^20,x, algorithm="giac")

[Out]

-1/831402*(92378*b^10*x^10 + 831402*a*b^9*x^9 + 3401190*a^2*b^8*x^8 + 8314020*a^3*b^7*x^7 + 13430340*a^4*b^6*x
^6 + 14965236*a^5*b^5*x^5 + 11639628*a^6*b^4*x^4 + 6235515*a^7*b^3*x^3 + 2200770*a^8*b^2*x^2 + 461890*a^9*b*x
+ 43758*a^10)/x^19