3.2.92 \(\int \frac {1}{x^5 (a+b x)^3} \, dx\) [192]

Optimal. Leaf size=97 \[ -\frac {1}{4 a^3 x^4}+\frac {b}{a^4 x^3}-\frac {3 b^2}{a^5 x^2}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}+\frac {5 b^4}{a^6 (a+b x)}+\frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7} \]

[Out]

-1/4/a^3/x^4+b/a^4/x^3-3*b^2/a^5/x^2+10*b^3/a^6/x+1/2*b^4/a^5/(b*x+a)^2+5*b^4/a^6/(b*x+a)+15*b^4*ln(x)/a^7-15*
b^4*ln(b*x+a)/a^7

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Rubi [A]
time = 0.04, antiderivative size = 97, 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 = {46} \begin {gather*} \frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7}+\frac {5 b^4}{a^6 (a+b x)}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}-\frac {3 b^2}{a^5 x^2}+\frac {b}{a^4 x^3}-\frac {1}{4 a^3 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)^3),x]

[Out]

-1/4*1/(a^3*x^4) + b/(a^4*x^3) - (3*b^2)/(a^5*x^2) + (10*b^3)/(a^6*x) + b^4/(2*a^5*(a + b*x)^2) + (5*b^4)/(a^6
*(a + b*x)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a + b*x])/a^7

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 90, normalized size = 0.93 \begin {gather*} \frac {\frac {a \left (-a^5+2 a^4 b x-5 a^3 b^2 x^2+20 a^2 b^3 x^3+90 a b^4 x^4+60 b^5 x^5\right )}{x^4 (a+b x)^2}+60 b^4 \log (x)-60 b^4 \log (a+b x)}{4 a^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)^3),x]

[Out]

((a*(-a^5 + 2*a^4*b*x - 5*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 90*a*b^4*x^4 + 60*b^5*x^5))/(x^4*(a + b*x)^2) + 60*b^
4*Log[x] - 60*b^4*Log[a + b*x])/(4*a^7)

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Mathics [A]
time = 2.85, size = 120, normalized size = 1.24 \begin {gather*} \frac {a \left (-a^5+2 a^4 b x-5 a^3 b^2 x^2+20 a^2 b^3 x^3+90 a b^4 x^4+60 b^5 x^5\right )+60 b^4 x^4 \left (a^2+2 a b x+b^2 x^2\right ) \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )}{4 a^7 x^4 \left (a^2+2 a b x+b^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^5*(a + b*x)^3),x]')

[Out]

(a (-a ^ 5 + 2 a ^ 4 b x - 5 a ^ 3 b ^ 2 x ^ 2 + 20 a ^ 2 b ^ 3 x ^ 3 + 90 a b ^ 4 x ^ 4 + 60 b ^ 5 x ^ 5) + 6
0 b ^ 4 x ^ 4 (a ^ 2 + 2 a b x + b ^ 2 x ^ 2) (Log[x] - Log[(a + b x) / b])) / (4 a ^ 7 x ^ 4 (a ^ 2 + 2 a b x
+ b ^ 2 x ^ 2))

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Maple [A]
time = 0.08, size = 94, normalized size = 0.97

method	result	size
default	\(-\frac {1}{4 a^{3} x^{4}}+\frac {b}{a^{4} x^{3}}-\frac {3 b^{2}}{a^{5} x^{2}}+\frac {10 b^{3}}{a^{6} x}+\frac {b^{4}}{2 a^{5} \left (b x +a \right )^{2}}+\frac {5 b^{4}}{a^{6} \left (b x +a \right )}+\frac {15 b^{4} \ln \left (x \right )}{a^{7}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}\)	\(94\)
norman	\(\frac {-\frac {1}{4 a}+\frac {b x}{2 a^{2}}-\frac {5 b^{2} x^{2}}{4 a^{3}}+\frac {5 b^{3} x^{3}}{a^{4}}-\frac {30 b^{5} x^{5}}{a^{6}}-\frac {45 b^{6} x^{6}}{2 a^{7}}}{x^{4} \left (b x +a \right )^{2}}+\frac {15 b^{4} \ln \left (x \right )}{a^{7}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}\)	\(94\)
risch	\(\frac {\frac {15 b^{5} x^{5}}{a^{6}}+\frac {45 b^{4} x^{4}}{2 a^{5}}+\frac {5 b^{3} x^{3}}{a^{4}}-\frac {5 b^{2} x^{2}}{4 a^{3}}+\frac {b x}{2 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b x +a \right )^{2}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}+\frac {15 b^{4} \ln \left (-x \right )}{a^{7}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4/a^3/x^4+b/a^4/x^3-3*b^2/a^5/x^2+10*b^3/a^6/x+1/2*b^4/a^5/(b*x+a)^2+5*b^4/a^6/(b*x+a)+15*b^4*ln(x)/a^7-15*
b^4*ln(b*x+a)/a^7

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Maxima [A]
time = 0.25, size = 108, normalized size = 1.11 \begin {gather*} \frac {60 \, b^{5} x^{5} + 90 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 5 \, a^{3} b^{2} x^{2} + 2 \, a^{4} b x - a^{5}}{4 \, {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} - \frac {15 \, b^{4} \log \left (b x + a\right )}{a^{7}} + \frac {15 \, b^{4} \log \left (x\right )}{a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^3,x, algorithm="maxima")

[Out]

1/4*(60*b^5*x^5 + 90*a*b^4*x^4 + 20*a^2*b^3*x^3 - 5*a^3*b^2*x^2 + 2*a^4*b*x - a^5)/(a^6*b^2*x^6 + 2*a^7*b*x^5
+ a^8*x^4) - 15*b^4*log(b*x + a)/a^7 + 15*b^4*log(x)/a^7

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Fricas [A]
time = 0.31, size = 152, normalized size = 1.57 \begin {gather*} \frac {60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6} - 60 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(60*a*b^5*x^5 + 90*a^2*b^4*x^4 + 20*a^3*b^3*x^3 - 5*a^4*b^2*x^2 + 2*a^5*b*x - a^6 - 60*(b^6*x^6 + 2*a*b^5*
x^5 + a^2*b^4*x^4)*log(b*x + a) + 60*(b^6*x^6 + 2*a*b^5*x^5 + a^2*b^4*x^4)*log(x))/(a^7*b^2*x^6 + 2*a^8*b*x^5
+ a^9*x^4)

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Sympy [A]
time = 0.23, size = 102, normalized size = 1.05 \begin {gather*} \frac {- a^{5} + 2 a^{4} b x - 5 a^{3} b^{2} x^{2} + 20 a^{2} b^{3} x^{3} + 90 a b^{4} x^{4} + 60 b^{5} x^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{5} + 4 a^{6} b^{2} x^{6}} + \frac {15 b^{4} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x+a)**3,x)

[Out]

(-a**5 + 2*a**4*b*x - 5*a**3*b**2*x**2 + 20*a**2*b**3*x**3 + 90*a*b**4*x**4 + 60*b**5*x**5)/(4*a**8*x**4 + 8*a
**7*b*x**5 + 4*a**6*b**2*x**6) + 15*b**4*(log(x) - log(a/b + x))/a**7

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Giac [A]
time = 0.00, size = 107, normalized size = 1.10 \begin {gather*} -\frac {15 b^{5} \ln \left |x b+a\right |}{b a^{7}}+\frac {15 b^{4} \ln \left |x\right |}{a^{7}}+\frac {\frac {1}{4} \left (60 b^{5} a x^{5}+90 b^{4} a^{2} x^{4}+20 b^{3} a^{3} x^{3}-5 b^{2} a^{4} x^{2}+2 b a^{5} x-a^{6}\right )}{a^{7} x^{4} \left (b x+a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^3,x)

[Out]

-15*b^4*log(abs(b*x + a))/a^7 + 15*b^4*log(abs(x))/a^7 + 1/4*(60*a*b^5*x^5 + 90*a^2*b^4*x^4 + 20*a^3*b^3*x^3 -
5*a^4*b^2*x^2 + 2*a^5*b*x - a^6)/((b*x + a)^2*a^7*x^4)

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Mupad [B]
time = 0.09, size = 101, normalized size = 1.04 \begin {gather*} \frac {\frac {5\,b^3\,x^3}{a^4}-\frac {5\,b^2\,x^2}{4\,a^3}-\frac {1}{4\,a}+\frac {45\,b^4\,x^4}{2\,a^5}+\frac {15\,b^5\,x^5}{a^6}+\frac {b\,x}{2\,a^2}}{a^2\,x^4+2\,a\,b\,x^5+b^2\,x^6}-\frac {30\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x)^3),x)

[Out]

((5*b^3*x^3)/a^4 - (5*b^2*x^2)/(4*a^3) - 1/(4*a) + (45*b^4*x^4)/(2*a^5) + (15*b^5*x^5)/a^6 + (b*x)/(2*a^2))/(a
^2*x^4 + b^2*x^6 + 2*a*b*x^5) - (30*b^4*atanh((2*b*x)/a + 1))/a^7

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