3.3.6 \(\int \frac {1}{x^5 (a+b x)^4} \, dx\)

Optimal. Leaf size=117 \[ \frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8}+\frac {15 b^4}{a^7 (a+b x)}+\frac {20 b^3}{a^7 x}+\frac {5 b^4}{2 a^6 (a+b x)^2}-\frac {5 b^2}{a^6 x^2}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {4 b}{3 a^5 x^3}-\frac {1}{4 a^4 x^4} \]

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Rubi [A]  time = 0.07, antiderivative size = 117, 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 {5 b^2}{a^6 x^2}+\frac {15 b^4}{a^7 (a+b x)}+\frac {5 b^4}{2 a^6 (a+b x)^2}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {20 b^3}{a^7 x}+\frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8}+\frac {4 b}{3 a^5 x^3}-\frac {1}{4 a^4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)^4),x]

[Out]

-1/(4*a^4*x^4) + (4*b)/(3*a^5*x^3) - (5*b^2)/(a^6*x^2) + (20*b^3)/(a^7*x) + b^4/(3*a^5*(a + b*x)^3) + (5*b^4)/
(2*a^6*(a + b*x)^2) + (15*b^4)/(a^7*(a + b*x)) + (35*b^4*Log[x])/a^8 - (35*b^4*Log[a + b*x])/a^8

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^5 (a+b x)^4} \, dx &=\int \left (\frac {1}{a^4 x^5}-\frac {4 b}{a^5 x^4}+\frac {10 b^2}{a^6 x^3}-\frac {20 b^3}{a^7 x^2}+\frac {35 b^4}{a^8 x}-\frac {b^5}{a^5 (a+b x)^4}-\frac {5 b^5}{a^6 (a+b x)^3}-\frac {15 b^5}{a^7 (a+b x)^2}-\frac {35 b^5}{a^8 (a+b x)}\right ) \, dx\\ &=-\frac {1}{4 a^4 x^4}+\frac {4 b}{3 a^5 x^3}-\frac {5 b^2}{a^6 x^2}+\frac {20 b^3}{a^7 x}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {5 b^4}{2 a^6 (a+b x)^2}+\frac {15 b^4}{a^7 (a+b x)}+\frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 101, normalized size = 0.86 \begin {gather*} \frac {\frac {a \left (-3 a^6+7 a^5 b x-21 a^4 b^2 x^2+105 a^3 b^3 x^3+770 a^2 b^4 x^4+1050 a b^5 x^5+420 b^6 x^6\right )}{x^4 (a+b x)^3}-420 b^4 \log (a+b x)+420 b^4 \log (x)}{12 a^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)^4),x]

[Out]

((a*(-3*a^6 + 7*a^5*b*x - 21*a^4*b^2*x^2 + 105*a^3*b^3*x^3 + 770*a^2*b^4*x^4 + 1050*a*b^5*x^5 + 420*b^6*x^6))/
(x^4*(a + b*x)^3) + 420*b^4*Log[x] - 420*b^4*Log[a + b*x])/(12*a^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^5 (a+b x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^5*(a + b*x)^4),x]

[Out]

IntegrateAlgebraic[1/(x^5*(a + b*x)^4), x]

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fricas [A]  time = 1.13, size = 196, normalized size = 1.68 \begin {gather*} \frac {420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7} - 420 \, {\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 420 \, {\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \relax (x)}{12 \, {\left (a^{8} b^{3} x^{7} + 3 \, a^{9} b^{2} x^{6} + 3 \, a^{10} b x^{5} + a^{11} x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/12*(420*a*b^6*x^6 + 1050*a^2*b^5*x^5 + 770*a^3*b^4*x^4 + 105*a^4*b^3*x^3 - 21*a^5*b^2*x^2 + 7*a^6*b*x - 3*a^
7 - 420*(b^7*x^7 + 3*a*b^6*x^6 + 3*a^2*b^5*x^5 + a^3*b^4*x^4)*log(b*x + a) + 420*(b^7*x^7 + 3*a*b^6*x^6 + 3*a^
2*b^5*x^5 + a^3*b^4*x^4)*log(x))/(a^8*b^3*x^7 + 3*a^9*b^2*x^6 + 3*a^10*b*x^5 + a^11*x^4)

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giac [A]  time = 1.00, size = 108, normalized size = 0.92 \begin {gather*} -\frac {35 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{8}} + \frac {35 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7}}{12 \, {\left (b x + a\right )}^{3} a^{8} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^4,x, algorithm="giac")

[Out]

-35*b^4*log(abs(b*x + a))/a^8 + 35*b^4*log(abs(x))/a^8 + 1/12*(420*a*b^6*x^6 + 1050*a^2*b^5*x^5 + 770*a^3*b^4*
x^4 + 105*a^4*b^3*x^3 - 21*a^5*b^2*x^2 + 7*a^6*b*x - 3*a^7)/((b*x + a)^3*a^8*x^4)

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maple [A]  time = 0.01, size = 110, normalized size = 0.94 \begin {gather*} \frac {b^{4}}{3 \left (b x +a \right )^{3} a^{5}}+\frac {5 b^{4}}{2 \left (b x +a \right )^{2} a^{6}}+\frac {15 b^{4}}{\left (b x +a \right ) a^{7}}+\frac {35 b^{4} \ln \relax (x )}{a^{8}}-\frac {35 b^{4} \ln \left (b x +a \right )}{a^{8}}+\frac {20 b^{3}}{a^{7} x}-\frac {5 b^{2}}{a^{6} x^{2}}+\frac {4 b}{3 a^{5} x^{3}}-\frac {1}{4 a^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x+a)^4,x)

[Out]

-1/4/a^4/x^4+4/3*b/a^5/x^3-5*b^2/a^6/x^2+20*b^3/a^7/x+1/3*b^4/a^5/(b*x+a)^3+5/2*b^4/a^6/(b*x+a)^2+15*b^4/a^7/(
b*x+a)+35*b^4*ln(x)/a^8-35*b^4*ln(b*x+a)/a^8

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maxima [A]  time = 1.39, size = 130, normalized size = 1.11 \begin {gather*} \frac {420 \, b^{6} x^{6} + 1050 \, a b^{5} x^{5} + 770 \, a^{2} b^{4} x^{4} + 105 \, a^{3} b^{3} x^{3} - 21 \, a^{4} b^{2} x^{2} + 7 \, a^{5} b x - 3 \, a^{6}}{12 \, {\left (a^{7} b^{3} x^{7} + 3 \, a^{8} b^{2} x^{6} + 3 \, a^{9} b x^{5} + a^{10} x^{4}\right )}} - \frac {35 \, b^{4} \log \left (b x + a\right )}{a^{8}} + \frac {35 \, b^{4} \log \relax (x)}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^4,x, algorithm="maxima")

[Out]

1/12*(420*b^6*x^6 + 1050*a*b^5*x^5 + 770*a^2*b^4*x^4 + 105*a^3*b^3*x^3 - 21*a^4*b^2*x^2 + 7*a^5*b*x - 3*a^6)/(
a^7*b^3*x^7 + 3*a^8*b^2*x^6 + 3*a^9*b*x^5 + a^10*x^4) - 35*b^4*log(b*x + a)/a^8 + 35*b^4*log(x)/a^8

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mupad [B]  time = 0.17, size = 123, normalized size = 1.05 \begin {gather*} \frac {\frac {35\,b^3\,x^3}{4\,a^4}-\frac {7\,b^2\,x^2}{4\,a^3}-\frac {1}{4\,a}+\frac {385\,b^4\,x^4}{6\,a^5}+\frac {175\,b^5\,x^5}{2\,a^6}+\frac {35\,b^6\,x^6}{a^7}+\frac {7\,b\,x}{12\,a^2}}{a^3\,x^4+3\,a^2\,b\,x^5+3\,a\,b^2\,x^6+b^3\,x^7}-\frac {70\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x)^4),x)

[Out]

((35*b^3*x^3)/(4*a^4) - (7*b^2*x^2)/(4*a^3) - 1/(4*a) + (385*b^4*x^4)/(6*a^5) + (175*b^5*x^5)/(2*a^6) + (35*b^
6*x^6)/a^7 + (7*b*x)/(12*a^2))/(a^3*x^4 + b^3*x^7 + 3*a^2*b*x^5 + 3*a*b^2*x^6) - (70*b^4*atanh((2*b*x)/a + 1))
/a^8

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sympy [A]  time = 0.58, size = 128, normalized size = 1.09 \begin {gather*} \frac {- 3 a^{6} + 7 a^{5} b x - 21 a^{4} b^{2} x^{2} + 105 a^{3} b^{3} x^{3} + 770 a^{2} b^{4} x^{4} + 1050 a b^{5} x^{5} + 420 b^{6} x^{6}}{12 a^{10} x^{4} + 36 a^{9} b x^{5} + 36 a^{8} b^{2} x^{6} + 12 a^{7} b^{3} x^{7}} + \frac {35 b^{4} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x+a)**4,x)

[Out]

(-3*a**6 + 7*a**5*b*x - 21*a**4*b**2*x**2 + 105*a**3*b**3*x**3 + 770*a**2*b**4*x**4 + 1050*a*b**5*x**5 + 420*b
**6*x**6)/(12*a**10*x**4 + 36*a**9*b*x**5 + 36*a**8*b**2*x**6 + 12*a**7*b**3*x**7) + 35*b**4*(log(x) - log(a/b
+ x))/a**8

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