Optimal. Leaf size=81 \[ -\frac {1}{490 x^2}-\frac {69}{1715 x}-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}-\frac {234 \tan ^{-1}\left (\frac {3-2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}}+\frac {80 \log (x)}{2401}-\frac {40 \log \left (7-6 x+2 x^2\right )}{2401} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {754, 814, 648,
632, 210, 642} \begin {gather*} -\frac {234 \text {ArcTan}\left (\frac {3-2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}}-\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}-\frac {1}{490 x^2}-\frac {40 \log \left (2 x^2-6 x+7\right )}{2401}-\frac {69}{1715 x}+\frac {80 \log (x)}{2401} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx &=-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {1}{140} \int \frac {4+36 x}{x^3 \left (7-6 x+2 x^2\right )} \, dx\\ &=-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {1}{140} \int \left (\frac {4}{7 x^3}+\frac {276}{49 x^2}+\frac {1600}{343 x}-\frac {8 (-717+400 x)}{343 \left (7-6 x+2 x^2\right )}\right ) \, dx\\ &=-\frac {1}{490 x^2}-\frac {69}{1715 x}-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {80 \log (x)}{2401}-\frac {2 \int \frac {-717+400 x}{7-6 x+2 x^2} \, dx}{12005}\\ &=-\frac {1}{490 x^2}-\frac {69}{1715 x}-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {80 \log (x)}{2401}-\frac {40 \int \frac {-6+4 x}{7-6 x+2 x^2} \, dx}{2401}+\frac {234 \int \frac {1}{7-6 x+2 x^2} \, dx}{12005}\\ &=-\frac {1}{490 x^2}-\frac {69}{1715 x}-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {80 \log (x)}{2401}-\frac {40 \log \left (7-6 x+2 x^2\right )}{2401}-\frac {468 \text {Subst}\left (\int \frac {1}{-20-x^2} \, dx,x,-6+4 x\right )}{12005}\\ &=-\frac {1}{490 x^2}-\frac {69}{1715 x}-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}-\frac {234 \tan ^{-1}\left (\frac {3-2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}}+\frac {80 \log (x)}{2401}-\frac {40 \log \left (7-6 x+2 x^2\right )}{2401}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 70, normalized size = 0.86 \begin {gather*} \frac {-\frac {1225}{x^2}-\frac {4200}{x}-\frac {140 (-41+9 x)}{7-6 x+2 x^2}+468 \sqrt {5} \tan ^{-1}\left (\frac {-3+2 x}{\sqrt {5}}\right )+4000 \log (x)-2000 \log \left (7-6 x+2 x^2\right )}{120050} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 62, normalized size = 0.77
method | result | size |
default | \(-\frac {4 \left (\frac {63 x}{20}-\frac {287}{20}\right )}{2401 \left (x^{2}-3 x +\frac {7}{2}\right )}-\frac {40 \ln \left (2 x^{2}-6 x +7\right )}{2401}+\frac {234 \sqrt {5}\, \arctan \left (\frac {\left (4 x -6\right ) \sqrt {5}}{10}\right )}{60025}-\frac {1}{98 x^{2}}-\frac {12}{343 x}+\frac {80 \ln \left (x \right )}{2401}\) | \(62\) |
risch | \(\frac {-\frac {138}{1715} x^{3}+\frac {407}{1715} x^{2}-\frac {9}{49} x -\frac {1}{14}}{x^{2} \left (2 x^{2}-6 x +7\right )}-\frac {40 \ln \left (4 x^{2}-12 x +14\right )}{2401}+\frac {234 \sqrt {5}\, \arctan \left (\frac {\left (2 x -3\right ) \sqrt {5}}{5}\right )}{60025}+\frac {80 \ln \left (x \right )}{2401}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.37, size = 69, normalized size = 0.85 \begin {gather*} \frac {234}{60025} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - \frac {276 \, x^{3} - 814 \, x^{2} + 630 \, x + 245}{3430 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )}} - \frac {40}{2401} \, \log \left (2 \, x^{2} - 6 \, x + 7\right ) + \frac {80}{2401} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 116, normalized size = 1.43 \begin {gather*} -\frac {9660 \, x^{3} - 468 \, \sqrt {5} {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - 28490 \, x^{2} + 2000 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \log \left (2 \, x^{2} - 6 \, x + 7\right ) - 4000 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \log \left (x\right ) + 22050 \, x + 8575}{120050 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.09, size = 80, normalized size = 0.99 \begin {gather*} \frac {80 \log {\left (x \right )}}{2401} - \frac {40 \log {\left (x^{2} - 3 x + \frac {7}{2} \right )}}{2401} + \frac {234 \sqrt {5} \operatorname {atan}{\left (\frac {2 \sqrt {5} x}{5} - \frac {3 \sqrt {5}}{5} \right )}}{60025} + \frac {- 276 x^{3} + 814 x^{2} - 630 x - 245}{6860 x^{4} - 20580 x^{3} + 24010 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.42, size = 67, normalized size = 0.83 \begin {gather*} \frac {234}{60025} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - \frac {276 \, x^{3} - 814 \, x^{2} + 630 \, x + 245}{3430 \, {\left (2 \, x^{2} - 6 \, x + 7\right )} x^{2}} - \frac {40}{2401} \, \log \left (2 \, x^{2} - 6 \, x + 7\right ) + \frac {80}{2401} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.11, size = 77, normalized size = 0.95 \begin {gather*} \frac {80\,\ln \left (x\right )}{2401}-\frac {\frac {69\,x^3}{1715}-\frac {407\,x^2}{3430}+\frac {9\,x}{98}+\frac {1}{28}}{x^4-3\,x^3+\frac {7\,x^2}{2}}-\ln \left (x-\frac {3}{2}-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {40}{2401}+\frac {\sqrt {5}\,117{}\mathrm {i}}{60025}\right )+\ln \left (x-\frac {3}{2}+\frac {\sqrt {5}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {40}{2401}+\frac {\sqrt {5}\,117{}\mathrm {i}}{60025}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________