3.3.53 \(\int \frac {1}{x^{10} (a+b x)^2} \, dx\)

Optimal. Leaf size=146 \[ -\frac {10 b^9 \log (x)}{a^{11}}+\frac {10 b^9 \log (a+b x)}{a^{11}}-\frac {b^9}{a^{10} (a+b x)}-\frac {9 b^8}{a^{10} x}+\frac {4 b^7}{a^9 x^2}-\frac {7 b^6}{3 a^8 x^3}+\frac {3 b^5}{2 a^7 x^4}-\frac {b^4}{a^6 x^5}+\frac {2 b^3}{3 a^5 x^6}-\frac {3 b^2}{7 a^4 x^7}+\frac {b}{4 a^3 x^8}-\frac {1}{9 a^2 x^9} \]

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Rubi [A]  time = 0.09, antiderivative size = 146, 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 = {44} \begin {gather*} \frac {4 b^7}{a^9 x^2}-\frac {7 b^6}{3 a^8 x^3}+\frac {3 b^5}{2 a^7 x^4}-\frac {b^4}{a^6 x^5}+\frac {2 b^3}{3 a^5 x^6}-\frac {3 b^2}{7 a^4 x^7}-\frac {b^9}{a^{10} (a+b x)}-\frac {9 b^8}{a^{10} x}-\frac {10 b^9 \log (x)}{a^{11}}+\frac {10 b^9 \log (a+b x)}{a^{11}}+\frac {b}{4 a^3 x^8}-\frac {1}{9 a^2 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a + b*x)^2),x]

[Out]

-1/(9*a^2*x^9) + b/(4*a^3*x^8) - (3*b^2)/(7*a^4*x^7) + (2*b^3)/(3*a^5*x^6) - b^4/(a^6*x^5) + (3*b^5)/(2*a^7*x^
4) - (7*b^6)/(3*a^8*x^3) + (4*b^7)/(a^9*x^2) - (9*b^8)/(a^10*x) - b^9/(a^10*(a + b*x)) - (10*b^9*Log[x])/a^11
+ (10*b^9*Log[a + b*x])/a^11

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^{10} (a+b x)^2} \, dx &=\int \left (\frac {1}{a^2 x^{10}}-\frac {2 b}{a^3 x^9}+\frac {3 b^2}{a^4 x^8}-\frac {4 b^3}{a^5 x^7}+\frac {5 b^4}{a^6 x^6}-\frac {6 b^5}{a^7 x^5}+\frac {7 b^6}{a^8 x^4}-\frac {8 b^7}{a^9 x^3}+\frac {9 b^8}{a^{10} x^2}-\frac {10 b^9}{a^{11} x}+\frac {b^{10}}{a^{10} (a+b x)^2}+\frac {10 b^{10}}{a^{11} (a+b x)}\right ) \, dx\\ &=-\frac {1}{9 a^2 x^9}+\frac {b}{4 a^3 x^8}-\frac {3 b^2}{7 a^4 x^7}+\frac {2 b^3}{3 a^5 x^6}-\frac {b^4}{a^6 x^5}+\frac {3 b^5}{2 a^7 x^4}-\frac {7 b^6}{3 a^8 x^3}+\frac {4 b^7}{a^9 x^2}-\frac {9 b^8}{a^{10} x}-\frac {b^9}{a^{10} (a+b x)}-\frac {10 b^9 \log (x)}{a^{11}}+\frac {10 b^9 \log (a+b x)}{a^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 134, normalized size = 0.92 \begin {gather*} -\frac {\frac {a \left (28 a^9-35 a^8 b x+45 a^7 b^2 x^2-60 a^6 b^3 x^3+84 a^5 b^4 x^4-126 a^4 b^5 x^5+210 a^3 b^6 x^6-420 a^2 b^7 x^7+1260 a b^8 x^8+2520 b^9 x^9\right )}{x^9 (a+b x)}-2520 b^9 \log (a+b x)+2520 b^9 \log (x)}{252 a^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(a + b*x)^2),x]

[Out]

-1/252*((a*(28*a^9 - 35*a^8*b*x + 45*a^7*b^2*x^2 - 60*a^6*b^3*x^3 + 84*a^5*b^4*x^4 - 126*a^4*b^5*x^5 + 210*a^3
*b^6*x^6 - 420*a^2*b^7*x^7 + 1260*a*b^8*x^8 + 2520*b^9*x^9))/(x^9*(a + b*x)) + 2520*b^9*Log[x] - 2520*b^9*Log[
a + b*x])/a^11

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{10} (a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^10*(a + b*x)^2),x]

[Out]

IntegrateAlgebraic[1/(x^10*(a + b*x)^2), x]

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fricas [A]  time = 0.74, size = 163, normalized size = 1.12 \begin {gather*} -\frac {2520 \, a b^{9} x^{9} + 1260 \, a^{2} b^{8} x^{8} - 420 \, a^{3} b^{7} x^{7} + 210 \, a^{4} b^{6} x^{6} - 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} - 60 \, a^{7} b^{3} x^{3} + 45 \, a^{8} b^{2} x^{2} - 35 \, a^{9} b x + 28 \, a^{10} - 2520 \, {\left (b^{10} x^{10} + a b^{9} x^{9}\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{10} x^{10} + a b^{9} x^{9}\right )} \log \relax (x)}{252 \, {\left (a^{11} b x^{10} + a^{12} x^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/252*(2520*a*b^9*x^9 + 1260*a^2*b^8*x^8 - 420*a^3*b^7*x^7 + 210*a^4*b^6*x^6 - 126*a^5*b^5*x^5 + 84*a^6*b^4*x
^4 - 60*a^7*b^3*x^3 + 45*a^8*b^2*x^2 - 35*a^9*b*x + 28*a^10 - 2520*(b^10*x^10 + a*b^9*x^9)*log(b*x + a) + 2520
*(b^10*x^10 + a*b^9*x^9)*log(x))/(a^11*b*x^10 + a^12*x^9)

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giac [A]  time = 1.28, size = 180, normalized size = 1.23 \begin {gather*} -\frac {10 \, b^{9} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{11}} - \frac {b^{9}}{{\left (b x + a\right )} a^{10}} - \frac {\frac {41481 \, a b^{9}}{b x + a} - \frac {155844 \, a^{2} b^{9}}{{\left (b x + a\right )}^{2}} + \frac {337176 \, a^{3} b^{9}}{{\left (b x + a\right )}^{3}} - \frac {460404 \, a^{4} b^{9}}{{\left (b x + a\right )}^{4}} + \frac {407484 \, a^{5} b^{9}}{{\left (b x + a\right )}^{5}} - \frac {229320 \, a^{6} b^{9}}{{\left (b x + a\right )}^{6}} + \frac {75600 \, a^{7} b^{9}}{{\left (b x + a\right )}^{7}} - \frac {11340 \, a^{8} b^{9}}{{\left (b x + a\right )}^{8}} - 4861 \, b^{9}}{252 \, a^{11} {\left (\frac {a}{b x + a} - 1\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^2,x, algorithm="giac")

[Out]

-10*b^9*log(abs(-a/(b*x + a) + 1))/a^11 - b^9/((b*x + a)*a^10) - 1/252*(41481*a*b^9/(b*x + a) - 155844*a^2*b^9
/(b*x + a)^2 + 337176*a^3*b^9/(b*x + a)^3 - 460404*a^4*b^9/(b*x + a)^4 + 407484*a^5*b^9/(b*x + a)^5 - 229320*a
^6*b^9/(b*x + a)^6 + 75600*a^7*b^9/(b*x + a)^7 - 11340*a^8*b^9/(b*x + a)^8 - 4861*b^9)/(a^11*(a/(b*x + a) - 1)
^9)

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maple [A]  time = 0.01, size = 135, normalized size = 0.92 \begin {gather*} -\frac {b^{9}}{\left (b x +a \right ) a^{10}}-\frac {10 b^{9} \ln \relax (x )}{a^{11}}+\frac {10 b^{9} \ln \left (b x +a \right )}{a^{11}}-\frac {9 b^{8}}{a^{10} x}+\frac {4 b^{7}}{a^{9} x^{2}}-\frac {7 b^{6}}{3 a^{8} x^{3}}+\frac {3 b^{5}}{2 a^{7} x^{4}}-\frac {b^{4}}{a^{6} x^{5}}+\frac {2 b^{3}}{3 a^{5} x^{6}}-\frac {3 b^{2}}{7 a^{4} x^{7}}+\frac {b}{4 a^{3} x^{8}}-\frac {1}{9 a^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(b*x+a)^2,x)

[Out]

-1/9/a^2/x^9+1/4*b/a^3/x^8-3/7*b^2/a^4/x^7+2/3*b^3/a^5/x^6-b^4/a^6/x^5+3/2*b^5/a^7/x^4-7/3*b^6/a^8/x^3+4*b^7/a
^9/x^2-9*b^8/a^10/x-b^9/a^10/(b*x+a)-10*b^9*ln(x)/a^11+10*b^9*ln(b*x+a)/a^11

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maxima [A]  time = 1.40, size = 141, normalized size = 0.97 \begin {gather*} -\frac {2520 \, b^{9} x^{9} + 1260 \, a b^{8} x^{8} - 420 \, a^{2} b^{7} x^{7} + 210 \, a^{3} b^{6} x^{6} - 126 \, a^{4} b^{5} x^{5} + 84 \, a^{5} b^{4} x^{4} - 60 \, a^{6} b^{3} x^{3} + 45 \, a^{7} b^{2} x^{2} - 35 \, a^{8} b x + 28 \, a^{9}}{252 \, {\left (a^{10} b x^{10} + a^{11} x^{9}\right )}} + \frac {10 \, b^{9} \log \left (b x + a\right )}{a^{11}} - \frac {10 \, b^{9} \log \relax (x)}{a^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/252*(2520*b^9*x^9 + 1260*a*b^8*x^8 - 420*a^2*b^7*x^7 + 210*a^3*b^6*x^6 - 126*a^4*b^5*x^5 + 84*a^5*b^4*x^4 -
60*a^6*b^3*x^3 + 45*a^7*b^2*x^2 - 35*a^8*b*x + 28*a^9)/(a^10*b*x^10 + a^11*x^9) + 10*b^9*log(b*x + a)/a^11 -
10*b^9*log(x)/a^11

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mupad [B]  time = 0.08, size = 135, normalized size = 0.92 \begin {gather*} \frac {20\,b^9\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{11}}-\frac {\frac {1}{9\,a}+\frac {5\,b^2\,x^2}{28\,a^3}-\frac {5\,b^3\,x^3}{21\,a^4}+\frac {b^4\,x^4}{3\,a^5}-\frac {b^5\,x^5}{2\,a^6}+\frac {5\,b^6\,x^6}{6\,a^7}-\frac {5\,b^7\,x^7}{3\,a^8}+\frac {5\,b^8\,x^8}{a^9}+\frac {10\,b^9\,x^9}{a^{10}}-\frac {5\,b\,x}{36\,a^2}}{b\,x^{10}+a\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(a + b*x)^2),x)

[Out]

(20*b^9*atanh((2*b*x)/a + 1))/a^11 - (1/(9*a) + (5*b^2*x^2)/(28*a^3) - (5*b^3*x^3)/(21*a^4) + (b^4*x^4)/(3*a^5
) - (b^5*x^5)/(2*a^6) + (5*b^6*x^6)/(6*a^7) - (5*b^7*x^7)/(3*a^8) + (5*b^8*x^8)/a^9 + (10*b^9*x^9)/a^10 - (5*b
*x)/(36*a^2))/(a*x^9 + b*x^10)

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sympy [A]  time = 0.61, size = 139, normalized size = 0.95 \begin {gather*} \frac {- 28 a^{9} + 35 a^{8} b x - 45 a^{7} b^{2} x^{2} + 60 a^{6} b^{3} x^{3} - 84 a^{5} b^{4} x^{4} + 126 a^{4} b^{5} x^{5} - 210 a^{3} b^{6} x^{6} + 420 a^{2} b^{7} x^{7} - 1260 a b^{8} x^{8} - 2520 b^{9} x^{9}}{252 a^{11} x^{9} + 252 a^{10} b x^{10}} + \frac {10 b^{9} \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(b*x+a)**2,x)

[Out]

(-28*a**9 + 35*a**8*b*x - 45*a**7*b**2*x**2 + 60*a**6*b**3*x**3 - 84*a**5*b**4*x**4 + 126*a**4*b**5*x**5 - 210
*a**3*b**6*x**6 + 420*a**2*b**7*x**7 - 1260*a*b**8*x**8 - 2520*b**9*x**9)/(252*a**11*x**9 + 252*a**10*b*x**10)
+ 10*b**9*(-log(x) + log(a/b + x))/a**11

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