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

Optimal. Leaf size=122 \[ \frac{45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac{105}{2} a^6 b^4 x^4+\frac{252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac{120}{7} a^3 b^7 x^7+\frac{45}{8} a^2 b^8 x^8+10 a^9 b x+a^{10} \log (x)+\frac{10}{9} a b^9 x^9+\frac{b^{10} x^{10}}{10} \]

[Out]

10*a^9*b*x + (45*a^8*b^2*x^2)/2 + 40*a^7*b^3*x^3 + (105*a^6*b^4*x^4)/2 + (252*a^5*b^5*x^5)/5 + 35*a^4*b^6*x^6
+ (120*a^3*b^7*x^7)/7 + (45*a^2*b^8*x^8)/8 + (10*a*b^9*x^9)/9 + (b^10*x^10)/10 + a^10*Log[x]

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Rubi [A]  time = 0.0433098, antiderivative size = 122, 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} \[ \frac{45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac{105}{2} a^6 b^4 x^4+\frac{252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac{120}{7} a^3 b^7 x^7+\frac{45}{8} a^2 b^8 x^8+10 a^9 b x+a^{10} \log (x)+\frac{10}{9} a b^9 x^9+\frac{b^{10} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x,x]

[Out]

10*a^9*b*x + (45*a^8*b^2*x^2)/2 + 40*a^7*b^3*x^3 + (105*a^6*b^4*x^4)/2 + (252*a^5*b^5*x^5)/5 + 35*a^4*b^6*x^6
+ (120*a^3*b^7*x^7)/7 + (45*a^2*b^8*x^8)/8 + (10*a*b^9*x^9)/9 + (b^10*x^10)/10 + a^10*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} \, dx &=\int \left (10 a^9 b+\frac{a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx\\ &=10 a^9 b x+\frac{45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac{105}{2} a^6 b^4 x^4+\frac{252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac{120}{7} a^3 b^7 x^7+\frac{45}{8} a^2 b^8 x^8+\frac{10}{9} a b^9 x^9+\frac{b^{10} x^{10}}{10}+a^{10} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0048686, size = 122, normalized size = 1. \[ \frac{45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac{105}{2} a^6 b^4 x^4+\frac{252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac{120}{7} a^3 b^7 x^7+\frac{45}{8} a^2 b^8 x^8+10 a^9 b x+a^{10} \log (x)+\frac{10}{9} a b^9 x^9+\frac{b^{10} x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x,x]

[Out]

10*a^9*b*x + (45*a^8*b^2*x^2)/2 + 40*a^7*b^3*x^3 + (105*a^6*b^4*x^4)/2 + (252*a^5*b^5*x^5)/5 + 35*a^4*b^6*x^6
+ (120*a^3*b^7*x^7)/7 + (45*a^2*b^8*x^8)/8 + (10*a*b^9*x^9)/9 + (b^10*x^10)/10 + a^10*Log[x]

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Maple [A]  time = 0.003, size = 109, normalized size = 0.9 \begin{align*} 10\,{a}^{9}bx+{\frac{45\,{a}^{8}{b}^{2}{x}^{2}}{2}}+40\,{a}^{7}{b}^{3}{x}^{3}+{\frac{105\,{a}^{6}{b}^{4}{x}^{4}}{2}}+{\frac{252\,{a}^{5}{b}^{5}{x}^{5}}{5}}+35\,{a}^{4}{b}^{6}{x}^{6}+{\frac{120\,{a}^{3}{b}^{7}{x}^{7}}{7}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{8}}{8}}+{\frac{10\,a{b}^{9}{x}^{9}}{9}}+{\frac{{b}^{10}{x}^{10}}{10}}+{a}^{10}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*a^9*b*x+45/2*a^8*b^2*x^2+40*a^7*b^3*x^3+105/2*a^6*b^4*x^4+252/5*a^5*b^5*x^5+35*a^4*b^6*x^6+120/7*a^3*b^7*x^
7+45/8*a^2*b^8*x^8+10/9*a*b^9*x^9+1/10*b^10*x^10+a^10*ln(x)

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Maxima [A]  time = 1.04289, size = 146, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{10} x^{10} + \frac{10}{9} \, a b^{9} x^{9} + \frac{45}{8} \, a^{2} b^{8} x^{8} + \frac{120}{7} \, a^{3} b^{7} x^{7} + 35 \, a^{4} b^{6} x^{6} + \frac{252}{5} \, a^{5} b^{5} x^{5} + \frac{105}{2} \, a^{6} b^{4} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac{45}{2} \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^10*x^10 + 10/9*a*b^9*x^9 + 45/8*a^2*b^8*x^8 + 120/7*a^3*b^7*x^7 + 35*a^4*b^6*x^6 + 252/5*a^5*b^5*x^5 +
105/2*a^6*b^4*x^4 + 40*a^7*b^3*x^3 + 45/2*a^8*b^2*x^2 + 10*a^9*b*x + a^10*log(x)

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Fricas [A]  time = 1.8874, size = 259, normalized size = 2.12 \begin{align*} \frac{1}{10} \, b^{10} x^{10} + \frac{10}{9} \, a b^{9} x^{9} + \frac{45}{8} \, a^{2} b^{8} x^{8} + \frac{120}{7} \, a^{3} b^{7} x^{7} + 35 \, a^{4} b^{6} x^{6} + \frac{252}{5} \, a^{5} b^{5} x^{5} + \frac{105}{2} \, a^{6} b^{4} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac{45}{2} \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^10*x^10 + 10/9*a*b^9*x^9 + 45/8*a^2*b^8*x^8 + 120/7*a^3*b^7*x^7 + 35*a^4*b^6*x^6 + 252/5*a^5*b^5*x^5 +
105/2*a^6*b^4*x^4 + 40*a^7*b^3*x^3 + 45/2*a^8*b^2*x^2 + 10*a^9*b*x + a^10*log(x)

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Sympy [A]  time = 0.553125, size = 126, normalized size = 1.03 \begin{align*} a^{10} \log{\left (x \right )} + 10 a^{9} b x + \frac{45 a^{8} b^{2} x^{2}}{2} + 40 a^{7} b^{3} x^{3} + \frac{105 a^{6} b^{4} x^{4}}{2} + \frac{252 a^{5} b^{5} x^{5}}{5} + 35 a^{4} b^{6} x^{6} + \frac{120 a^{3} b^{7} x^{7}}{7} + \frac{45 a^{2} b^{8} x^{8}}{8} + \frac{10 a b^{9} x^{9}}{9} + \frac{b^{10} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*log(x) + 10*a**9*b*x + 45*a**8*b**2*x**2/2 + 40*a**7*b**3*x**3 + 105*a**6*b**4*x**4/2 + 252*a**5*b**5*x*
*5/5 + 35*a**4*b**6*x**6 + 120*a**3*b**7*x**7/7 + 45*a**2*b**8*x**8/8 + 10*a*b**9*x**9/9 + b**10*x**10/10

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Giac [A]  time = 1.13069, size = 147, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{10} x^{10} + \frac{10}{9} \, a b^{9} x^{9} + \frac{45}{8} \, a^{2} b^{8} x^{8} + \frac{120}{7} \, a^{3} b^{7} x^{7} + 35 \, a^{4} b^{6} x^{6} + \frac{252}{5} \, a^{5} b^{5} x^{5} + \frac{105}{2} \, a^{6} b^{4} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac{45}{2} \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^10*x^10 + 10/9*a*b^9*x^9 + 45/8*a^2*b^8*x^8 + 120/7*a^3*b^7*x^7 + 35*a^4*b^6*x^6 + 252/5*a^5*b^5*x^5 +
105/2*a^6*b^4*x^4 + 40*a^7*b^3*x^3 + 45/2*a^8*b^2*x^2 + 10*a^9*b*x + a^10*log(abs(x))