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

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

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Rubi [A]  time = 0.05, antiderivative size = 115, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 115, normalized size = 1.00 \begin {gather*} -\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}+120 a^3 b^7 \log (x)+45 a^2 b^8 x+5 a b^9 x^2+\frac {b^{10} x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^8,x]

[Out]

-1/7*a^10/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]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10}}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^10/x^8,x]

[Out]

IntegrateAlgebraic[(a + b*x)^10/x^8, x]

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fricas [A]  time = 1.31, size = 114, normalized size = 0.99 \begin {gather*} \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 \relax (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}}{21 \, x^{7}} \end {gather*}

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

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giac [A]  time = 1.10, size = 111, normalized size = 0.97 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 110, normalized size = 0.96 \begin {gather*} \frac {b^{10} x^{3}}{3}+5 a \,b^{9} x^{2}+120 a^{3} b^{7} \ln \relax (x )+45 a^{2} b^{8} x -\frac {210 a^{4} b^{6}}{x}-\frac {126 a^{5} b^{5}}{x^{2}}-\frac {70 a^{6} b^{4}}{x^{3}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {9 a^{8} b^{2}}{x^{5}}-\frac {5 a^{9} b}{3 x^{6}}-\frac {a^{10}}{7 x^{7}} \end {gather*}

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)

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maxima [A]  time = 1.30, size = 110, normalized size = 0.96 \begin {gather*} \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 \relax (x) - \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 {gather*}

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

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mupad [B]  time = 0.10, size = 110, normalized size = 0.96 \begin {gather*} \frac {b^{10}\,x^3}{3}-\frac {\frac {a^{10}}{7}+\frac {5\,a^9\,b\,x}{3}+9\,a^8\,b^2\,x^2+30\,a^7\,b^3\,x^3+70\,a^6\,b^4\,x^4+126\,a^5\,b^5\,x^5+210\,a^4\,b^6\,x^6}{x^7}+45\,a^2\,b^8\,x+5\,a\,b^9\,x^2+120\,a^3\,b^7\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^8,x)

[Out]

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

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sympy [A]  time = 0.70, size = 119, normalized size = 1.03 \begin {gather*} 120 a^{3} b^{7} \log {\relax (x )} + 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 {gather*}

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)

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