3.3.38 \(\int \frac {1}{x^4 (a+b x)^{10}} \, dx\)

Optimal. Leaf size=198 \[ -\frac {220 b^3 \log (x)}{a^{13}}+\frac {220 b^3 \log (a+b x)}{a^{13}}-\frac {165 b^3}{a^{12} (a+b x)}-\frac {55 b^2}{a^{12} x}-\frac {60 b^3}{a^{11} (a+b x)^2}+\frac {5 b}{a^{11} x^2}-\frac {28 b^3}{a^{10} (a+b x)^3}-\frac {1}{3 a^{10} x^3}-\frac {14 b^3}{a^9 (a+b x)^4}-\frac {7 b^3}{a^8 (a+b x)^5}-\frac {10 b^3}{3 a^7 (a+b x)^6}-\frac {10 b^3}{7 a^6 (a+b x)^7}-\frac {b^3}{2 a^5 (a+b x)^8}-\frac {b^3}{9 a^4 (a+b x)^9} \]

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 198, 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 {165 b^3}{a^{12} (a+b x)}-\frac {60 b^3}{a^{11} (a+b x)^2}-\frac {28 b^3}{a^{10} (a+b x)^3}-\frac {14 b^3}{a^9 (a+b x)^4}-\frac {7 b^3}{a^8 (a+b x)^5}-\frac {10 b^3}{3 a^7 (a+b x)^6}-\frac {10 b^3}{7 a^6 (a+b x)^7}-\frac {b^3}{2 a^5 (a+b x)^8}-\frac {b^3}{9 a^4 (a+b x)^9}-\frac {55 b^2}{a^{12} x}-\frac {220 b^3 \log (x)}{a^{13}}+\frac {220 b^3 \log (a+b x)}{a^{13}}+\frac {5 b}{a^{11} x^2}-\frac {1}{3 a^{10} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^10),x]

[Out]

-1/(3*a^10*x^3) + (5*b)/(a^11*x^2) - (55*b^2)/(a^12*x) - b^3/(9*a^4*(a + b*x)^9) - b^3/(2*a^5*(a + b*x)^8) - (
10*b^3)/(7*a^6*(a + b*x)^7) - (10*b^3)/(3*a^7*(a + b*x)^6) - (7*b^3)/(a^8*(a + b*x)^5) - (14*b^3)/(a^9*(a + b*
x)^4) - (28*b^3)/(a^10*(a + b*x)^3) - (60*b^3)/(a^11*(a + b*x)^2) - (165*b^3)/(a^12*(a + b*x)) - (220*b^3*Log[
x])/a^13 + (220*b^3*Log[a + b*x])/a^13

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^4 (a+b x)^{10}} \, dx &=\int \left (\frac {1}{a^{10} x^4}-\frac {10 b}{a^{11} x^3}+\frac {55 b^2}{a^{12} x^2}-\frac {220 b^3}{a^{13} x}+\frac {b^4}{a^4 (a+b x)^{10}}+\frac {4 b^4}{a^5 (a+b x)^9}+\frac {10 b^4}{a^6 (a+b x)^8}+\frac {20 b^4}{a^7 (a+b x)^7}+\frac {35 b^4}{a^8 (a+b x)^6}+\frac {56 b^4}{a^9 (a+b x)^5}+\frac {84 b^4}{a^{10} (a+b x)^4}+\frac {120 b^4}{a^{11} (a+b x)^3}+\frac {165 b^4}{a^{12} (a+b x)^2}+\frac {220 b^4}{a^{13} (a+b x)}\right ) \, dx\\ &=-\frac {1}{3 a^{10} x^3}+\frac {5 b}{a^{11} x^2}-\frac {55 b^2}{a^{12} x}-\frac {b^3}{9 a^4 (a+b x)^9}-\frac {b^3}{2 a^5 (a+b x)^8}-\frac {10 b^3}{7 a^6 (a+b x)^7}-\frac {10 b^3}{3 a^7 (a+b x)^6}-\frac {7 b^3}{a^8 (a+b x)^5}-\frac {14 b^3}{a^9 (a+b x)^4}-\frac {28 b^3}{a^{10} (a+b x)^3}-\frac {60 b^3}{a^{11} (a+b x)^2}-\frac {165 b^3}{a^{12} (a+b x)}-\frac {220 b^3 \log (x)}{a^{13}}+\frac {220 b^3 \log (a+b x)}{a^{13}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 156, normalized size = 0.79 \begin {gather*} -\frac {-27720 b^3 \log (a+b x)+\frac {a \left (42 a^{11}-252 a^{10} b x+2772 a^9 b^2 x^2+78419 a^8 b^3 x^3+456291 a^7 b^4 x^4+1326204 a^6 b^5 x^5+2318316 a^5 b^6 x^6+2604294 a^4 b^7 x^7+1905750 a^3 b^8 x^8+882420 a^2 b^9 x^9+235620 a b^{10} x^{10}+27720 b^{11} x^{11}\right )}{x^3 (a+b x)^9}+27720 b^3 \log (x)}{126 a^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^10),x]

[Out]

-1/126*((a*(42*a^11 - 252*a^10*b*x + 2772*a^9*b^2*x^2 + 78419*a^8*b^3*x^3 + 456291*a^7*b^4*x^4 + 1326204*a^6*b
^5*x^5 + 2318316*a^5*b^6*x^6 + 2604294*a^4*b^7*x^7 + 1905750*a^3*b^8*x^8 + 882420*a^2*b^9*x^9 + 235620*a*b^10*
x^10 + 27720*b^11*x^11))/(x^3*(a + b*x)^9) + 27720*b^3*Log[x] - 27720*b^3*Log[a + b*x])/a^13

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 (a+b x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^4*(a + b*x)^10),x]

[Out]

IntegrateAlgebraic[1/(x^4*(a + b*x)^10), x]

________________________________________________________________________________________

fricas [B]  time = 1.26, size = 449, normalized size = 2.27 \begin {gather*} -\frac {27720 \, a b^{11} x^{11} + 235620 \, a^{2} b^{10} x^{10} + 882420 \, a^{3} b^{9} x^{9} + 1905750 \, a^{4} b^{8} x^{8} + 2604294 \, a^{5} b^{7} x^{7} + 2318316 \, a^{6} b^{6} x^{6} + 1326204 \, a^{7} b^{5} x^{5} + 456291 \, a^{8} b^{4} x^{4} + 78419 \, a^{9} b^{3} x^{3} + 2772 \, a^{10} b^{2} x^{2} - 252 \, a^{11} b x + 42 \, a^{12} - 27720 \, {\left (b^{12} x^{12} + 9 \, a b^{11} x^{11} + 36 \, a^{2} b^{10} x^{10} + 84 \, a^{3} b^{9} x^{9} + 126 \, a^{4} b^{8} x^{8} + 126 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{5} + 9 \, a^{8} b^{4} x^{4} + a^{9} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 27720 \, {\left (b^{12} x^{12} + 9 \, a b^{11} x^{11} + 36 \, a^{2} b^{10} x^{10} + 84 \, a^{3} b^{9} x^{9} + 126 \, a^{4} b^{8} x^{8} + 126 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{5} + 9 \, a^{8} b^{4} x^{4} + a^{9} b^{3} x^{3}\right )} \log \relax (x)}{126 \, {\left (a^{13} b^{9} x^{12} + 9 \, a^{14} b^{8} x^{11} + 36 \, a^{15} b^{7} x^{10} + 84 \, a^{16} b^{6} x^{9} + 126 \, a^{17} b^{5} x^{8} + 126 \, a^{18} b^{4} x^{7} + 84 \, a^{19} b^{3} x^{6} + 36 \, a^{20} b^{2} x^{5} + 9 \, a^{21} b x^{4} + a^{22} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/126*(27720*a*b^11*x^11 + 235620*a^2*b^10*x^10 + 882420*a^3*b^9*x^9 + 1905750*a^4*b^8*x^8 + 2604294*a^5*b^7*
x^7 + 2318316*a^6*b^6*x^6 + 1326204*a^7*b^5*x^5 + 456291*a^8*b^4*x^4 + 78419*a^9*b^3*x^3 + 2772*a^10*b^2*x^2 -
252*a^11*b*x + 42*a^12 - 27720*(b^12*x^12 + 9*a*b^11*x^11 + 36*a^2*b^10*x^10 + 84*a^3*b^9*x^9 + 126*a^4*b^8*x
^8 + 126*a^5*b^7*x^7 + 84*a^6*b^6*x^6 + 36*a^7*b^5*x^5 + 9*a^8*b^4*x^4 + a^9*b^3*x^3)*log(b*x + a) + 27720*(b^
12*x^12 + 9*a*b^11*x^11 + 36*a^2*b^10*x^10 + 84*a^3*b^9*x^9 + 126*a^4*b^8*x^8 + 126*a^5*b^7*x^7 + 84*a^6*b^6*x
^6 + 36*a^7*b^5*x^5 + 9*a^8*b^4*x^4 + a^9*b^3*x^3)*log(x))/(a^13*b^9*x^12 + 9*a^14*b^8*x^11 + 36*a^15*b^7*x^10
+ 84*a^16*b^6*x^9 + 126*a^17*b^5*x^8 + 126*a^18*b^4*x^7 + 84*a^19*b^3*x^6 + 36*a^20*b^2*x^5 + 9*a^21*b*x^4 +
a^22*x^3)

________________________________________________________________________________________

giac [A]  time = 1.35, size = 163, normalized size = 0.82 \begin {gather*} \frac {220 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{13}} - \frac {220 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{13}} - \frac {27720 \, a b^{11} x^{11} + 235620 \, a^{2} b^{10} x^{10} + 882420 \, a^{3} b^{9} x^{9} + 1905750 \, a^{4} b^{8} x^{8} + 2604294 \, a^{5} b^{7} x^{7} + 2318316 \, a^{6} b^{6} x^{6} + 1326204 \, a^{7} b^{5} x^{5} + 456291 \, a^{8} b^{4} x^{4} + 78419 \, a^{9} b^{3} x^{3} + 2772 \, a^{10} b^{2} x^{2} - 252 \, a^{11} b x + 42 \, a^{12}}{126 \, {\left (b x + a\right )}^{9} a^{13} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^10,x, algorithm="giac")

[Out]

220*b^3*log(abs(b*x + a))/a^13 - 220*b^3*log(abs(x))/a^13 - 1/126*(27720*a*b^11*x^11 + 235620*a^2*b^10*x^10 +
882420*a^3*b^9*x^9 + 1905750*a^4*b^8*x^8 + 2604294*a^5*b^7*x^7 + 2318316*a^6*b^6*x^6 + 1326204*a^7*b^5*x^5 + 4
56291*a^8*b^4*x^4 + 78419*a^9*b^3*x^3 + 2772*a^10*b^2*x^2 - 252*a^11*b*x + 42*a^12)/((b*x + a)^9*a^13*x^3)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 189, normalized size = 0.95 \begin {gather*} -\frac {b^{3}}{9 \left (b x +a \right )^{9} a^{4}}-\frac {b^{3}}{2 \left (b x +a \right )^{8} a^{5}}-\frac {10 b^{3}}{7 \left (b x +a \right )^{7} a^{6}}-\frac {10 b^{3}}{3 \left (b x +a \right )^{6} a^{7}}-\frac {7 b^{3}}{\left (b x +a \right )^{5} a^{8}}-\frac {14 b^{3}}{\left (b x +a \right )^{4} a^{9}}-\frac {28 b^{3}}{\left (b x +a \right )^{3} a^{10}}-\frac {60 b^{3}}{\left (b x +a \right )^{2} a^{11}}-\frac {165 b^{3}}{\left (b x +a \right ) a^{12}}-\frac {220 b^{3} \ln \relax (x )}{a^{13}}+\frac {220 b^{3} \ln \left (b x +a \right )}{a^{13}}-\frac {55 b^{2}}{a^{12} x}+\frac {5 b}{a^{11} x^{2}}-\frac {1}{3 a^{10} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^10,x)

[Out]

-1/3/a^10/x^3+5*b/a^11/x^2-55*b^2/a^12/x-1/9*b^3/a^4/(b*x+a)^9-1/2*b^3/a^5/(b*x+a)^8-10/7*b^3/a^6/(b*x+a)^7-10
/3*b^3/a^7/(b*x+a)^6-7*b^3/a^8/(b*x+a)^5-14*b^3/a^9/(b*x+a)^4-28*b^3/a^10/(b*x+a)^3-60*b^3/a^11/(b*x+a)^2-165*
b^3/a^12/(b*x+a)-220*b^3*ln(x)/a^13+220*b^3*ln(b*x+a)/a^13

________________________________________________________________________________________

maxima [A]  time = 1.72, size = 251, normalized size = 1.27 \begin {gather*} -\frac {27720 \, b^{11} x^{11} + 235620 \, a b^{10} x^{10} + 882420 \, a^{2} b^{9} x^{9} + 1905750 \, a^{3} b^{8} x^{8} + 2604294 \, a^{4} b^{7} x^{7} + 2318316 \, a^{5} b^{6} x^{6} + 1326204 \, a^{6} b^{5} x^{5} + 456291 \, a^{7} b^{4} x^{4} + 78419 \, a^{8} b^{3} x^{3} + 2772 \, a^{9} b^{2} x^{2} - 252 \, a^{10} b x + 42 \, a^{11}}{126 \, {\left (a^{12} b^{9} x^{12} + 9 \, a^{13} b^{8} x^{11} + 36 \, a^{14} b^{7} x^{10} + 84 \, a^{15} b^{6} x^{9} + 126 \, a^{16} b^{5} x^{8} + 126 \, a^{17} b^{4} x^{7} + 84 \, a^{18} b^{3} x^{6} + 36 \, a^{19} b^{2} x^{5} + 9 \, a^{20} b x^{4} + a^{21} x^{3}\right )}} + \frac {220 \, b^{3} \log \left (b x + a\right )}{a^{13}} - \frac {220 \, b^{3} \log \relax (x)}{a^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/126*(27720*b^11*x^11 + 235620*a*b^10*x^10 + 882420*a^2*b^9*x^9 + 1905750*a^3*b^8*x^8 + 2604294*a^4*b^7*x^7
+ 2318316*a^5*b^6*x^6 + 1326204*a^6*b^5*x^5 + 456291*a^7*b^4*x^4 + 78419*a^8*b^3*x^3 + 2772*a^9*b^2*x^2 - 252*
a^10*b*x + 42*a^11)/(a^12*b^9*x^12 + 9*a^13*b^8*x^11 + 36*a^14*b^7*x^10 + 84*a^15*b^6*x^9 + 126*a^16*b^5*x^8 +
126*a^17*b^4*x^7 + 84*a^18*b^3*x^6 + 36*a^19*b^2*x^5 + 9*a^20*b*x^4 + a^21*x^3) + 220*b^3*log(b*x + a)/a^13 -
220*b^3*log(x)/a^13

________________________________________________________________________________________

mupad [B]  time = 0.59, size = 245, normalized size = 1.24 \begin {gather*} \frac {440\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{13}}-\frac {\frac {1}{3\,a}+\frac {22\,b^2\,x^2}{a^3}+\frac {78419\,b^3\,x^3}{126\,a^4}+\frac {50699\,b^4\,x^4}{14\,a^5}+\frac {73678\,b^5\,x^5}{7\,a^6}+\frac {55198\,b^6\,x^6}{3\,a^7}+\frac {20669\,b^7\,x^7}{a^8}+\frac {15125\,b^8\,x^8}{a^9}+\frac {21010\,b^9\,x^9}{3\,a^{10}}+\frac {1870\,b^{10}\,x^{10}}{a^{11}}+\frac {220\,b^{11}\,x^{11}}{a^{12}}-\frac {2\,b\,x}{a^2}}{a^9\,x^3+9\,a^8\,b\,x^4+36\,a^7\,b^2\,x^5+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^7+126\,a^4\,b^5\,x^8+84\,a^3\,b^6\,x^9+36\,a^2\,b^7\,x^{10}+9\,a\,b^8\,x^{11}+b^9\,x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x)^10),x)

[Out]

(440*b^3*atanh((2*b*x)/a + 1))/a^13 - (1/(3*a) + (22*b^2*x^2)/a^3 + (78419*b^3*x^3)/(126*a^4) + (50699*b^4*x^4
)/(14*a^5) + (73678*b^5*x^5)/(7*a^6) + (55198*b^6*x^6)/(3*a^7) + (20669*b^7*x^7)/a^8 + (15125*b^8*x^8)/a^9 + (
21010*b^9*x^9)/(3*a^10) + (1870*b^10*x^10)/a^11 + (220*b^11*x^11)/a^12 - (2*b*x)/a^2)/(a^9*x^3 + b^9*x^12 + 9*
a^8*b*x^4 + 9*a*b^8*x^11 + 36*a^7*b^2*x^5 + 84*a^6*b^3*x^6 + 126*a^5*b^4*x^7 + 126*a^4*b^5*x^8 + 84*a^3*b^6*x^
9 + 36*a^2*b^7*x^10)

________________________________________________________________________________________

sympy [A]  time = 1.24, size = 258, normalized size = 1.30 \begin {gather*} \frac {- 42 a^{11} + 252 a^{10} b x - 2772 a^{9} b^{2} x^{2} - 78419 a^{8} b^{3} x^{3} - 456291 a^{7} b^{4} x^{4} - 1326204 a^{6} b^{5} x^{5} - 2318316 a^{5} b^{6} x^{6} - 2604294 a^{4} b^{7} x^{7} - 1905750 a^{3} b^{8} x^{8} - 882420 a^{2} b^{9} x^{9} - 235620 a b^{10} x^{10} - 27720 b^{11} x^{11}}{126 a^{21} x^{3} + 1134 a^{20} b x^{4} + 4536 a^{19} b^{2} x^{5} + 10584 a^{18} b^{3} x^{6} + 15876 a^{17} b^{4} x^{7} + 15876 a^{16} b^{5} x^{8} + 10584 a^{15} b^{6} x^{9} + 4536 a^{14} b^{7} x^{10} + 1134 a^{13} b^{8} x^{11} + 126 a^{12} b^{9} x^{12}} + \frac {220 b^{3} \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**10,x)

[Out]

(-42*a**11 + 252*a**10*b*x - 2772*a**9*b**2*x**2 - 78419*a**8*b**3*x**3 - 456291*a**7*b**4*x**4 - 1326204*a**6
*b**5*x**5 - 2318316*a**5*b**6*x**6 - 2604294*a**4*b**7*x**7 - 1905750*a**3*b**8*x**8 - 882420*a**2*b**9*x**9
- 235620*a*b**10*x**10 - 27720*b**11*x**11)/(126*a**21*x**3 + 1134*a**20*b*x**4 + 4536*a**19*b**2*x**5 + 10584
*a**18*b**3*x**6 + 15876*a**17*b**4*x**7 + 15876*a**16*b**5*x**8 + 10584*a**15*b**6*x**9 + 4536*a**14*b**7*x**
10 + 1134*a**13*b**8*x**11 + 126*a**12*b**9*x**12) + 220*b**3*(-log(x) + log(a/b + x))/a**13

________________________________________________________________________________________