3.13.75 \(\int \frac {1}{x^{16} (a+b x^5)} \, dx\) [1275]

Optimal. Leaf size=63 \[ -\frac {1}{15 a x^{15}}+\frac {b}{10 a^2 x^{10}}-\frac {b^2}{5 a^3 x^5}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^5\right )}{5 a^4} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46} \begin {gather*} \frac {b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac {b^3 \log (x)}{a^4}-\frac {b^2}{5 a^3 x^5}+\frac {b}{10 a^2 x^{10}}-\frac {1}{15 a x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^16*(a + b*x^5)),x]

[Out]

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

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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x^{16} \left (a+b x^5\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^4 (a+b x)} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {1}{15 a x^{15}}+\frac {b}{10 a^2 x^{10}}-\frac {b^2}{5 a^3 x^5}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^5\right )}{5 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 63, normalized size = 1.00 \begin {gather*} -\frac {1}{15 a x^{15}}+\frac {b}{10 a^2 x^{10}}-\frac {b^2}{5 a^3 x^5}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^5\right )}{5 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^16*(a + b*x^5)),x]

[Out]

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

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Maple [A]
time = 0.21, size = 56, normalized size = 0.89

method	result	size
default	\(-\frac {1}{15 a \,x^{15}}+\frac {b}{10 a^{2} x^{10}}-\frac {b^{2}}{5 a^{3} x^{5}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (b \,x^{5}+a \right )}{5 a^{4}}\)	\(56\)
norman	\(\frac {-\frac {1}{15 a}+\frac {b \,x^{5}}{10 a^{2}}-\frac {b^{2} x^{10}}{5 a^{3}}}{x^{15}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (b \,x^{5}+a \right )}{5 a^{4}}\)	\(58\)
risch	\(\frac {-\frac {1}{15 a}+\frac {b \,x^{5}}{10 a^{2}}-\frac {b^{2} x^{10}}{5 a^{3}}}{x^{15}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (-b \,x^{5}-a \right )}{5 a^{4}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^16/(b*x^5+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 58, normalized size = 0.92 \begin {gather*} \frac {b^{3} \log \left (b x^{5} + a\right )}{5 \, a^{4}} - \frac {b^{3} \log \left (x^{5}\right )}{5 \, a^{4}} - \frac {6 \, b^{2} x^{10} - 3 \, a b x^{5} + 2 \, a^{2}}{30 \, a^{3} x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^5+a),x, algorithm="maxima")

[Out]

1/5*b^3*log(b*x^5 + a)/a^4 - 1/5*b^3*log(x^5)/a^4 - 1/30*(6*b^2*x^10 - 3*a*b*x^5 + 2*a^2)/(a^3*x^15)

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Fricas [A]
time = 0.38, size = 58, normalized size = 0.92 \begin {gather*} \frac {6 \, b^{3} x^{15} \log \left (b x^{5} + a\right ) - 30 \, b^{3} x^{15} \log \left (x\right ) - 6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - 2 \, a^{3}}{30 \, a^{4} x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^5+a),x, algorithm="fricas")

[Out]

1/30*(6*b^3*x^15*log(b*x^5 + a) - 30*b^3*x^15*log(x) - 6*a*b^2*x^10 + 3*a^2*b*x^5 - 2*a^3)/(a^4*x^15)

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Sympy [A]
time = 0.31, size = 56, normalized size = 0.89 \begin {gather*} \frac {- 2 a^{2} + 3 a b x^{5} - 6 b^{2} x^{10}}{30 a^{3} x^{15}} - \frac {b^{3} \log {\left (x \right )}}{a^{4}} + \frac {b^{3} \log {\left (\frac {a}{b} + x^{5} \right )}}{5 a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**16/(b*x**5+a),x)

[Out]

(-2*a**2 + 3*a*b*x**5 - 6*b**2*x**10)/(30*a**3*x**15) - b**3*log(x)/a**4 + b**3*log(a/b + x**5)/(5*a**4)

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Giac [A]
time = 1.65, size = 69, normalized size = 1.10 \begin {gather*} \frac {b^{3} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{4}} - \frac {b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {11 \, b^{3} x^{15} - 6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - 2 \, a^{3}}{30 \, a^{4} x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^5+a),x, algorithm="giac")

[Out]

1/5*b^3*log(abs(b*x^5 + a))/a^4 - b^3*log(abs(x))/a^4 + 1/30*(11*b^3*x^15 - 6*a*b^2*x^10 + 3*a^2*b*x^5 - 2*a^3
)/(a^4*x^15)

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Mupad [B]
time = 1.09, size = 58, normalized size = 0.92 \begin {gather*} \frac {b^3\,\ln \left (b\,x^5+a\right )}{5\,a^4}-\frac {\frac {1}{15\,a}-\frac {b\,x^5}{10\,a^2}+\frac {b^2\,x^{10}}{5\,a^3}}{x^{15}}-\frac {b^3\,\ln \left (x\right )}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^16*(a + b*x^5)),x)

[Out]

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

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