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

Optimal. Leaf size=77 \[ \frac{80 x^{7/2}}{567}+\frac{400 x^{5/2}}{6561}+\frac{50000 x^{3/2}}{1594323}+\frac{2 x^4}{9}+\frac{200 x^3}{2187}+\frac{2500 x^2}{59049}+\frac{125000 x}{4782969}-\frac{56145628 \sqrt{x}}{43046721}-\frac{280728140 \log \left (5-9 \sqrt{x}\right )}{387420489} \]

[Out]

(-56145628*Sqrt[x])/43046721 + (125000*x)/4782969 + (50000*x^(3/2))/1594323 + (2
500*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

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Rubi [A]  time = 0.115842, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{80 x^{7/2}}{567}+\frac{400 x^{5/2}}{6561}+\frac{50000 x^{3/2}}{1594323}+\frac{2 x^4}{9}+\frac{200 x^3}{2187}+\frac{2500 x^2}{59049}+\frac{125000 x}{4782969}-\frac{56145628 \sqrt{x}}{43046721}-\frac{280728140 \log \left (5-9 \sqrt{x}\right )}{387420489} \]

Antiderivative was successfully verified.

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

[Out]

(-56145628*Sqrt[x])/43046721 + (125000*x)/4782969 + (50000*x^(3/2))/1594323 + (2
500*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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{80 x^{\frac{7}{2}}}{567} + \frac{400 x^{\frac{5}{2}}}{6561} + \frac{50000 x^{\frac{3}{2}}}{1594323} + \frac{2 x^{4}}{9} + \frac{200 x^{3}}{2187} + \frac{2500 x^{2}}{59049} - \frac{280728140 \log{\left (- 9 \sqrt{x} + 5 \right )}}{387420489} + 16 \int ^{\sqrt{x}} \frac{78125}{43046721}\, dx - 12 \int ^{\sqrt{x}} \frac{1}{9}\, dx + \frac{250000 \int ^{\sqrt{x}} x\, dx}{4782969} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((6-8*x**(7/2))/(5-9*x**(1/2)),x)

[Out]

80*x**(7/2)/567 + 400*x**(5/2)/6561 + 50000*x**(3/2)/1594323 + 2*x**4/9 + 200*x*
*3/2187 + 2500*x**2/59049 - 280728140*log(-9*sqrt(x) + 5)/387420489 + 16*Integra
l(78125/43046721, (x, sqrt(x))) - 12*Integral(1/9, (x, sqrt(x))) + 250000*Integr
al(x, (x, sqrt(x)))/4782969

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Mathematica [A]  time = 0.0322572, size = 66, normalized size = 0.86 \[ \frac{2 \left (9 \left (21257640 x^{7/2}+9185400 x^{5/2}+4725000 x^{3/2}[phone] x^4+13778100 x^3+6378750 x^2+3937500 x-196509698 \sqrt{x}\right )-982548490 \log \left (5-9 \sqrt{x}\right )\right )}{2711943423} \]

Antiderivative was successfully verified.

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

[Out]

(2*(9*(-196509698*Sqrt[x] + 3937500*x + 4725000*x^(3/2) + 6378750*x^2 + 9185400*
x^(5/2) + 13778100*x^3 + 21257640*x^(7/2) + 33480783*x^4) - 982548490*Log[5 - 9*
Sqrt[x]]))/2711943423

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Maple [A]  time = 0.006, size = 50, normalized size = 0.7 \[{\frac{2\,{x}^{4}}{9}}+{\frac{80}{567}{x}^{{\frac{7}{2}}}}+{\frac{200\,{x}^{3}}{2187}}+{\frac{400}{6561}{x}^{{\frac{5}{2}}}}+{\frac{2500\,{x}^{2}}{59049}}+{\frac{50000}{1594323}{x}^{{\frac{3}{2}}}}+{\frac{125000\,x}{4782969}}-{\frac{56145628}{43046721}\sqrt{x}}-{\frac{280728140}{387420489}\ln \left ( -5+9\,\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*x^4+80/567*x^(7/2)+200/2187*x^3+400/6561*x^(5/2)+2500/59049*x^2+50000/159432
3*x^(3/2)+125000/4782969*x-56145628/43046721*x^(1/2)-280728140/387420489*ln(-5+9
*x^(1/2))

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Maxima [A]  time = 0.722884, size = 66, normalized size = 0.86 \[ \frac{2}{9} \, x^{4} + \frac{80}{567} \, x^{\frac{7}{2}} + \frac{200}{2187} \, x^{3} + \frac{400}{6561} \, x^{\frac{5}{2}} + \frac{2500}{59049} \, x^{2} + \frac{50000}{1594323} \, x^{\frac{3}{2}} + \frac{125000}{4782969} \, x - \frac{56145628}{43046721} \, \sqrt{x} - \frac{280728140}{387420489} \, \log \left (9 \, \sqrt{x} - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(4*x^(7/2) - 3)/(9*sqrt(x) - 5),x, algorithm="maxima")

[Out]

2/9*x^4 + 80/567*x^(7/2) + 200/2187*x^3 + 400/6561*x^(5/2) + 2500/59049*x^2 + 50
000/1594323*x^(3/2) + 125000/4782969*x - 56145628/43046721*sqrt(x) - 280728140/3
87420489*log(9*sqrt(x) - 5)

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Fricas [A]  time = 0.265334, size = 66, normalized size = 0.86 \[ \frac{2}{9} \, x^{4} + \frac{200}{2187} \, x^{3} + \frac{2500}{59049} \, x^{2} + \frac{4}{301327047} \,{\left (10628820 \, x^{3} + 4592700 \, x^{2} + 2362500 \, x - 98254849\right )} \sqrt{x} + \frac{125000}{4782969} \, x - \frac{280728140}{387420489} \, \log \left (9 \, \sqrt{x} - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(4*x^(7/2) - 3)/(9*sqrt(x) - 5),x, algorithm="fricas")

[Out]

2/9*x^4 + 200/2187*x^3 + 2500/59049*x^2 + 4/301327047*(10628820*x^3 + 4592700*x^
2 + 2362500*x - 98254849)*sqrt(x) + 125000/4782969*x - 280728140/387420489*log(9
*sqrt(x) - 5)

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Sympy [A]  time = 10.3873, size = 71, normalized size = 0.92 \[ \frac{80 x^{\frac{7}{2}}}{567} + \frac{400 x^{\frac{5}{2}}}{6561} + \frac{50000 x^{\frac{3}{2}}}{1594323} - \frac{56145628 \sqrt{x}}{43046721} + \frac{2 x^{4}}{9} + \frac{200 x^{3}}{2187} + \frac{2500 x^{2}}{59049} + \frac{125000 x}{4782969} - \frac{280728140 \log{\left (\sqrt{x} - \frac{5}{9} \right )}}{387420489} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

80*x**(7/2)/567 + 400*x**(5/2)/6561 + 50000*x**(3/2)/1594323 - 56145628*sqrt(x)/
43046721 + 2*x**4/9 + 200*x**3/2187 + 2500*x**2/59049 + 125000*x/4782969 - 28072
8140*log(sqrt(x) - 5/9)/387420489

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GIAC/XCAS [A]  time = 0.266366, size = 68, normalized size = 0.88 \[ \frac{2}{9} \, x^{4} + \frac{80}{567} \, x^{\frac{7}{2}} + \frac{200}{2187} \, x^{3} + \frac{400}{6561} \, x^{\frac{5}{2}} + \frac{2500}{59049} \, x^{2} + \frac{50000}{1594323} \, x^{\frac{3}{2}} + \frac{125000}{4782969} \, x - \frac{56145628}{43046721} \, \sqrt{x} - \frac{280728140}{387420489} \,{\rm ln}\left ({\left | 9 \, \sqrt{x} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(4*x^(7/2) - 3)/(9*sqrt(x) - 5),x, algorithm="giac")

[Out]

2/9*x^4 + 80/567*x^(7/2) + 200/2187*x^3 + 400/6561*x^(5/2) + 2500/59049*x^2 + 50
000/1594323*x^(3/2) + 125000/4782969*x - 56145628/43046721*sqrt(x) - 280728140/3
87420489*ln(abs(9*sqrt(x) - 5))