3.250 \(\int \frac{1}{x (a-b x^2)^5} \, dx\)

Optimal. Leaf size=91 \[ \frac{1}{2 a^4 \left (a-b x^2\right )}+\frac{1}{4 a^3 \left (a-b x^2\right )^2}+\frac{1}{6 a^2 \left (a-b x^2\right )^3}-\frac{\log \left (a-b x^2\right )}{2 a^5}+\frac{\log (x)}{a^5}+\frac{1}{8 a \left (a-b x^2\right )^4} \]

[Out]

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

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Rubi [A]  time = 0.063872, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 44} \[ \frac{1}{2 a^4 \left (a-b x^2\right )}+\frac{1}{4 a^3 \left (a-b x^2\right )^2}+\frac{1}{6 a^2 \left (a-b x^2\right )^3}-\frac{\log \left (a-b x^2\right )}{2 a^5}+\frac{\log (x)}{a^5}+\frac{1}{8 a \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a - b*x^2)^5),x]

[Out]

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

Rule 266

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]]

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

Mathematica [A]  time = 0.0322645, size = 67, normalized size = 0.74 \[ \frac{\frac{a \left (-52 a^2 b x^2+25 a^3+42 a b^2 x^4-12 b^3 x^6\right )}{\left (a-b x^2\right )^4}-12 \log \left (a-b x^2\right )+24 \log (x)}{24 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a - b*x^2)^5),x]

[Out]

((a*(25*a^3 - 52*a^2*b*x^2 + 42*a*b^2*x^4 - 12*b^3*x^6))/(a - b*x^2)^4 + 24*Log[x] - 12*Log[a - b*x^2])/(24*a^
5)

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Maple [A]  time = 0.013, size = 87, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{5}}}-{\frac{1}{2\,{a}^{4} \left ( b{x}^{2}-a \right ) }}+{\frac{1}{8\,a \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{1}{4\,{a}^{3} \left ( b{x}^{2}-a \right ) ^{2}}}-{\frac{1}{6\,{a}^{2} \left ( b{x}^{2}-a \right ) ^{3}}}-{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-b*x^2+a)^5,x)

[Out]

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

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Maxima [A]  time = 1.36549, size = 143, normalized size = 1.57 \begin{align*} -\frac{12 \, b^{3} x^{6} - 42 \, a b^{2} x^{4} + 52 \, a^{2} b x^{2} - 25 \, a^{3}}{24 \,{\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac{\log \left (b x^{2} - a\right )}{2 \, a^{5}} + \frac{\log \left (x^{2}\right )}{2 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b*x^2+a)^5,x, algorithm="maxima")

[Out]

-1/24*(12*b^3*x^6 - 42*a*b^2*x^4 + 52*a^2*b*x^2 - 25*a^3)/(a^4*b^4*x^8 - 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 - 4*a^7
*b*x^2 + a^8) - 1/2*log(b*x^2 - a)/a^5 + 1/2*log(x^2)/a^5

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Fricas [B]  time = 1.25652, size = 379, normalized size = 4.16 \begin{align*} -\frac{12 \, a b^{3} x^{6} - 42 \, a^{2} b^{2} x^{4} + 52 \, a^{3} b x^{2} - 25 \, a^{4} + 12 \,{\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} - a\right ) - 24 \,{\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (x\right )}{24 \,{\left (a^{5} b^{4} x^{8} - 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} - 4 \, a^{8} b x^{2} + a^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b*x^2+a)^5,x, algorithm="fricas")

[Out]

-1/24*(12*a*b^3*x^6 - 42*a^2*b^2*x^4 + 52*a^3*b*x^2 - 25*a^4 + 12*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a
^3*b*x^2 + a^4)*log(b*x^2 - a) - 24*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a^3*b*x^2 + a^4)*log(x))/(a^5*b
^4*x^8 - 4*a^6*b^3*x^6 + 6*a^7*b^2*x^4 - 4*a^8*b*x^2 + a^9)

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Sympy [A]  time = 1.40765, size = 104, normalized size = 1.14 \begin{align*} - \frac{- 25 a^{3} + 52 a^{2} b x^{2} - 42 a b^{2} x^{4} + 12 b^{3} x^{6}}{24 a^{8} - 96 a^{7} b x^{2} + 144 a^{6} b^{2} x^{4} - 96 a^{5} b^{3} x^{6} + 24 a^{4} b^{4} x^{8}} + \frac{\log{\left (x \right )}}{a^{5}} - \frac{\log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b*x**2+a)**5,x)

[Out]

-(-25*a**3 + 52*a**2*b*x**2 - 42*a*b**2*x**4 + 12*b**3*x**6)/(24*a**8 - 96*a**7*b*x**2 + 144*a**6*b**2*x**4 -
96*a**5*b**3*x**6 + 24*a**4*b**4*x**8) + log(x)/a**5 - log(-a/b + x**2)/(2*a**5)

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Giac [A]  time = 2.94426, size = 115, normalized size = 1.26 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a^{5}} - \frac{\log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{5}} + \frac{25 \, b^{4} x^{8} - 112 \, a b^{3} x^{6} + 192 \, a^{2} b^{2} x^{4} - 152 \, a^{3} b x^{2} + 50 \, a^{4}}{24 \,{\left (b x^{2} - a\right )}^{4} a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-b*x^2+a)^5,x, algorithm="giac")

[Out]

1/2*log(x^2)/a^5 - 1/2*log(abs(b*x^2 - a))/a^5 + 1/24*(25*b^4*x^8 - 112*a*b^3*x^6 + 192*a^2*b^2*x^4 - 152*a^3*
b*x^2 + 50*a^4)/((b*x^2 - a)^4*a^5)