\(\int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx\) [128]

Optimal result

Integrand size = 21, antiderivative size = 77 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=-\frac {56145628 \sqrt {x}}{43046721}+\frac {125000 x}{4782969}+\frac {50000 x^{3/2}}{1594323}+\frac {2500 x^2}{59049}+\frac {400 x^{5/2}}{6561}+\frac {200 x^3}{2187}+\frac {80 x^{7/2}}{567}+\frac {2 x^4}{9}-\frac {280728140 \log \left (5-9 \sqrt {x}\right )}{387420489} \] Output:

-56145628/43046721*x^(1/2)+125000/4782969*x+50000/1594323*x^(3/2)+2500/590 
49*x^2+400/6561*x^(5/2)+200/2187*x^3+80/567*x^(7/2)+2/9*x^4-280728140/3874 
20489*ln(5-9*x^(1/2))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (-196509698+3937500 \sqrt {x}+4725000 x+6378750 x^{3/2}+9185400 x^2+13778100 x^{5/2}[phone] x^3+33480783 x^{7/2}\right )}{301327047}-\frac {280728140 \log \left (-5+9 \sqrt {x}\right )}{387420489} \] Input:

Integrate[(6 - 8*x^(7/2))/(5 - 9*Sqrt[x]),x]
 

Output:

(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
 

Rubi [A] (verified)

Time = 0.38 (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

\(\displaystyle \frac {80 x^{7/2}}{567}+\frac {400 x^{5/2}}{6561}+\frac {50000 x^{3/2}}{1594323}+\frac {2 x^4}{9}+\frac {200 x^3}{2187}+\frac {2500 x^2}{59049}+\frac {125000 x}{4782969}-\frac {56145628 \sqrt {x}}{43046721}-\frac {280728140 \log \left (5-9 \sqrt {x}\right )}{387420489}\)

Input:

Int[(6 - 8*x^(7/2))/(5 - 9*Sqrt[x]),x]
 

Output:

(-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
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2432

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])
 

Maple [A] (verified)

Time = 0.07 (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	\(\frac {2 \left (531441 x^{3}+750141 x^{2}+851391 x +913891\right ) \left (-1+x \right )}{4782969}+2 \left (\frac {40}{567} x^{3}+\frac {200}{6561} x^{2}+\frac {25000}{1594323} x -\frac {28072814}{43046721}\right ) \sqrt {x}-\frac {140364070 \ln \left (90 \sqrt {x}-25-81 x \right )}{387420489}\)	\(55\)
meijerg	\(-\frac {4 \sqrt {x}}{3}-\frac {280728140 \ln \left (1-\frac {9 \sqrt {x}}{5}\right )}{387420489}+\frac {31250 \sqrt {x}\, \left (\frac {301327047 x^{\frac {7}{2}}}{15625}+\frac {38263752 x^{3}}{3125}+\frac {4960116 x^{\frac {5}{2}}}{625}+\frac {3306744 x^{2}}{625}+\frac {91854 x^{\frac {3}{2}}}{25}+\frac {13608 x}{5}+2268 \sqrt {x}+2520\right )}{2711943423}\)	\(57\)

Input:

int((6-8*x^(7/2))/(5-9*x^(1/2)),x,method=_RETURNVERBOSE)
 

Output:

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))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2}{9} \, x^{4} + \frac {200}{2187} \, x^{3} + \frac {2500}{59049} \, x^{2} + \frac {4}{301327047} \, {\left (10628820 \, x^{3} + 4592700 \, x^{2} + 2362500 \, x - 98254849\right )} \sqrt {x} + \frac {125000}{4782969} \, x - \frac {280728140}{387420489} \, \log \left (9 \, \sqrt {x} - 5\right ) \] Input:

integrate((6-8*x^(7/2))/(5-9*x^(1/2)),x, algorithm="fricas")
 

Output:

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)
 

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {80 x^{\frac {7}{2}}}{567} + \frac {400 x^{\frac {5}{2}}}{6561} + \frac {50000 x^{\frac {3}{2}}}{1594323} - \frac {56145628 \sqrt {x}}{43046721} + \frac {2 x^{4}}{9} + \frac {200 x^{3}}{2187} + \frac {2500 x^{2}}{59049} + \frac {125000 x}{4782969} - \frac {280728140 \log {\left (9 \sqrt {x} - 5 \right )}}{387420489} \] Input:

integrate((6-8*x**(7/2))/(5-9*x**(1/2)),x)
                                                                                    
                                                                                    
 

Output:

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
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2}{9} \, x^{4} + \frac {80}{567} \, x^{\frac {7}{2}} + \frac {200}{2187} \, x^{3} + \frac {400}{6561} \, x^{\frac {5}{2}} + \frac {2500}{59049} \, x^{2} + \frac {50000}{1594323} \, x^{\frac {3}{2}} + \frac {125000}{4782969} \, x - \frac {56145628}{43046721} \, \sqrt {x} - \frac {280728140}{387420489} \, \log \left (9 \, \sqrt {x} - 5\right ) \] Input:

integrate((6-8*x^(7/2))/(5-9*x^(1/2)),x, algorithm="maxima")
 

Output:

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)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.65 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {2}{9} \, x^{4} + \frac {80}{567} \, x^{\frac {7}{2}} + \frac {200}{2187} \, x^{3} + \frac {400}{6561} \, x^{\frac {5}{2}} + \frac {2500}{59049} \, x^{2} + \frac {50000}{1594323} \, x^{\frac {3}{2}} + \frac {125000}{4782969} \, x - \frac {56145628}{43046721} \, \sqrt {x} - \frac {280728140}{387420489} \, \log \left ({\left | 9 \, \sqrt {x} - 5 \right |}\right ) \] Input:

integrate((6-8*x^(7/2))/(5-9*x^(1/2)),x, algorithm="giac")
 

Output:

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))
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {125000\,x}{4782969}-\frac {280728140\,\ln \left (\sqrt {x}-\frac {5}{9}\right )}{387420489}+\frac {2500\,x^2}{59049}-\frac {56145628\,\sqrt {x}}{43046721}+\frac {200\,x^3}{2187}+\frac {2\,x^4}{9}+\frac {50000\,x^{3/2}}{1594323}+\frac {400\,x^{5/2}}{6561}+\frac {80\,x^{7/2}}{567} \] Input:

int((8*x^(7/2) - 6)/(9*x^(1/2) - 5),x)
 

Output:

(125000*x)/4782969 - (280728140*log(x^(1/2) - 5/9))/387420489 + (2500*x^2) 
/59049 - (56145628*x^(1/2))/43046721 + (200*x^3)/2187 + (2*x^4)/9 + (50000 
*x^(3/2))/1594323 + (400*x^(5/2))/6561 + (80*x^(7/2))/567
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66 \[ \int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx=\frac {80 \sqrt {x}\, x^{3}}{567}+\frac {400 \sqrt {x}\, x^{2}}{6561}+\frac {50000 \sqrt {x}\, x}{1594323}-\frac {56145628 \sqrt {x}}{43046721}-\frac {280728140 \,\mathrm {log}\left (9 \sqrt {x}-5\right )}{387420489}+\frac {2 x^{4}}{9}+\frac {200 x^{3}}{2187}+\frac {2500 x^{2}}{59049}+\frac {125000 x}{4782969} \] Input:

int((6-8*x^(7/2))/(5-9*x^(1/2)),x)
 

Output:

(2*(191318760*sqrt(x)*x**3 + 82668600*sqrt(x)*x**2 + 42525000*sqrt(x)*x - 
1768587282*sqrt(x) - 982548490*log(9*sqrt(x) - 5) + 301327047*x**4 + 12400 
2900*x**3 + 57408750*x**2 + 35437500*x))/2711943423