3.142 \(\int \frac{(a+b x)^{10}}{x^8} \, dx\)

Optimal. Leaf size=115 \[ -\frac{9 a^8 b^2}{x^5}-\frac{30 a^7 b^3}{x^4}-\frac{70 a^6 b^4}{x^3}-\frac{126 a^5 b^5}{x^2}-\frac{210 a^4 b^6}{x}+45 a^2 b^8 x+120 a^3 b^7 \log (x)-\frac{5 a^9 b}{3 x^6}-\frac{a^{10}}{7 x^7}+5 a b^9 x^2+\frac{b^{10} x^3}{3} \]

[Out]

-a^10/(7*x^7) - (5*a^9*b)/(3*x^6) - (9*a^8*b^2)/x^5 - (30*a^7*b^3)/x^4 - (70*a^6*b^4)/x^3 - (126*a^5*b^5)/x^2
- (210*a^4*b^6)/x + 45*a^2*b^8*x + 5*a*b^9*x^2 + (b^10*x^3)/3 + 120*a^3*b^7*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0496511, antiderivative size = 115, 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} \[ -\frac{9 a^8 b^2}{x^5}-\frac{30 a^7 b^3}{x^4}-\frac{70 a^6 b^4}{x^3}-\frac{126 a^5 b^5}{x^2}-\frac{210 a^4 b^6}{x}+45 a^2 b^8 x+120 a^3 b^7 \log (x)-\frac{5 a^9 b}{3 x^6}-\frac{a^{10}}{7 x^7}+5 a b^9 x^2+\frac{b^{10} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^8,x]

[Out]

-a^10/(7*x^7) - (5*a^9*b)/(3*x^6) - (9*a^8*b^2)/x^5 - (30*a^7*b^3)/x^4 - (70*a^6*b^4)/x^3 - (126*a^5*b^5)/x^2
- (210*a^4*b^6)/x + 45*a^2*b^8*x + 5*a*b^9*x^2 + (b^10*x^3)/3 + 120*a^3*b^7*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^8} \, dx &=\int \left (45 a^2 b^8+\frac{a^{10}}{x^8}+\frac{10 a^9 b}{x^7}+\frac{45 a^8 b^2}{x^6}+\frac{120 a^7 b^3}{x^5}+\frac{210 a^6 b^4}{x^4}+\frac{252 a^5 b^5}{x^3}+\frac{210 a^4 b^6}{x^2}+\frac{120 a^3 b^7}{x}+10 a b^9 x+b^{10} x^2\right ) \, dx\\ &=-\frac{a^{10}}{7 x^7}-\frac{5 a^9 b}{3 x^6}-\frac{9 a^8 b^2}{x^5}-\frac{30 a^7 b^3}{x^4}-\frac{70 a^6 b^4}{x^3}-\frac{126 a^5 b^5}{x^2}-\frac{210 a^4 b^6}{x}+45 a^2 b^8 x+5 a b^9 x^2+\frac{b^{10} x^3}{3}+120 a^3 b^7 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0159778, size = 115, normalized size = 1. \[ -\frac{9 a^8 b^2}{x^5}-\frac{30 a^7 b^3}{x^4}-\frac{70 a^6 b^4}{x^3}-\frac{126 a^5 b^5}{x^2}-\frac{210 a^4 b^6}{x}+45 a^2 b^8 x+120 a^3 b^7 \log (x)-\frac{5 a^9 b}{3 x^6}-\frac{a^{10}}{7 x^7}+5 a b^9 x^2+\frac{b^{10} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^8,x]

[Out]

-a^10/(7*x^7) - (5*a^9*b)/(3*x^6) - (9*a^8*b^2)/x^5 - (30*a^7*b^3)/x^4 - (70*a^6*b^4)/x^3 - (126*a^5*b^5)/x^2
- (210*a^4*b^6)/x + 45*a^2*b^8*x + 5*a*b^9*x^2 + (b^10*x^3)/3 + 120*a^3*b^7*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 110, normalized size = 1. \begin{align*} -{\frac{{a}^{10}}{7\,{x}^{7}}}-{\frac{5\,{a}^{9}b}{3\,{x}^{6}}}-9\,{\frac{{a}^{8}{b}^{2}}{{x}^{5}}}-30\,{\frac{{a}^{7}{b}^{3}}{{x}^{4}}}-70\,{\frac{{a}^{6}{b}^{4}}{{x}^{3}}}-126\,{\frac{{a}^{5}{b}^{5}}{{x}^{2}}}-210\,{\frac{{a}^{4}{b}^{6}}{x}}+45\,{a}^{2}{b}^{8}x+5\,a{b}^{9}{x}^{2}+{\frac{{b}^{10}{x}^{3}}{3}}+120\,{a}^{3}{b}^{7}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^8,x)

[Out]

-1/7*a^10/x^7-5/3*a^9*b/x^6-9*a^8*b^2/x^5-30*a^7*b^3/x^4-70*a^6*b^4/x^3-126*a^5*b^5/x^2-210*a^4*b^6/x+45*a^2*b
^8*x+5*a*b^9*x^2+1/3*b^10*x^3+120*a^3*b^7*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 1.05544, size = 149, normalized size = 1.3 \begin{align*} \frac{1}{3} \, b^{10} x^{3} + 5 \, a b^{9} x^{2} + 45 \, a^{2} b^{8} x + 120 \, a^{3} b^{7} \log \left (x\right ) - \frac{4410 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 1470 \, a^{6} b^{4} x^{4} + 630 \, a^{7} b^{3} x^{3} + 189 \, a^{8} b^{2} x^{2} + 35 \, a^{9} b x + 3 \, a^{10}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^8,x, algorithm="maxima")

[Out]

1/3*b^10*x^3 + 5*a*b^9*x^2 + 45*a^2*b^8*x + 120*a^3*b^7*log(x) - 1/21*(4410*a^4*b^6*x^6 + 2646*a^5*b^5*x^5 + 1
470*a^6*b^4*x^4 + 630*a^7*b^3*x^3 + 189*a^8*b^2*x^2 + 35*a^9*b*x + 3*a^10)/x^7

________________________________________________________________________________________

Fricas [A]  time = 1.59277, size = 269, normalized size = 2.34 \begin{align*} \frac{7 \, b^{10} x^{10} + 105 \, a b^{9} x^{9} + 945 \, a^{2} b^{8} x^{8} + 2520 \, a^{3} b^{7} x^{7} \log \left (x\right ) - 4410 \, a^{4} b^{6} x^{6} - 2646 \, a^{5} b^{5} x^{5} - 1470 \, a^{6} b^{4} x^{4} - 630 \, a^{7} b^{3} x^{3} - 189 \, a^{8} b^{2} x^{2} - 35 \, a^{9} b x - 3 \, a^{10}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^8,x, algorithm="fricas")

[Out]

1/21*(7*b^10*x^10 + 105*a*b^9*x^9 + 945*a^2*b^8*x^8 + 2520*a^3*b^7*x^7*log(x) - 4410*a^4*b^6*x^6 - 2646*a^5*b^
5*x^5 - 1470*a^6*b^4*x^4 - 630*a^7*b^3*x^3 - 189*a^8*b^2*x^2 - 35*a^9*b*x - 3*a^10)/x^7

________________________________________________________________________________________

Sympy [A]  time = 0.99561, size = 117, normalized size = 1.02 \begin{align*} 120 a^{3} b^{7} \log{\left (x \right )} + 45 a^{2} b^{8} x + 5 a b^{9} x^{2} + \frac{b^{10} x^{3}}{3} - \frac{3 a^{10} + 35 a^{9} b x + 189 a^{8} b^{2} x^{2} + 630 a^{7} b^{3} x^{3} + 1470 a^{6} b^{4} x^{4} + 2646 a^{5} b^{5} x^{5} + 4410 a^{4} b^{6} x^{6}}{21 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**8,x)

[Out]

120*a**3*b**7*log(x) + 45*a**2*b**8*x + 5*a*b**9*x**2 + b**10*x**3/3 - (3*a**10 + 35*a**9*b*x + 189*a**8*b**2*
x**2 + 630*a**7*b**3*x**3 + 1470*a**6*b**4*x**4 + 2646*a**5*b**5*x**5 + 4410*a**4*b**6*x**6)/(21*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.23328, size = 150, normalized size = 1.3 \begin{align*} \frac{1}{3} \, b^{10} x^{3} + 5 \, a b^{9} x^{2} + 45 \, a^{2} b^{8} x + 120 \, a^{3} b^{7} \log \left ({\left | x \right |}\right ) - \frac{4410 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 1470 \, a^{6} b^{4} x^{4} + 630 \, a^{7} b^{3} x^{3} + 189 \, a^{8} b^{2} x^{2} + 35 \, a^{9} b x + 3 \, a^{10}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^8,x, algorithm="giac")

[Out]

1/3*b^10*x^3 + 5*a*b^9*x^2 + 45*a^2*b^8*x + 120*a^3*b^7*log(abs(x)) - 1/21*(4410*a^4*b^6*x^6 + 2646*a^5*b^5*x^
5 + 1470*a^6*b^4*x^4 + 630*a^7*b^3*x^3 + 189*a^8*b^2*x^2 + 35*a^9*b*x + 3*a^10)/x^7