3.3.37 \(\int \frac {1}{x^3 (a+b x)^{10}} \, dx\)

Optimal. Leaf size=191 \[ \frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {10 b}{a^{11} x}+\frac {18 b^2}{a^{10} (a+b x)^2}-\frac {1}{2 a^{10} x^2}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {b^2}{9 a^3 (a+b x)^9} \]

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Rubi [A]  time = 0.14, antiderivative size = 191, 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 {45 b^2}{a^{11} (a+b x)}+\frac {18 b^2}{a^{10} (a+b x)^2}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {b^2}{9 a^3 (a+b x)^9}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}}+\frac {10 b}{a^{11} x}-\frac {1}{2 a^{10} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)^10),x]

[Out]

-1/(2*a^10*x^2) + (10*b)/(a^11*x) + b^2/(9*a^3*(a + b*x)^9) + (3*b^2)/(8*a^4*(a + b*x)^8) + (6*b^2)/(7*a^5*(a
+ b*x)^7) + (5*b^2)/(3*a^6*(a + b*x)^6) + (3*b^2)/(a^7*(a + b*x)^5) + (21*b^2)/(4*a^8*(a + b*x)^4) + (28*b^2)/
(3*a^9*(a + b*x)^3) + (18*b^2)/(a^10*(a + b*x)^2) + (45*b^2)/(a^11*(a + b*x)) + (55*b^2*Log[x])/a^12 - (55*b^2
*Log[a + b*x])/a^12

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^3 (a+b x)^{10}} \, dx &=\int \left (\frac {1}{a^{10} x^3}-\frac {10 b}{a^{11} x^2}+\frac {55 b^2}{a^{12} x}-\frac {b^3}{a^3 (a+b x)^{10}}-\frac {3 b^3}{a^4 (a+b x)^9}-\frac {6 b^3}{a^5 (a+b x)^8}-\frac {10 b^3}{a^6 (a+b x)^7}-\frac {15 b^3}{a^7 (a+b x)^6}-\frac {21 b^3}{a^8 (a+b x)^5}-\frac {28 b^3}{a^9 (a+b x)^4}-\frac {36 b^3}{a^{10} (a+b x)^3}-\frac {45 b^3}{a^{11} (a+b x)^2}-\frac {55 b^3}{a^{12} (a+b x)}\right ) \, dx\\ &=-\frac {1}{2 a^{10} x^2}+\frac {10 b}{a^{11} x}+\frac {b^2}{9 a^3 (a+b x)^9}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {18 b^2}{a^{10} (a+b x)^2}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 145, normalized size = 0.76 \begin {gather*} \frac {\frac {a \left (-252 a^{10}+2772 a^9 b x+78419 a^8 b^2 x^2+456291 a^7 b^3 x^3+1326204 a^6 b^4 x^4+2318316 a^5 b^5 x^5+2604294 a^4 b^6 x^6+1905750 a^3 b^7 x^7+882420 a^2 b^8 x^8+235620 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^2 (a+b x)^9}-27720 b^2 \log (a+b x)+27720 b^2 \log (x)}{504 a^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x)^10),x]

[Out]

((a*(-252*a^10 + 2772*a^9*b*x + 78419*a^8*b^2*x^2 + 456291*a^7*b^3*x^3 + 1326204*a^6*b^4*x^4 + 2318316*a^5*b^5
*x^5 + 2604294*a^4*b^6*x^6 + 1905750*a^3*b^7*x^7 + 882420*a^2*b^8*x^8 + 235620*a*b^9*x^9 + 27720*b^10*x^10))/(
x^2*(a + b*x)^9) + 27720*b^2*Log[x] - 27720*b^2*Log[a + b*x])/(504*a^12)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 (a+b x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^3*(a + b*x)^10),x]

[Out]

IntegrateAlgebraic[1/(x^3*(a + b*x)^10), x]

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fricas [B]  time = 0.61, size = 438, normalized size = 2.29 \begin {gather*} \frac {27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11} - 27720 \, {\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 27720 \, {\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \relax (x)}{504 \, {\left (a^{12} b^{9} x^{11} + 9 \, a^{13} b^{8} x^{10} + 36 \, a^{14} b^{7} x^{9} + 84 \, a^{15} b^{6} x^{8} + 126 \, a^{16} b^{5} x^{7} + 126 \, a^{17} b^{4} x^{6} + 84 \, a^{18} b^{3} x^{5} + 36 \, a^{19} b^{2} x^{4} + 9 \, a^{20} b x^{3} + a^{21} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^10,x, algorithm="fricas")

[Out]

1/504*(27720*a*b^10*x^10 + 235620*a^2*b^9*x^9 + 882420*a^3*b^8*x^8 + 1905750*a^4*b^7*x^7 + 2604294*a^5*b^6*x^6
+ 2318316*a^6*b^5*x^5 + 1326204*a^7*b^4*x^4 + 456291*a^8*b^3*x^3 + 78419*a^9*b^2*x^2 + 2772*a^10*b*x - 252*a^
11 - 27720*(b^11*x^11 + 9*a*b^10*x^10 + 36*a^2*b^9*x^9 + 84*a^3*b^8*x^8 + 126*a^4*b^7*x^7 + 126*a^5*b^6*x^6 +
84*a^6*b^5*x^5 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^3 + a^9*b^2*x^2)*log(b*x + a) + 27720*(b^11*x^11 + 9*a*b^10*x^10
+ 36*a^2*b^9*x^9 + 84*a^3*b^8*x^8 + 126*a^4*b^7*x^7 + 126*a^5*b^6*x^6 + 84*a^6*b^5*x^5 + 36*a^7*b^4*x^4 + 9*a
^8*b^3*x^3 + a^9*b^2*x^2)*log(x))/(a^12*b^9*x^11 + 9*a^13*b^8*x^10 + 36*a^14*b^7*x^9 + 84*a^15*b^6*x^8 + 126*a
^16*b^5*x^7 + 126*a^17*b^4*x^6 + 84*a^18*b^3*x^5 + 36*a^19*b^2*x^4 + 9*a^20*b*x^3 + a^21*x^2)

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giac [A]  time = 1.10, size = 152, normalized size = 0.80 \begin {gather*} -\frac {55 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{12}} + \frac {55 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac {27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11}}{504 \, {\left (b x + a\right )}^{9} a^{12} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^10,x, algorithm="giac")

[Out]

-55*b^2*log(abs(b*x + a))/a^12 + 55*b^2*log(abs(x))/a^12 + 1/504*(27720*a*b^10*x^10 + 235620*a^2*b^9*x^9 + 882
420*a^3*b^8*x^8 + 1905750*a^4*b^7*x^7 + 2604294*a^5*b^6*x^6 + 2318316*a^6*b^5*x^5 + 1326204*a^7*b^4*x^4 + 4562
91*a^8*b^3*x^3 + 78419*a^9*b^2*x^2 + 2772*a^10*b*x - 252*a^11)/((b*x + a)^9*a^12*x^2)

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maple [A]  time = 0.02, size = 178, normalized size = 0.93 \begin {gather*} \frac {b^{2}}{9 \left (b x +a \right )^{9} a^{3}}+\frac {3 b^{2}}{8 \left (b x +a \right )^{8} a^{4}}+\frac {6 b^{2}}{7 \left (b x +a \right )^{7} a^{5}}+\frac {5 b^{2}}{3 \left (b x +a \right )^{6} a^{6}}+\frac {3 b^{2}}{\left (b x +a \right )^{5} a^{7}}+\frac {21 b^{2}}{4 \left (b x +a \right )^{4} a^{8}}+\frac {28 b^{2}}{3 \left (b x +a \right )^{3} a^{9}}+\frac {18 b^{2}}{\left (b x +a \right )^{2} a^{10}}+\frac {45 b^{2}}{\left (b x +a \right ) a^{11}}+\frac {55 b^{2} \ln \relax (x )}{a^{12}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}+\frac {10 b}{a^{11} x}-\frac {1}{2 a^{10} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x+a)^10,x)

[Out]

-1/2/a^10/x^2+10*b/a^11/x+1/9*b^2/a^3/(b*x+a)^9+3/8*b^2/a^4/(b*x+a)^8+6/7*b^2/a^5/(b*x+a)^7+5/3*b^2/a^6/(b*x+a
)^6+3*b^2/a^7/(b*x+a)^5+21/4*b^2/a^8/(b*x+a)^4+28/3*b^2/a^9/(b*x+a)^3+18*b^2/a^10/(b*x+a)^2+45*b^2/a^11/(b*x+a
)+55*b^2*ln(x)/a^12-55*b^2*ln(b*x+a)/a^12

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maxima [A]  time = 1.71, size = 240, normalized size = 1.26 \begin {gather*} \frac {27720 \, b^{10} x^{10} + 235620 \, a b^{9} x^{9} + 882420 \, a^{2} b^{8} x^{8} + 1905750 \, a^{3} b^{7} x^{7} + 2604294 \, a^{4} b^{6} x^{6} + 2318316 \, a^{5} b^{5} x^{5} + 1326204 \, a^{6} b^{4} x^{4} + 456291 \, a^{7} b^{3} x^{3} + 78419 \, a^{8} b^{2} x^{2} + 2772 \, a^{9} b x - 252 \, a^{10}}{504 \, {\left (a^{11} b^{9} x^{11} + 9 \, a^{12} b^{8} x^{10} + 36 \, a^{13} b^{7} x^{9} + 84 \, a^{14} b^{6} x^{8} + 126 \, a^{15} b^{5} x^{7} + 126 \, a^{16} b^{4} x^{6} + 84 \, a^{17} b^{3} x^{5} + 36 \, a^{18} b^{2} x^{4} + 9 \, a^{19} b x^{3} + a^{20} x^{2}\right )}} - \frac {55 \, b^{2} \log \left (b x + a\right )}{a^{12}} + \frac {55 \, b^{2} \log \relax (x)}{a^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^10,x, algorithm="maxima")

[Out]

1/504*(27720*b^10*x^10 + 235620*a*b^9*x^9 + 882420*a^2*b^8*x^8 + 1905750*a^3*b^7*x^7 + 2604294*a^4*b^6*x^6 + 2
318316*a^5*b^5*x^5 + 1326204*a^6*b^4*x^4 + 456291*a^7*b^3*x^3 + 78419*a^8*b^2*x^2 + 2772*a^9*b*x - 252*a^10)/(
a^11*b^9*x^11 + 9*a^12*b^8*x^10 + 36*a^13*b^7*x^9 + 84*a^14*b^6*x^8 + 126*a^15*b^5*x^7 + 126*a^16*b^4*x^6 + 84
*a^17*b^3*x^5 + 36*a^18*b^2*x^4 + 9*a^19*b*x^3 + a^20*x^2) - 55*b^2*log(b*x + a)/a^12 + 55*b^2*log(x)/a^12

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mupad [B]  time = 0.44, size = 233, normalized size = 1.22 \begin {gather*} \frac {\frac {78419\,b^2\,x^2}{504\,a^3}-\frac {1}{2\,a}+\frac {50699\,b^3\,x^3}{56\,a^4}+\frac {36839\,b^4\,x^4}{14\,a^5}+\frac {27599\,b^5\,x^5}{6\,a^6}+\frac {20669\,b^6\,x^6}{4\,a^7}+\frac {15125\,b^7\,x^7}{4\,a^8}+\frac {10505\,b^8\,x^8}{6\,a^9}+\frac {935\,b^9\,x^9}{2\,a^{10}}+\frac {55\,b^{10}\,x^{10}}{a^{11}}+\frac {11\,b\,x}{2\,a^2}}{a^9\,x^2+9\,a^8\,b\,x^3+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^5+126\,a^5\,b^4\,x^6+126\,a^4\,b^5\,x^7+84\,a^3\,b^6\,x^8+36\,a^2\,b^7\,x^9+9\,a\,b^8\,x^{10}+b^9\,x^{11}}-\frac {110\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x)^10),x)

[Out]

((78419*b^2*x^2)/(504*a^3) - 1/(2*a) + (50699*b^3*x^3)/(56*a^4) + (36839*b^4*x^4)/(14*a^5) + (27599*b^5*x^5)/(
6*a^6) + (20669*b^6*x^6)/(4*a^7) + (15125*b^7*x^7)/(4*a^8) + (10505*b^8*x^8)/(6*a^9) + (935*b^9*x^9)/(2*a^10)
+ (55*b^10*x^10)/a^11 + (11*b*x)/(2*a^2))/(a^9*x^2 + b^9*x^11 + 9*a^8*b*x^3 + 9*a*b^8*x^10 + 36*a^7*b^2*x^4 +
84*a^6*b^3*x^5 + 126*a^5*b^4*x^6 + 126*a^4*b^5*x^7 + 84*a^3*b^6*x^8 + 36*a^2*b^7*x^9) - (110*b^2*atanh((2*b*x)
/a + 1))/a^12

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sympy [A]  time = 1.18, size = 246, normalized size = 1.29 \begin {gather*} \frac {- 252 a^{10} + 2772 a^{9} b x + 78419 a^{8} b^{2} x^{2} + 456291 a^{7} b^{3} x^{3} + 1326204 a^{6} b^{4} x^{4} + 2318316 a^{5} b^{5} x^{5} + 2604294 a^{4} b^{6} x^{6} + 1905750 a^{3} b^{7} x^{7} + 882420 a^{2} b^{8} x^{8} + 235620 a b^{9} x^{9} + 27720 b^{10} x^{10}}{504 a^{20} x^{2} + 4536 a^{19} b x^{3} + 18144 a^{18} b^{2} x^{4} + 42336 a^{17} b^{3} x^{5} + 63504 a^{16} b^{4} x^{6} + 63504 a^{15} b^{5} x^{7} + 42336 a^{14} b^{6} x^{8} + 18144 a^{13} b^{7} x^{9} + 4536 a^{12} b^{8} x^{10} + 504 a^{11} b^{9} x^{11}} + \frac {55 b^{2} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x+a)**10,x)

[Out]

(-252*a**10 + 2772*a**9*b*x + 78419*a**8*b**2*x**2 + 456291*a**7*b**3*x**3 + 1326204*a**6*b**4*x**4 + 2318316*
a**5*b**5*x**5 + 2604294*a**4*b**6*x**6 + 1905750*a**3*b**7*x**7 + 882420*a**2*b**8*x**8 + 235620*a*b**9*x**9
+ 27720*b**10*x**10)/(504*a**20*x**2 + 4536*a**19*b*x**3 + 18144*a**18*b**2*x**4 + 42336*a**17*b**3*x**5 + 635
04*a**16*b**4*x**6 + 63504*a**15*b**5*x**7 + 42336*a**14*b**6*x**8 + 18144*a**13*b**7*x**9 + 4536*a**12*b**8*x
**10 + 504*a**11*b**9*x**11) + 55*b**2*(log(x) - log(a/b + x))/a**12

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