3.252 \(\int \frac{1}{x^3 (a-b x^2)^5} \, dx\)

Optimal. Leaf size=106 \[ \frac{2 b}{a^5 \left (a-b x^2\right )}+\frac{3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac{b}{3 a^3 \left (a-b x^2\right )^3}+\frac{b}{8 a^2 \left (a-b x^2\right )^4}-\frac{5 b \log \left (a-b x^2\right )}{2 a^6}+\frac{5 b \log (x)}{a^6}-\frac{1}{2 a^5 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0870354, antiderivative size = 106, 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} \[ \frac{2 b}{a^5 \left (a-b x^2\right )}+\frac{3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac{b}{3 a^3 \left (a-b x^2\right )^3}+\frac{b}{8 a^2 \left (a-b x^2\right )^4}-\frac{5 b \log \left (a-b x^2\right )}{2 a^6}+\frac{5 b \log (x)}{a^6}-\frac{1}{2 a^5 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a - b*x^2)^5),x]

[Out]

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

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

Mathematica [A]  time = 0.0631878, size = 83, normalized size = 0.78 \[ \frac{\frac{a \left (-260 a^2 b^2 x^4+125 a^3 b x^2-12 a^4+210 a b^3 x^6-60 b^4 x^8\right )}{x^2 \left (a-b x^2\right )^4}-60 b \log \left (a-b x^2\right )+120 b \log (x)}{24 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a - b*x^2)^5),x]

[Out]

((a*(-12*a^4 + 125*a^3*b*x^2 - 260*a^2*b^2*x^4 + 210*a*b^3*x^6 - 60*b^4*x^8))/(x^2*(a - b*x^2)^4) + 120*b*Log[
x] - 60*b*Log[a - b*x^2])/(24*a^6)

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Maple [A]  time = 0.015, size = 102, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{5}{x}^{2}}}+5\,{\frac{b\ln \left ( x \right ) }{{a}^{6}}}-2\,{\frac{b}{{a}^{5} \left ( b{x}^{2}-a \right ) }}-{\frac{b}{3\,{a}^{3} \left ( b{x}^{2}-a \right ) ^{3}}}+{\frac{b}{8\,{a}^{2} \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{3\,b}{4\,{a}^{4} \left ( b{x}^{2}-a \right ) ^{2}}}-{\frac{5\,b\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-b*x^2+a)^5,x)

[Out]

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

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Maxima [A]  time = 2.12544, size = 166, normalized size = 1.57 \begin{align*} -\frac{60 \, b^{4} x^{8} - 210 \, a b^{3} x^{6} + 260 \, a^{2} b^{2} x^{4} - 125 \, a^{3} b x^{2} + 12 \, a^{4}}{24 \,{\left (a^{5} b^{4} x^{10} - 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} - 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} - \frac{5 \, b \log \left (b x^{2} - a\right )}{2 \, a^{6}} + \frac{5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-b*x^2+a)^5,x, algorithm="maxima")

[Out]

-1/24*(60*b^4*x^8 - 210*a*b^3*x^6 + 260*a^2*b^2*x^4 - 125*a^3*b*x^2 + 12*a^4)/(a^5*b^4*x^10 - 4*a^6*b^3*x^8 +
6*a^7*b^2*x^6 - 4*a^8*b*x^4 + a^9*x^2) - 5/2*b*log(b*x^2 - a)/a^6 + 5/2*b*log(x^2)/a^6

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Fricas [B]  time = 1.27491, size = 440, normalized size = 4.15 \begin{align*} -\frac{60 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} - 125 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \,{\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 120 \,{\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (x\right )}{24 \,{\left (a^{6} b^{4} x^{10} - 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} - 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-b*x^2+a)^5,x, algorithm="fricas")

[Out]

-1/24*(60*a*b^4*x^8 - 210*a^2*b^3*x^6 + 260*a^3*b^2*x^4 - 125*a^4*b*x^2 + 12*a^5 + 60*(b^5*x^10 - 4*a*b^4*x^8
+ 6*a^2*b^3*x^6 - 4*a^3*b^2*x^4 + a^4*b*x^2)*log(b*x^2 - a) - 120*(b^5*x^10 - 4*a*b^4*x^8 + 6*a^2*b^3*x^6 - 4*
a^3*b^2*x^4 + a^4*b*x^2)*log(x))/(a^6*b^4*x^10 - 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 - 4*a^9*b*x^4 + a^10*x^2)

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Sympy [A]  time = 2.81119, size = 126, normalized size = 1.19 \begin{align*} - \frac{12 a^{4} - 125 a^{3} b x^{2} + 260 a^{2} b^{2} x^{4} - 210 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{9} x^{2} - 96 a^{8} b x^{4} + 144 a^{7} b^{2} x^{6} - 96 a^{6} b^{3} x^{8} + 24 a^{5} b^{4} x^{10}} + \frac{5 b \log{\left (x \right )}}{a^{6}} - \frac{5 b \log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(-b*x**2+a)**5,x)

[Out]

-(12*a**4 - 125*a**3*b*x**2 + 260*a**2*b**2*x**4 - 210*a*b**3*x**6 + 60*b**4*x**8)/(24*a**9*x**2 - 96*a**8*b*x
**4 + 144*a**7*b**2*x**6 - 96*a**6*b**3*x**8 + 24*a**5*b**4*x**10) + 5*b*log(x)/a**6 - 5*b*log(-a/b + x**2)/(2
*a**6)

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Giac [A]  time = 2.55494, size = 143, normalized size = 1.35 \begin{align*} \frac{5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} - \frac{5 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{6}} - \frac{5 \, b x^{2} + a}{2 \, a^{6} x^{2}} + \frac{125 \, b^{5} x^{8} - 548 \, a b^{4} x^{6} + 912 \, a^{2} b^{3} x^{4} - 688 \, a^{3} b^{2} x^{2} + 202 \, a^{4} b}{24 \,{\left (b x^{2} - a\right )}^{4} a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-b*x^2+a)^5,x, algorithm="giac")

[Out]

5/2*b*log(x^2)/a^6 - 5/2*b*log(abs(b*x^2 - a))/a^6 - 1/2*(5*b*x^2 + a)/(a^6*x^2) + 1/24*(125*b^5*x^8 - 548*a*b
^4*x^6 + 912*a^2*b^3*x^4 - 688*a^3*b^2*x^2 + 202*a^4*b)/((b*x^2 - a)^4*a^6)