3.297 \(\int \frac {x}{(3-2 \sqrt {x})^{3/4}} \, dx\)
Optimal. Leaf size=69 \[ \frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used =
{266, 43} \[ \frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}} \]
Antiderivative was successfully verified.
[In]
Int[x/(3 - 2*Sqrt[x])^(3/4),x]
[Out]
(-27*(3 - 2*Sqrt[x])^(1/4))/2 + (27*(3 - 2*Sqrt[x])^(5/4))/10 - (3 - 2*Sqrt[x])^(9/4)/2 + (3 - 2*Sqrt[x])^(13/
4)/26
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 {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{(3-2 x)^{3/4}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {27}{8 (3-2 x)^{3/4}}-\frac {27}{8} \sqrt [4]{3-2 x}+\frac {9}{8} (3-2 x)^{5/4}-\frac {1}{8} (3-2 x)^{9/4}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.52 \[ -\frac {4}{65} \sqrt [4]{3-2 \sqrt {x}} \left (5 x^{3/2}+10 x+24 \sqrt {x}+144\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x/(3 - 2*Sqrt[x])^(3/4),x]
[Out]
(-4*(3 - 2*Sqrt[x])^(1/4)*(144 + 24*Sqrt[x] + 10*x + 5*x^(3/2)))/65
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IntegrateAlgebraic [A] time = 0.30, size = 80, normalized size = 1.16 \[ -\frac {4}{13} \sqrt [4]{3-2 \sqrt {x}} x^{3/2}-\frac {8}{13} \sqrt [4]{3-2 \sqrt {x}} x-\frac {96}{65} \sqrt [4]{3-2 \sqrt {x}} \sqrt {x}-\frac {576}{65} \sqrt [4]{3-2 \sqrt {x}} \]
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x/(3 - 2*Sqrt[x])^(3/4),x]
[Out]
(-576*(3 - 2*Sqrt[x])^(1/4))/65 - (96*(3 - 2*Sqrt[x])^(1/4)*Sqrt[x])/65 - (8*(3 - 2*Sqrt[x])^(1/4)*x)/13 - (4*
(3 - 2*Sqrt[x])^(1/4)*x^(3/2))/13
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fricas [A] time = 0.58, size = 25, normalized size = 0.36 \[ -\frac {4}{65} \, {\left ({\left (5 \, x + 24\right )} \sqrt {x} + 10 \, x + 144\right )} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(3-2*x^(1/2))^(3/4),x, algorithm="fricas")
[Out]
-4/65*((5*x + 24)*sqrt(x) + 10*x + 144)*(-2*sqrt(x) + 3)^(1/4)
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giac [A] time = 0.60, size = 63, normalized size = 0.91 \[ -\frac {1}{26} \, {\left (2 \, \sqrt {x} - 3\right )}^{3} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} - \frac {1}{2} \, {\left (2 \, \sqrt {x} - 3\right )}^{2} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} + \frac {27}{10} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {5}{4}} - \frac {27}{2} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(3-2*x^(1/2))^(3/4),x, algorithm="giac")
[Out]
-1/26*(2*sqrt(x) - 3)^3*(-2*sqrt(x) + 3)^(1/4) - 1/2*(2*sqrt(x) - 3)^2*(-2*sqrt(x) + 3)^(1/4) + 27/10*(-2*sqrt
(x) + 3)^(5/4) - 27/2*(-2*sqrt(x) + 3)^(1/4)
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maple [C] time = 0.29, size = 20, normalized size = 0.29
|
|
|
| method |
result |
size |
|
|
|
| meijerg |
\(\frac {3^{\frac {1}{4}} x^{2} \hypergeom \left (\left [\frac {3}{4}, 4\right ], \relax [5], \frac {2 \sqrt {x}}{3}\right )}{6}\) |
\(20\) |
| derivativedivides |
\(-\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {1}{4}}}{2}+\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {5}{4}}}{10}-\frac {\left (3-2 \sqrt {x}\right )^{\frac {9}{4}}}{2}+\frac {\left (3-2 \sqrt {x}\right )^{\frac {13}{4}}}{26}\) |
\(46\) |
| default |
\(-\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {1}{4}}}{2}+\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {5}{4}}}{10}-\frac {\left (3-2 \sqrt {x}\right )^{\frac {9}{4}}}{2}+\frac {\left (3-2 \sqrt {x}\right )^{\frac {13}{4}}}{26}\) |
\(46\) |
|
|
|
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(3-2*x^(1/2))^(3/4),x,method=_RETURNVERBOSE)
[Out]
1/6*3^(1/4)*x^2*hypergeom([3/4,4],[5],2/3*x^(1/2))
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maxima [A] time = 0.47, size = 45, normalized size = 0.65 \[ \frac {1}{26} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {13}{4}} - \frac {1}{2} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {9}{4}} + \frac {27}{10} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {5}{4}} - \frac {27}{2} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(3-2*x^(1/2))^(3/4),x, algorithm="maxima")
[Out]
1/26*(-2*sqrt(x) + 3)^(13/4) - 1/2*(-2*sqrt(x) + 3)^(9/4) + 27/10*(-2*sqrt(x) + 3)^(5/4) - 27/2*(-2*sqrt(x) +
3)^(1/4)
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mupad [B] time = 0.30, size = 45, normalized size = 0.65 \[ \frac {27\,{\left (3-2\,\sqrt {x}\right )}^{5/4}}{10}-\frac {27\,{\left (3-2\,\sqrt {x}\right )}^{1/4}}{2}-\frac {{\left (3-2\,\sqrt {x}\right )}^{9/4}}{2}+\frac {{\left (3-2\,\sqrt {x}\right )}^{13/4}}{26} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(3 - 2*x^(1/2))^(3/4),x)
[Out]
(27*(3 - 2*x^(1/2))^(5/4))/10 - (27*(3 - 2*x^(1/2))^(1/4))/2 - (3 - 2*x^(1/2))^(9/4)/2 + (3 - 2*x^(1/2))^(13/4
)/26
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sympy [B] time = 2.25, size = 3305, normalized size = 47.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(3-2*x**(1/2))**(3/4),x)
[Out]
Piecewise((1280*3**(1/4)*x**(25/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3
**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*
x**9 + 47385*3**(1/4)*x**8) + 26304*3**(1/4)*x**(23/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*
x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**1
0 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 200016*3**(1/4)*x**(21/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*I*pi
/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 +
140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 331776*sqrt(3)*x**(21/2)/(-37440*3**(1/
4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x
**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 2123820*3**(1/4)*x**(19/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*
I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**
11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 2488320*sqrt(3)*x**(19/2)/(-37440*3
**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1
/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 1609632*3**(1/4)*x**(17/2)*(2*sqrt(x) - 3)**(1/4)*ex
p(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4
)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 1679616*sqrt(3)*x**(17/2)/(-37
440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*
3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 8960*3**(1/4)*x**12*(2*sqrt(x) - 3)**(1/4)*exp(
-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*
x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 18432*3**(1/4)*x**11*(2*sqrt(x)
- 3)**(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2)
+ 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 36864*sqrt(3)*x*
*11/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 +
140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 965520*3**(1/4)*x**10*(2*sqrt(x) - 3)**
(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160
*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 1244160*sqrt(3)*x**10/
(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140
400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 2548584*3**(1/4)*x**9*(2*sqrt(x) - 3)**(1/4
)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**
(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 2799360*sqrt(3)*x**9/(-374
40*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3
**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 419904*3**(1/4)*x**8*(2*sqrt(x) - 3)**(1/4)*exp(
-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*
x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 419904*sqrt(3)*x**8/(-37440*3**(
1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)
*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8), 2*Abs(sqrt(x))/3 > 1), (-1280*3**(1/4)*x**(25/2)*(3 - 2*
sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(
1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 26304*3**(1/4)*x**(23/2)*(3
- 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160
*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 200016*3**(1/4)*x**(21
/2)*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2)
+ 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 331776*sqrt(3)*x
**(21/2)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x*
*11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 2123820*3**(1/4)*x**(19/2)*(3 - 2*
sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(
1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 2488320*sqrt(3)*x**(19/2)/(
-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 1404
00*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 1609632*3**(1/4)*x**(17/2)*(3 - 2*sqrt(x))**
(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11
+ 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 1679616*sqrt(3)*x**(17/2)/(-37440*3**
(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4
)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 8960*3**(1/4)*x**12*(3 - 2*sqrt(x))**(1/4)/(-37440*3**
(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4
)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 18432*3**(1/4)*x**11*(3 - 2*sqrt(x))**(1/4)/(-37440*3*
*(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/
4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 36864*sqrt(3)*x**11/(-37440*3**(1/4)*x**(21/2) - 2808
00*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1
/4)*x**9 + 47385*3**(1/4)*x**8) - 965520*3**(1/4)*x**10*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 28
0800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**
(1/4)*x**9 + 47385*3**(1/4)*x**8) + 1244160*sqrt(3)*x**10/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/
2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3*
*(1/4)*x**8) - 2548584*3**(1/4)*x**9*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(1
9/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*
3**(1/4)*x**8) + 2799360*sqrt(3)*x**9/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)
*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 41990
4*3**(1/4)*x**8*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4
)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 4199
04*sqrt(3)*x**8/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(
1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8), True))
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