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

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

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.0489553, antiderivative size = 119, 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} \[ -\frac{15 a^8 b^2}{2 x^6}-\frac{24 a^7 b^3}{x^5}-\frac{105 a^6 b^4}{2 x^4}-\frac{84 a^5 b^5}{x^3}-\frac{105 a^4 b^6}{x^2}-\frac{120 a^3 b^7}{x}+45 a^2 b^8 \log (x)-\frac{10 a^9 b}{7 x^7}-\frac{a^{10}}{8 x^8}+10 a b^9 x+\frac{b^{10} x^2}{2} \]

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*}

Mathematica [A]  time = 0.0058256, size = 119, normalized size = 1. \[ -\frac{15 a^8 b^2}{2 x^6}-\frac{24 a^7 b^3}{x^5}-\frac{105 a^6 b^4}{2 x^4}-\frac{84 a^5 b^5}{x^3}-\frac{105 a^4 b^6}{x^2}-\frac{120 a^3 b^7}{x}+45 a^2 b^8 \log (x)-\frac{10 a^9 b}{7 x^7}-\frac{a^{10}}{8 x^8}+10 a b^9 x+\frac{b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{8\,{x}^{8}}}-{\frac{10\,{a}^{9}b}{7\,{x}^{7}}}-{\frac{15\,{a}^{8}{b}^{2}}{2\,{x}^{6}}}-24\,{\frac{{a}^{7}{b}^{3}}{{x}^{5}}}-{\frac{105\,{a}^{6}{b}^{4}}{2\,{x}^{4}}}-84\,{\frac{{a}^{5}{b}^{5}}{{x}^{3}}}-105\,{\frac{{a}^{4}{b}^{6}}{{x}^{2}}}-120\,{\frac{{a}^{3}{b}^{7}}{x}}+10\,a{b}^{9}x+{\frac{{b}^{10}{x}^{2}}{2}}+45\,{a}^{2}{b}^{8}\ln \left ( x \right ) \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.03382, size = 149, normalized size = 1.25 \begin{align*} \frac{1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8} \log \left (x\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}} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.61997, size = 273, normalized size = 2.29 \begin{align*} \frac{28 \, b^{10} x^{10} + 560 \, a b^{9} x^{9} + 2520 \, a^{2} b^{8} x^{8} \log \left (x\right ) - 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}} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 1.10422, size = 117, normalized size = 0.98 \begin{align*} 45 a^{2} b^{8} \log{\left (x \right )} + 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}} \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.17808, size = 150, normalized size = 1.26 \begin{align*} \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}} \end{align*}

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