Integrand size = 21, antiderivative size = 77 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=-\frac {56145628 \sqrt {x}}{43046721}+\frac {125000 x}{4782969}+\frac {50000 x^{3/2}}{1594323}+\frac {2500 x^2}{59049}+\frac {400 x^{5/2}}{6561}+\frac {200 x^3}{2187}+\frac {80 x^{7/2}}{567}+\frac {2 x^4}{9}-\frac {280728140 \log \left (5-9 \sqrt {x}\right )}{387420489} \]
125000/4782969*x+50000/1594323*x^(3/2)+2500/59049*x^2+400/6561*x^(5/2)+200 /2187*x^3+80/567*x^(7/2)+2/9*x^4-280728140/387420489*ln(5-9*x^(1/2))-56145 628/43046721*x^(1/2)
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (-196509698+3937500 \sqrt {x}+4725000 x+6378750 x^{3/2}+9185400 x^2+13778100 x^{5/2}+21257640 x^3+33480783 x^{7/2}\right )}{301327047}-\frac {280728140 \log \left (-5+9 \sqrt {x}\right )}{387420489} \]
(2*Sqrt[x]*(-196509698 + 3937500*Sqrt[x] + 4725000*x + 6378750*x^(3/2) + 9 185400*x^2 + 13778100*x^(5/2) + 21257640*x^3 + 33480783*x^(7/2)))/30132704 7 - (280728140*Log[-5 + 9*Sqrt[x]])/387420489
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2432, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx\) |
\(\Big \downarrow \) 2432 |
\(\displaystyle \int \left (\frac {8 x^{7/2}}{9 \sqrt {x}-5}-\frac {6}{9 \sqrt {x}-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {80 x^{7/2}}{567}+\frac {400 x^{5/2}}{6561}+\frac {50000 x^{3/2}}{1594323}+\frac {2 x^4}{9}+\frac {200 x^3}{2187}+\frac {2500 x^2}{59049}+\frac {125000 x}{4782969}-\frac {56145628 \sqrt {x}}{43046721}-\frac {280728140 \log \left (5-9 \sqrt {x}\right )}{387420489}\) |
(-56145628*Sqrt[x])/43046721 + (125000*x)/4782969 + (50000*x^(3/2))/159432 3 + (2500*x^2)/59049 + (400*x^(5/2))/6561 + (200*x^3)/2187 + (80*x^(7/2))/ 567 + (2*x^4)/9 - (280728140*Log[5 - 9*Sqrt[x]])/387420489
3.8.25.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[ Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || Poly Q[Pq, x^n])
Time = 1.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {2 x^{4}}{9}+\frac {80 x^{\frac {7}{2}}}{567}+\frac {200 x^{3}}{2187}+\frac {400 x^{\frac {5}{2}}}{6561}+\frac {2500 x^{2}}{59049}+\frac {50000 x^{\frac {3}{2}}}{1594323}+\frac {125000 x}{4782969}-\frac {56145628 \sqrt {x}}{43046721}-\frac {280728140 \ln \left (-5+9 \sqrt {x}\right )}{387420489}\) | \(50\) |
default | \(\frac {2 x^{4}}{9}+\frac {80 x^{\frac {7}{2}}}{567}+\frac {200 x^{3}}{2187}+\frac {400 x^{\frac {5}{2}}}{6561}+\frac {2500 x^{2}}{59049}+\frac {50000 x^{\frac {3}{2}}}{1594323}+\frac {125000 x}{4782969}-\frac {56145628 \sqrt {x}}{43046721}-\frac {280728140 \ln \left (-5+9 \sqrt {x}\right )}{387420489}\) | \(50\) |
trager | \(\frac {2 \left (531441 x^{3}+750141 x^{2}+851391 x +913891\right ) \left (x -1\right )}{4782969}+2 \left (\frac {40}{567} x^{3}+\frac {200}{6561} x^{2}+\frac {25000}{1594323} x -\frac {28072814}{43046721}\right ) \sqrt {x}-\frac {140364070 \ln \left (90 \sqrt {x}-25-81 x \right )}{387420489}\) | \(55\) |
meijerg | \(-\frac {4 \sqrt {x}}{3}-\frac {280728140 \ln \left (1-\frac {9 \sqrt {x}}{5}\right )}{387420489}+\frac {31250 \sqrt {x}\, \left (\frac {301327047 x^{\frac {7}{2}}}{15625}+\frac {38263752 x^{3}}{3125}+\frac {4960116 x^{\frac {5}{2}}}{625}+\frac {3306744 x^{2}}{625}+\frac {91854 x^{\frac {3}{2}}}{25}+\frac {13608 x}{5}+2268 \sqrt {x}+2520\right )}{2711943423}\) | \(57\) |
2/9*x^4+80/567*x^(7/2)+200/2187*x^3+400/6561*x^(5/2)+2500/59049*x^2+50000/ 1594323*x^(3/2)+125000/4782969*x-56145628/43046721*x^(1/2)-280728140/38742 0489*ln(-5+9*x^(1/2))
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2}{9} \, x^{4} + \frac {200}{2187} \, x^{3} + \frac {2500}{59049} \, x^{2} + \frac {4}{301327047} \, {\left (10628820 \, x^{3} + 4592700 \, x^{2} + 2362500 \, x - 98254849\right )} \sqrt {x} + \frac {125000}{4782969} \, x - \frac {280728140}{387420489} \, \log \left (9 \, \sqrt {x} - 5\right ) \]
2/9*x^4 + 200/2187*x^3 + 2500/59049*x^2 + 4/301327047*(10628820*x^3 + 4592 700*x^2 + 2362500*x - 98254849)*sqrt(x) + 125000/4782969*x - 280728140/387 420489*log(9*sqrt(x) - 5)
Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {80 x^{\frac {7}{2}}}{567} + \frac {400 x^{\frac {5}{2}}}{6561} + \frac {50000 x^{\frac {3}{2}}}{1594323} - \frac {56145628 \sqrt {x}}{43046721} + \frac {2 x^{4}}{9} + \frac {200 x^{3}}{2187} + \frac {2500 x^{2}}{59049} + \frac {125000 x}{4782969} - \frac {280728140 \log {\left (9 \sqrt {x} - 5 \right )}}{387420489} \]
80*x**(7/2)/567 + 400*x**(5/2)/6561 + 50000*x**(3/2)/1594323 - 56145628*sq rt(x)/43046721 + 2*x**4/9 + 200*x**3/2187 + 2500*x**2/59049 + 125000*x/478 2969 - 280728140*log(9*sqrt(x) - 5)/387420489
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2}{9} \, x^{4} + \frac {80}{567} \, x^{\frac {7}{2}} + \frac {200}{2187} \, x^{3} + \frac {400}{6561} \, x^{\frac {5}{2}} + \frac {2500}{59049} \, x^{2} + \frac {50000}{1594323} \, x^{\frac {3}{2}} + \frac {125000}{4782969} \, x - \frac {56145628}{43046721} \, \sqrt {x} - \frac {280728140}{387420489} \, \log \left (9 \, \sqrt {x} - 5\right ) \]
2/9*x^4 + 80/567*x^(7/2) + 200/2187*x^3 + 400/6561*x^(5/2) + 2500/59049*x^ 2 + 50000/1594323*x^(3/2) + 125000/4782969*x - 56145628/43046721*sqrt(x) - 280728140/387420489*log(9*sqrt(x) - 5)
Time = 0.36 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.65 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2}{9} \, x^{4} + \frac {80}{567} \, x^{\frac {7}{2}} + \frac {200}{2187} \, x^{3} + \frac {400}{6561} \, x^{\frac {5}{2}} + \frac {2500}{59049} \, x^{2} + \frac {50000}{1594323} \, x^{\frac {3}{2}} + \frac {125000}{4782969} \, x - \frac {56145628}{43046721} \, \sqrt {x} - \frac {280728140}{387420489} \, \log \left ({\left | 9 \, \sqrt {x} - 5 \right |}\right ) \]
2/9*x^4 + 80/567*x^(7/2) + 200/2187*x^3 + 400/6561*x^(5/2) + 2500/59049*x^ 2 + 50000/1594323*x^(3/2) + 125000/4782969*x - 56145628/43046721*sqrt(x) - 280728140/387420489*log(abs(9*sqrt(x) - 5))
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {125000\,x}{4782969}-\frac {280728140\,\ln \left (\sqrt {x}-\frac {5}{9}\right )}{387420489}+\frac {2500\,x^2}{59049}-\frac {56145628\,\sqrt {x}}{43046721}+\frac {200\,x^3}{2187}+\frac {2\,x^4}{9}+\frac {50000\,x^{3/2}}{1594323}+\frac {400\,x^{5/2}}{6561}+\frac {80\,x^{7/2}}{567} \]