Optimal. Leaf size=93 \[ -\frac {1}{8 a^2 x^8}+\frac {b}{3 a^3 x^6}-\frac {3 b^2}{4 a^4 x^4}+\frac {2 b^3}{a^5 x^2}+\frac {b^4}{2 a^5 \left (a+b x^2\right )}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log \left (a+b x^2\right )}{2 a^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 93, 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 {5 b^4 \log \left (a+b x^2\right )}{2 a^6}+\frac {5 b^4 \log (x)}{a^6}+\frac {b^4}{2 a^5 \left (a+b x^2\right )}+\frac {2 b^3}{a^5 x^2}-\frac {3 b^2}{4 a^4 x^4}+\frac {b}{3 a^3 x^6}-\frac {1}{8 a^2 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^5 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 x^5}-\frac {2 b}{a^3 x^4}+\frac {3 b^2}{a^4 x^3}-\frac {4 b^3}{a^5 x^2}+\frac {5 b^4}{a^6 x}-\frac {b^5}{a^5 (a+b x)^2}-\frac {5 b^5}{a^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{8 a^2 x^8}+\frac {b}{3 a^3 x^6}-\frac {3 b^2}{4 a^4 x^4}+\frac {2 b^3}{a^5 x^2}+\frac {b^4}{2 a^5 \left (a+b x^2\right )}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log \left (a+b x^2\right )}{2 a^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 79, normalized size = 0.85 \begin {gather*} \frac {a \left (-\frac {3 a^3}{x^8}+\frac {8 a^2 b}{x^6}-\frac {18 a b^2}{x^4}+12 b^3 \left (\frac {4}{x^2}+\frac {b}{a+b x^2}\right )\right )+120 b^4 \log (x)-60 b^4 \log \left (a+b x^2\right )}{24 a^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 88, normalized size = 0.95
method | result | size |
default | \(-\frac {b^{5} \left (-\frac {a}{b \left (b \,x^{2}+a \right )}+\frac {5 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{6}}-\frac {1}{8 a^{2} x^{8}}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{2}}{4 a^{4} x^{4}}+\frac {b}{3 a^{3} x^{6}}\) | \(88\) |
norman | \(\frac {-\frac {1}{8 a}+\frac {5 b \,x^{2}}{24 a^{2}}-\frac {5 b^{2} x^{4}}{12 a^{3}}+\frac {5 b^{3} x^{6}}{4 a^{4}}-\frac {5 b^{5} x^{10}}{2 a^{6}}}{x^{8} \left (b \,x^{2}+a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{6}}\) | \(89\) |
risch | \(\frac {\frac {5 b^{4} x^{8}}{2 a^{5}}+\frac {5 b^{3} x^{6}}{4 a^{4}}-\frac {5 b^{2} x^{4}}{12 a^{3}}+\frac {5 b \,x^{2}}{24 a^{2}}-\frac {1}{8 a}}{x^{8} \left (b \,x^{2}+a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{6}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 92, normalized size = 0.99 \begin {gather*} \frac {60 \, b^{4} x^{8} + 30 \, a b^{3} x^{6} - 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \, {\left (a^{5} b x^{10} + a^{6} x^{8}\right )}} - \frac {5 \, b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{6}} + \frac {5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.19, size = 112, normalized size = 1.20 \begin {gather*} \frac {60 \, a b^{4} x^{8} + 30 \, a^{2} b^{3} x^{6} - 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \, {\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (x\right )}{24 \, {\left (a^{6} b x^{10} + a^{7} x^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.24, size = 94, normalized size = 1.01 \begin {gather*} \frac {- 3 a^{4} + 5 a^{3} b x^{2} - 10 a^{2} b^{2} x^{4} + 30 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{6} x^{8} + 24 a^{5} b x^{10}} + \frac {5 b^{4} \log {\left (x \right )}}{a^{6}} - \frac {5 b^{4} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.79, size = 110, normalized size = 1.18 \begin {gather*} \frac {5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} - \frac {5 \, b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6}} + \frac {5 \, b^{5} x^{2} + 6 \, a b^{4}}{2 \, {\left (b x^{2} + a\right )} a^{6}} - \frac {125 \, b^{4} x^{8} - 48 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 8 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{6} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.73, size = 89, normalized size = 0.96 \begin {gather*} \frac {\frac {5\,b\,x^2}{24\,a^2}-\frac {1}{8\,a}-\frac {5\,b^2\,x^4}{12\,a^3}+\frac {5\,b^3\,x^6}{4\,a^4}+\frac {5\,b^4\,x^8}{2\,a^5}}{b\,x^{10}+a\,x^8}-\frac {5\,b^4\,\ln \left (b\,x^2+a\right )}{2\,a^6}+\frac {5\,b^4\,\ln \left (x\right )}{a^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________