3.143 \(\int \frac {(a+b x)^{10}}{x^9} \, dx\)

Optimal. Leaf size=119 \[ -\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+45 a^2 b^8 \log (x)+10 a b^9 x+\frac {b^{10} x^2}{2} \]

[Out]

-1/8*a^10/x^8-10/7*a^9*b/x^7-15/2*a^8*b^2/x^6-24*a^7*b^3/x^5-105/2*a^6*b^4/x^4-84*a^5*b^5/x^3-105*a^4*b^6/x^2-
120*a^3*b^7/x+10*a*b^9*x+1/2*b^10*x^2+45*a^2*b^8*ln(x)

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Rubi [A]  time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, 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} \[ -\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+45 a^2 b^8 \log (x)-\frac {10 a^9 b}{7 x^7}-\frac {a^{10}}{8 x^8}+10 a b^9 x+\frac {b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^9,x]

[Out]

-a^10/(8*x^8) - (10*a^9*b)/(7*x^7) - (15*a^8*b^2)/(2*x^6) - (24*a^7*b^3)/x^5 - (105*a^6*b^4)/(2*x^4) - (84*a^5
*b^5)/x^3 - (105*a^4*b^6)/x^2 - (120*a^3*b^7)/x + 10*a*b^9*x + (b^10*x^2)/2 + 45*a^2*b^8*Log[x]

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^9} \, dx &=\int \left (10 a b^9+\frac {a^{10}}{x^9}+\frac {10 a^9 b}{x^8}+\frac {45 a^8 b^2}{x^7}+\frac {120 a^7 b^3}{x^6}+\frac {210 a^6 b^4}{x^5}+\frac {252 a^5 b^5}{x^4}+\frac {210 a^4 b^6}{x^3}+\frac {120 a^3 b^7}{x^2}+\frac {45 a^2 b^8}{x}+b^{10} x\right ) \, dx\\ &=-\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+10 a b^9 x+\frac {b^{10} x^2}{2}+45 a^2 b^8 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 119, normalized size = 1.00 \[ -\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+45 a^2 b^8 \log (x)+10 a b^9 x+\frac {b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^9,x]

[Out]

-1/8*a^10/x^8 - (10*a^9*b)/(7*x^7) - (15*a^8*b^2)/(2*x^6) - (24*a^7*b^3)/x^5 - (105*a^6*b^4)/(2*x^4) - (84*a^5
*b^5)/x^3 - (105*a^4*b^6)/x^2 - (120*a^3*b^7)/x + 10*a*b^9*x + (b^10*x^2)/2 + 45*a^2*b^8*Log[x]

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fricas [A]  time = 0.45, size = 114, normalized size = 0.96 \[ \frac {28 \, b^{10} x^{10} + 560 \, a b^{9} x^{9} + 2520 \, a^{2} b^{8} x^{8} \log \relax (x) - 6720 \, a^{3} b^{7} x^{7} - 5880 \, a^{4} b^{6} x^{6} - 4704 \, a^{5} b^{5} x^{5} - 2940 \, a^{6} b^{4} x^{4} - 1344 \, a^{7} b^{3} x^{3} - 420 \, a^{8} b^{2} x^{2} - 80 \, a^{9} b x - 7 \, a^{10}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^9,x, algorithm="fricas")

[Out]

1/56*(28*b^10*x^10 + 560*a*b^9*x^9 + 2520*a^2*b^8*x^8*log(x) - 6720*a^3*b^7*x^7 - 5880*a^4*b^6*x^6 - 4704*a^5*
b^5*x^5 - 2940*a^6*b^4*x^4 - 1344*a^7*b^3*x^3 - 420*a^8*b^2*x^2 - 80*a^9*b*x - 7*a^10)/x^8

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giac [A]  time = 1.16, size = 111, normalized size = 0.93 \[ \frac {1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8} \log \left ({\left | x \right |}\right ) - \frac {6720 \, a^{3} b^{7} x^{7} + 5880 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 2940 \, a^{6} b^{4} x^{4} + 1344 \, a^{7} b^{3} x^{3} + 420 \, a^{8} b^{2} x^{2} + 80 \, a^{9} b x + 7 \, a^{10}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^9,x, algorithm="giac")

[Out]

1/2*b^10*x^2 + 10*a*b^9*x + 45*a^2*b^8*log(abs(x)) - 1/56*(6720*a^3*b^7*x^7 + 5880*a^4*b^6*x^6 + 4704*a^5*b^5*
x^5 + 2940*a^6*b^4*x^4 + 1344*a^7*b^3*x^3 + 420*a^8*b^2*x^2 + 80*a^9*b*x + 7*a^10)/x^8

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maple [A]  time = 0.01, size = 110, normalized size = 0.92 \[ \frac {b^{10} x^{2}}{2}+45 a^{2} b^{8} \ln \relax (x )+10 a \,b^{9} x -\frac {120 a^{3} b^{7}}{x}-\frac {105 a^{4} b^{6}}{x^{2}}-\frac {84 a^{5} b^{5}}{x^{3}}-\frac {105 a^{6} b^{4}}{2 x^{4}}-\frac {24 a^{7} b^{3}}{x^{5}}-\frac {15 a^{8} b^{2}}{2 x^{6}}-\frac {10 a^{9} b}{7 x^{7}}-\frac {a^{10}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^9,x)

[Out]

-1/8*a^10/x^8-10/7*a^9*b/x^7-15/2*a^8*b^2/x^6-24*a^7*b^3/x^5-105/2*a^6*b^4/x^4-84*a^5*b^5/x^3-105*a^4*b^6/x^2-
120*a^3*b^7/x+10*a*b^9*x+1/2*b^10*x^2+45*a^2*b^8*ln(x)

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maxima [A]  time = 1.42, size = 110, normalized size = 0.92 \[ \frac {1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8} \log \relax (x) - \frac {6720 \, a^{3} b^{7} x^{7} + 5880 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 2940 \, a^{6} b^{4} x^{4} + 1344 \, a^{7} b^{3} x^{3} + 420 \, a^{8} b^{2} x^{2} + 80 \, a^{9} b x + 7 \, a^{10}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^9,x, algorithm="maxima")

[Out]

1/2*b^10*x^2 + 10*a*b^9*x + 45*a^2*b^8*log(x) - 1/56*(6720*a^3*b^7*x^7 + 5880*a^4*b^6*x^6 + 4704*a^5*b^5*x^5 +
2940*a^6*b^4*x^4 + 1344*a^7*b^3*x^3 + 420*a^8*b^2*x^2 + 80*a^9*b*x + 7*a^10)/x^8

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mupad [B]  time = 0.07, size = 110, normalized size = 0.92 \[ \frac {b^{10}\,x^2}{2}-\frac {\frac {a^{10}}{8}+\frac {10\,a^9\,b\,x}{7}+\frac {15\,a^8\,b^2\,x^2}{2}+24\,a^7\,b^3\,x^3+\frac {105\,a^6\,b^4\,x^4}{2}+84\,a^5\,b^5\,x^5+105\,a^4\,b^6\,x^6+120\,a^3\,b^7\,x^7}{x^8}+45\,a^2\,b^8\,\ln \relax (x)+10\,a\,b^9\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^9,x)

[Out]

(b^10*x^2)/2 - (a^10/8 + (15*a^8*b^2*x^2)/2 + 24*a^7*b^3*x^3 + (105*a^6*b^4*x^4)/2 + 84*a^5*b^5*x^5 + 105*a^4*
b^6*x^6 + 120*a^3*b^7*x^7 + (10*a^9*b*x)/7)/x^8 + 45*a^2*b^8*log(x) + 10*a*b^9*x

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sympy [A]  time = 0.76, size = 119, normalized size = 1.00 \[ 45 a^{2} b^{8} \log {\relax (x )} + 10 a b^{9} x + \frac {b^{10} x^{2}}{2} + \frac {- 7 a^{10} - 80 a^{9} b x - 420 a^{8} b^{2} x^{2} - 1344 a^{7} b^{3} x^{3} - 2940 a^{6} b^{4} x^{4} - 4704 a^{5} b^{5} x^{5} - 5880 a^{4} b^{6} x^{6} - 6720 a^{3} b^{7} x^{7}}{56 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**9,x)

[Out]

45*a**2*b**8*log(x) + 10*a*b**9*x + b**10*x**2/2 + (-7*a**10 - 80*a**9*b*x - 420*a**8*b**2*x**2 - 1344*a**7*b*
*3*x**3 - 2940*a**6*b**4*x**4 - 4704*a**5*b**5*x**5 - 5880*a**4*b**6*x**6 - 6720*a**3*b**7*x**7)/(56*x**8)

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