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

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

[Out]

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

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Rubi [A]  time = 0.0656674, 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, Rules used = {1893, 190, 43, 266} \[ \frac{2 x^4}{9}+\frac{80 x^{7/2}}{567}+\frac{200 x^3}{2187}+\frac{400 x^{5/2}}{6561}+\frac{2500 x^2}{59049}+\frac{50000 x^{3/2}}{1594323}+\frac{125000 x}{4782969}-\frac{56145628 \sqrt{x}}{43046721}-\frac{280728140 \log \left (5-9 \sqrt{x}\right )}{387420489} \]

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 + (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

Rule 1893

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] || PolyQ[Pq, x^n])

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin{align*} \int \frac{6-8 x^{7/2}}{5-9 \sqrt{x}} \, dx &=\int \left (-\frac{6}{-5+9 \sqrt{x}}+\frac{8 x^{7/2}}{-5+9 \sqrt{x}}\right ) \, dx\\ &=-\left (6 \int \frac{1}{-5+9 \sqrt{x}} \, dx\right )+8 \int \frac{x^{7/2}}{-5+9 \sqrt{x}} \, dx\\ &=-\left (12 \operatorname{Subst}\left (\int \frac{x}{-5+9 x} \, dx,x,\sqrt{x}\right )\right )+16 \operatorname{Subst}\left (\int \frac{x^8}{-5+9 x} \, dx,x,\sqrt{x}\right )\\ &=-\left (12 \operatorname{Subst}\left (\int \left (\frac{1}{9}+\frac{5}{9 (-5+9 x)}\right ) \, dx,x,\sqrt{x}\right )\right )+16 \operatorname{Subst}\left (\int \left (\frac{78125}{43046721}+\frac{15625 x}{4782969}+\frac{3125 x^2}{531441}+\frac{625 x^3}{59049}+\frac{125 x^4}{6561}+\frac{25 x^5}{729}+\frac{5 x^6}{81}+\frac{x^7}{9}+\frac{390625}{43046721 (-5+9 x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\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}\\ \end{align*}

Mathematica [A]  time = 0.0556381, size = 66, normalized size = 0.86 \[ \frac{2 \left (9 \left (33480783 x^4+21257640 x^{7/2}[phone] x^3+9185400 x^{5/2}+6378750 x^2+4725000 x^{3/2}+3937500 x-196509698 \sqrt{x}\right )-982548490 \log \left (5-9 \sqrt{x}\right )\right )}{2711943423} \]

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 + 21257
640*x^(7/2) + 33480783*x^4) - 982548490*Log[5 - 9*Sqrt[x]]))/2711943423

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Maple [A]  time = 0.005, size = 50, normalized size = 0.7 \begin{align*}{\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 ) } \end{align*}

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+50000/1594323*x^(3/2)+125000/4782969*x-561
45628/43046721*x^(1/2)-280728140/387420489*ln(-5+9*x^(1/2))

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Maxima [A]  time = 1.0145, size = 66, normalized size = 0.86 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-8*x^(7/2))/(5-9*x^(1/2)),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 + 50000/1594323*x^(3/2) + 125000/4
782969*x - 56145628/43046721*sqrt(x) - 280728140/387420489*log(9*sqrt(x) - 5)

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Fricas [A]  time = 1.4554, size = 236, normalized size = 3.06 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-8*x^(7/2))/(5-9*x^(1/2)),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 = 27.6859, size = 71, normalized size = 0.92 \begin{align*} \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} \end{align*}

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 - 280728140*log(sqrt(x) - 5/9)/387420489

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Giac [A]  time = 1.15878, size = 68, normalized size = 0.88 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6-8*x^(7/2))/(5-9*x^(1/2)),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 + 50000/1594323*x^(3/2) + 125000/4
782969*x - 56145628/43046721*sqrt(x) - 280728140/387420489*log(abs(9*sqrt(x) - 5))