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

Optimal. Leaf size=114 \[ -\frac{45 a^8 b^2}{7 x^7}-\frac{20 a^7 b^3}{x^6}-\frac{42 a^6 b^4}{x^5}-\frac{63 a^5 b^5}{x^4}-\frac{70 a^4 b^6}{x^3}-\frac{60 a^3 b^7}{x^2}-\frac{45 a^2 b^8}{x}-\frac{5 a^9 b}{4 x^8}-\frac{a^{10}}{9 x^9}+10 a b^9 \log (x)+b^{10} x \]

[Out]

-a^10/(9*x^9) - (5*a^9*b)/(4*x^8) - (45*a^8*b^2)/(7*x^7) - (20*a^7*b^3)/x^6 - (42*a^6*b^4)/x^5 - (63*a^5*b^5)/
x^4 - (70*a^4*b^6)/x^3 - (60*a^3*b^7)/x^2 - (45*a^2*b^8)/x + b^10*x + 10*a*b^9*Log[x]

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Rubi [A]  time = 0.0530741, antiderivative size = 114, 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} \[ -\frac{45 a^8 b^2}{7 x^7}-\frac{20 a^7 b^3}{x^6}-\frac{42 a^6 b^4}{x^5}-\frac{63 a^5 b^5}{x^4}-\frac{70 a^4 b^6}{x^3}-\frac{60 a^3 b^7}{x^2}-\frac{45 a^2 b^8}{x}-\frac{5 a^9 b}{4 x^8}-\frac{a^{10}}{9 x^9}+10 a b^9 \log (x)+b^{10} x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^10,x]

[Out]

-a^10/(9*x^9) - (5*a^9*b)/(4*x^8) - (45*a^8*b^2)/(7*x^7) - (20*a^7*b^3)/x^6 - (42*a^6*b^4)/x^5 - (63*a^5*b^5)/
x^4 - (70*a^4*b^6)/x^3 - (60*a^3*b^7)/x^2 - (45*a^2*b^8)/x + b^10*x + 10*a*b^9*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^{10}} \, dx &=\int \left (b^{10}+\frac{a^{10}}{x^{10}}+\frac{10 a^9 b}{x^9}+\frac{45 a^8 b^2}{x^8}+\frac{120 a^7 b^3}{x^7}+\frac{210 a^6 b^4}{x^6}+\frac{252 a^5 b^5}{x^5}+\frac{210 a^4 b^6}{x^4}+\frac{120 a^3 b^7}{x^3}+\frac{45 a^2 b^8}{x^2}+\frac{10 a b^9}{x}\right ) \, dx\\ &=-\frac{a^{10}}{9 x^9}-\frac{5 a^9 b}{4 x^8}-\frac{45 a^8 b^2}{7 x^7}-\frac{20 a^7 b^3}{x^6}-\frac{42 a^6 b^4}{x^5}-\frac{63 a^5 b^5}{x^4}-\frac{70 a^4 b^6}{x^3}-\frac{60 a^3 b^7}{x^2}-\frac{45 a^2 b^8}{x}+b^{10} x+10 a b^9 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0077861, size = 114, normalized size = 1. \[ -\frac{45 a^8 b^2}{7 x^7}-\frac{20 a^7 b^3}{x^6}-\frac{42 a^6 b^4}{x^5}-\frac{63 a^5 b^5}{x^4}-\frac{70 a^4 b^6}{x^3}-\frac{60 a^3 b^7}{x^2}-\frac{45 a^2 b^8}{x}-\frac{5 a^9 b}{4 x^8}-\frac{a^{10}}{9 x^9}+10 a b^9 \log (x)+b^{10} x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^10,x]

[Out]

-a^10/(9*x^9) - (5*a^9*b)/(4*x^8) - (45*a^8*b^2)/(7*x^7) - (20*a^7*b^3)/x^6 - (42*a^6*b^4)/x^5 - (63*a^5*b^5)/
x^4 - (70*a^4*b^6)/x^3 - (60*a^3*b^7)/x^2 - (45*a^2*b^8)/x + b^10*x + 10*a*b^9*Log[x]

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Maple [A]  time = 0.008, size = 109, normalized size = 1. \begin{align*} -{\frac{{a}^{10}}{9\,{x}^{9}}}-{\frac{5\,{a}^{9}b}{4\,{x}^{8}}}-{\frac{45\,{a}^{8}{b}^{2}}{7\,{x}^{7}}}-20\,{\frac{{a}^{7}{b}^{3}}{{x}^{6}}}-42\,{\frac{{a}^{6}{b}^{4}}{{x}^{5}}}-63\,{\frac{{a}^{5}{b}^{5}}{{x}^{4}}}-70\,{\frac{{a}^{4}{b}^{6}}{{x}^{3}}}-60\,{\frac{{a}^{3}{b}^{7}}{{x}^{2}}}-45\,{\frac{{a}^{2}{b}^{8}}{x}}+{b}^{10}x+10\,a{b}^{9}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^10/x^9-5/4*a^9*b/x^8-45/7*a^8*b^2/x^7-20*a^7*b^3/x^6-42*a^6*b^4/x^5-63*a^5*b^5/x^4-70*a^4*b^6/x^3-60*a^
3*b^7/x^2-45*a^2*b^8/x+b^10*x+10*a*b^9*ln(x)

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Maxima [A]  time = 1.03206, size = 147, normalized size = 1.29 \begin{align*} b^{10} x + 10 \, a b^{9} \log \left (x\right ) - \frac{11340 \, a^{2} b^{8} x^{8} + 15120 \, a^{3} b^{7} x^{7} + 17640 \, a^{4} b^{6} x^{6} + 15876 \, a^{5} b^{5} x^{5} + 10584 \, a^{6} b^{4} x^{4} + 5040 \, a^{7} b^{3} x^{3} + 1620 \, a^{8} b^{2} x^{2} + 315 \, a^{9} b x + 28 \, a^{10}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^10*x + 10*a*b^9*log(x) - 1/252*(11340*a^2*b^8*x^8 + 15120*a^3*b^7*x^7 + 17640*a^4*b^6*x^6 + 15876*a^5*b^5*x^
5 + 10584*a^6*b^4*x^4 + 5040*a^7*b^3*x^3 + 1620*a^8*b^2*x^2 + 315*a^9*b*x + 28*a^10)/x^9

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Fricas [A]  time = 1.9087, size = 288, normalized size = 2.53 \begin{align*} \frac{252 \, b^{10} x^{10} + 2520 \, a b^{9} x^{9} \log \left (x\right ) - 11340 \, a^{2} b^{8} x^{8} - 15120 \, a^{3} b^{7} x^{7} - 17640 \, a^{4} b^{6} x^{6} - 15876 \, a^{5} b^{5} x^{5} - 10584 \, a^{6} b^{4} x^{4} - 5040 \, a^{7} b^{3} x^{3} - 1620 \, a^{8} b^{2} x^{2} - 315 \, a^{9} b x - 28 \, a^{10}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/252*(252*b^10*x^10 + 2520*a*b^9*x^9*log(x) - 11340*a^2*b^8*x^8 - 15120*a^3*b^7*x^7 - 17640*a^4*b^6*x^6 - 158
76*a^5*b^5*x^5 - 10584*a^6*b^4*x^4 - 5040*a^7*b^3*x^3 - 1620*a^8*b^2*x^2 - 315*a^9*b*x - 28*a^10)/x^9

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Sympy [A]  time = 1.11039, size = 116, normalized size = 1.02 \begin{align*} 10 a b^{9} \log{\left (x \right )} + b^{10} x - \frac{28 a^{10} + 315 a^{9} b x + 1620 a^{8} b^{2} x^{2} + 5040 a^{7} b^{3} x^{3} + 10584 a^{6} b^{4} x^{4} + 15876 a^{5} b^{5} x^{5} + 17640 a^{4} b^{6} x^{6} + 15120 a^{3} b^{7} x^{7} + 11340 a^{2} b^{8} x^{8}}{252 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*a*b**9*log(x) + b**10*x - (28*a**10 + 315*a**9*b*x + 1620*a**8*b**2*x**2 + 5040*a**7*b**3*x**3 + 10584*a**6
*b**4*x**4 + 15876*a**5*b**5*x**5 + 17640*a**4*b**6*x**6 + 15120*a**3*b**7*x**7 + 11340*a**2*b**8*x**8)/(252*x
**9)

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Giac [A]  time = 1.23941, size = 149, normalized size = 1.31 \begin{align*} b^{10} x + 10 \, a b^{9} \log \left ({\left | x \right |}\right ) - \frac{11340 \, a^{2} b^{8} x^{8} + 15120 \, a^{3} b^{7} x^{7} + 17640 \, a^{4} b^{6} x^{6} + 15876 \, a^{5} b^{5} x^{5} + 10584 \, a^{6} b^{4} x^{4} + 5040 \, a^{7} b^{3} x^{3} + 1620 \, a^{8} b^{2} x^{2} + 315 \, a^{9} b x + 28 \, a^{10}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^10*x + 10*a*b^9*log(abs(x)) - 1/252*(11340*a^2*b^8*x^8 + 15120*a^3*b^7*x^7 + 17640*a^4*b^6*x^6 + 15876*a^5*b
^5*x^5 + 10584*a^6*b^4*x^4 + 5040*a^7*b^3*x^3 + 1620*a^8*b^2*x^2 + 315*a^9*b*x + 28*a^10)/x^9