3.8.25 \(\int \frac {6-8 x^{7/2}}{5-9 \sqrt {x}} \, dx\) [725]

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

[Out]

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^4-280
728140/387420489*ln(5-9*x^(1/2))-56145628/43046721*x^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1907, 196, 45, 272} \begin {gather*} \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} \end {gather*}

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 45

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 196

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 272

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]]

Rule 1907

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])

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 \text {Subst}\left (\int \frac {x}{-5+9 x} \, dx,x,\sqrt {x}\right )\right )+16 \text {Subst}\left (\int \frac {x^8}{-5+9 x} \, dx,x,\sqrt {x}\right )\\ &=-\left (12 \text {Subst}\left (\int \left (\frac {1}{9}+\frac {5}{9 (-5+9 x)}\right ) \, dx,x,\sqrt {x}\right )\right )+16 \text {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*}

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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(-196509698 + 3937500*Sqrt[x] + 4725000*x + 6378750*x^(3/2) + 9185400*x^2 + 13778100*x^(5/2) + 2125
7640*x^3 + 33480783*x^(7/2)))/301327047 - (280728140*Log[-5 + 9*Sqrt[x]])/387420489

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Maple [A]
time = 0.40, size = 50, normalized size = 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	\(-\frac {4 \sqrt {x}}{3}-\frac {280728140 \ln \left (1-\frac {9 \sqrt {x}}{5}\right )}{387420489}+\frac {31250 \sqrt {x}\, \left (\frac {301327047 x^{\frac {7}{2}}}{15625}+\frac {38263752 x^{3}}{3125}+\frac {4960116 x^{\frac {5}{2}}}{625}+\frac {3306744 x^{2}}{625}+\frac {91854 x^{\frac {3}{2}}}{25}+\frac {13608 x}{5}+2268 \sqrt {x}+2520\right )}{2711943423}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.30, size = 49, normalized size = 0.64 \begin {gather*} \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 {gather*}

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 = 0.37, size = 49, normalized size = 0.64 \begin {gather*} \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 {gather*}

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 = 0.39, size = 71, normalized size = 0.92 \begin {gather*} \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} \end {gather*}

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(9*sqrt(x) - 5)/387420489

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Giac [A]
time = 2.18, size = 50, normalized size = 0.65 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.05, size = 47, normalized size = 0.61 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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

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