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

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

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.07, size = 83, normalized size = 0.78 \[ \frac {\frac {a \left (-12 a^4+125 a^3 b x^2-260 a^2 b^2 x^4+210 a b^3 x^6-60 b^4 x^8\right )}{x^2 \left (a-b x^2\right )^4}-60 b \log \left (a-b x^2\right )+120 b \log (x)}{24 a^6} \]

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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fricas [B]  time = 0.72, size = 209, normalized size = 1.97 \[ -\frac {60 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} - 125 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \, {\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 120 \, {\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \relax (x)}{24 \, {\left (a^{6} b^{4} x^{10} - 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} - 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \]

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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giac [A]  time = 0.64, size = 106, normalized size = 1.00 \[ \frac {5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} - \frac {5 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{6}} - \frac {5 \, b x^{2} + a}{2 \, a^{6} x^{2}} + \frac {125 \, b^{5} x^{8} - 548 \, a b^{4} x^{6} + 912 \, a^{2} b^{3} x^{4} - 688 \, a^{3} b^{2} x^{2} + 202 \, a^{4} b}{24 \, {\left (b x^{2} - a\right )}^{4} a^{6}} \]

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)

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maple [A]  time = 0.02, size = 102, normalized size = 0.96 \[ \frac {b}{8 \left (b \,x^{2}-a \right )^{4} a^{2}}-\frac {b}{3 \left (b \,x^{2}-a \right )^{3} a^{3}}+\frac {3 b}{4 \left (b \,x^{2}-a \right )^{2} a^{4}}-\frac {2 b}{\left (b \,x^{2}-a \right ) a^{5}}+\frac {5 b \ln \relax (x )}{a^{6}}-\frac {5 b \ln \left (b \,x^{2}-a \right )}{2 a^{6}}-\frac {1}{2 a^{5} x^{2}} \]

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

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maxima [A]  time = 1.35, size = 123, normalized size = 1.16 \[ -\frac {60 \, b^{4} x^{8} - 210 \, a b^{3} x^{6} + 260 \, a^{2} b^{2} x^{4} - 125 \, a^{3} b x^{2} + 12 \, a^{4}}{24 \, {\left (a^{5} b^{4} x^{10} - 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} - 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} - \frac {5 \, b \log \left (b x^{2} - a\right )}{2 \, a^{6}} + \frac {5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} \]

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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mupad [B]  time = 5.35, size = 120, normalized size = 1.13 \[ \frac {5\,b\,\ln \relax (x)}{a^6}-\frac {5\,b\,\ln \left (a-b\,x^2\right )}{2\,a^6}-\frac {\frac {1}{2\,a}-\frac {125\,b\,x^2}{24\,a^2}+\frac {65\,b^2\,x^4}{6\,a^3}-\frac {35\,b^3\,x^6}{4\,a^4}+\frac {5\,b^4\,x^8}{2\,a^5}}{a^4\,x^2-4\,a^3\,b\,x^4+6\,a^2\,b^2\,x^6-4\,a\,b^3\,x^8+b^4\,x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a - b*x^2)^5),x)

[Out]

(5*b*log(x))/a^6 - (5*b*log(a - b*x^2))/(2*a^6) - (1/(2*a) - (125*b*x^2)/(24*a^2) + (65*b^2*x^4)/(6*a^3) - (35
*b^3*x^6)/(4*a^4) + (5*b^4*x^8)/(2*a^5))/(a^4*x^2 + b^4*x^10 - 4*a^3*b*x^4 - 4*a*b^3*x^8 + 6*a^2*b^2*x^6)

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sympy [A]  time = 0.76, size = 126, normalized size = 1.19 \[ - \frac {12 a^{4} - 125 a^{3} b x^{2} + 260 a^{2} b^{2} x^{4} - 210 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{9} x^{2} - 96 a^{8} b x^{4} + 144 a^{7} b^{2} x^{6} - 96 a^{6} b^{3} x^{8} + 24 a^{5} b^{4} x^{10}} + \frac {5 b \log {\relax (x )}}{a^{6}} - \frac {5 b \log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{6}} \]

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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