3.6.13 \(\int x^{5/2} \sqrt {2-b x} \, dx\)

Optimal. Leaf size=112 \[ \frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{24 b^2}+\frac {1}{4} x^{7/2} \sqrt {2-b x}-\frac {x^{5/2} \sqrt {2-b x}}{12 b} \]

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Rubi [A]  time = 0.03, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {50, 54, 216} \begin {gather*} -\frac {5 x^{3/2} \sqrt {2-b x}}{24 b^2}-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b^3}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {1}{4} x^{7/2} \sqrt {2-b x}-\frac {x^{5/2} \sqrt {2-b x}}{12 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*Sqrt[2 - b*x],x]

[Out]

(-5*Sqrt[x]*Sqrt[2 - b*x])/(8*b^3) - (5*x^(3/2)*Sqrt[2 - b*x])/(24*b^2) - (x^(5/2)*Sqrt[2 - b*x])/(12*b) + (x^
(7/2)*Sqrt[2 - b*x])/4 + (5*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(7/2))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int x^{5/2} \sqrt {2-b x} \, dx &=\frac {1}{4} x^{7/2} \sqrt {2-b x}+\frac {1}{4} \int \frac {x^{5/2}}{\sqrt {2-b x}} \, dx\\ &=-\frac {x^{5/2} \sqrt {2-b x}}{12 b}+\frac {1}{4} x^{7/2} \sqrt {2-b x}+\frac {5 \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{12 b}\\ &=-\frac {5 x^{3/2} \sqrt {2-b x}}{24 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{12 b}+\frac {1}{4} x^{7/2} \sqrt {2-b x}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{8 b^2}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{24 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{12 b}+\frac {1}{4} x^{7/2} \sqrt {2-b x}+\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{8 b^3}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{24 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{12 b}+\frac {1}{4} x^{7/2} \sqrt {2-b x}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=-\frac {5 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{24 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{12 b}+\frac {1}{4} x^{7/2} \sqrt {2-b x}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 71, normalized size = 0.63 \begin {gather*} \frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {\sqrt {x} \sqrt {2-b x} \left (6 b^3 x^3-2 b^2 x^2-5 b x-15\right )}{24 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*Sqrt[2 - b*x],x]

[Out]

(Sqrt[x]*Sqrt[2 - b*x]*(-15 - 5*b*x - 2*b^2*x^2 + 6*b^3*x^3))/(24*b^3) + (5*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])
/(4*b^(7/2))

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IntegrateAlgebraic [A]  time = 0.13, size = 94, normalized size = 0.84 \begin {gather*} \frac {5 \sqrt {-b} \log \left (\sqrt {2-b x}-\sqrt {-b} \sqrt {x}\right )}{4 b^4}+\frac {\sqrt {2-b x} \left (6 b^3 x^{7/2}-2 b^2 x^{5/2}-5 b x^{3/2}-15 \sqrt {x}\right )}{24 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/2)*Sqrt[2 - b*x],x]

[Out]

(Sqrt[2 - b*x]*(-15*Sqrt[x] - 5*b*x^(3/2) - 2*b^2*x^(5/2) + 6*b^3*x^(7/2)))/(24*b^3) + (5*Sqrt[-b]*Log[-(Sqrt[
-b]*Sqrt[x]) + Sqrt[2 - b*x]])/(4*b^4)

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fricas [A]  time = 1.04, size = 141, normalized size = 1.26 \begin {gather*} \left [\frac {{\left (6 \, b^{4} x^{3} - 2 \, b^{3} x^{2} - 5 \, b^{2} x - 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 15 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{24 \, b^{4}}, \frac {{\left (6 \, b^{4} x^{3} - 2 \, b^{3} x^{2} - 5 \, b^{2} x - 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 30 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{24 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-b*x+2)^(1/2),x, algorithm="fricas")

[Out]

[1/24*((6*b^4*x^3 - 2*b^3*x^2 - 5*b^2*x - 15*b)*sqrt(-b*x + 2)*sqrt(x) - 15*sqrt(-b)*log(-b*x + sqrt(-b*x + 2)
*sqrt(-b)*sqrt(x) + 1))/b^4, 1/24*((6*b^4*x^3 - 2*b^3*x^2 - 5*b^2*x - 15*b)*sqrt(-b*x + 2)*sqrt(x) - 30*sqrt(b
)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))))/b^4]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-b*x+2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{4,[1,
1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%
{-4,[1,2]%%%}+%%%{-28,[1,1]%%%}+%%%{-8,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{8,[0,1]%%%}+%%%{24,[0,0]%%%},0,%%%{4,[3,3
]%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-64,[2,2]%%%}+%%%{20,[2,1]%%%}+%%%
{8,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-20,[1,2]%%%}+%%%{128,[1,1]%%%}+%%%{-16,[1,0]%%%}+%%%{-4,[0,3]%%%}+%%%{8,[0,
2]%%%}+%%%{16,[0,1]%%%}+%%%{-32,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,[4,1]%%%}+
%%%{1,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-12,[3,3]%%%}+%%%{20,[3,2]%%%}+%%%{-20,[3,1]%%%}+%%%{8,[3,0]%%%}+%%%{6,[2,
4]%%%}+%%%{-20,[2,3]%%%}+%%%{46,[2,2]%%%}+%%%{-40,[2,1]%%%}+%%%{24,[2,0]%%%}+%%%{4,[1,4]%%%}+%%%{-20,[1,3]%%%}
+%%%{40,[1,2]%%%}+%%%{-48,[1,1]%%%}+%%%{32,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-8,[0,3]%%%}+%%%{24,[0,2]%%%}+%%%{-32
,[0,1]%%%}+%%%{16,[0,0]%%%}] at parameters values [-41.1343540126,25.8388736797]Warning, choosing root of [1,0
,%%%{4,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,
0]%%%}+%%%{-4,[1,2]%%%}+%%%{-28,[1,1]%%%}+%%%{-8,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{8,[0,1]%%%}+%%%{24,[0,0]%%%},0,
%%%{4,[3,3]%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-64,[2,2]%%%}+%%%{20,[2,
1]%%%}+%%%{8,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-20,[1,2]%%%}+%%%{128,[1,1]%%%}+%%%{-16,[1,0]%%%}+%%%{-4,[0,3]%%%}
+%%%{8,[0,2]%%%}+%%%{16,[0,1]%%%}+%%%{-32,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,
[4,1]%%%}+%%%{1,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-12,[3,3]%%%}+%%%{20,[3,2]%%%}+%%%{-20,[3,1]%%%}+%%%{8,[3,0]%%%}
+%%%{6,[2,4]%%%}+%%%{-20,[2,3]%%%}+%%%{46,[2,2]%%%}+%%%{-40,[2,1]%%%}+%%%{24,[2,0]%%%}+%%%{4,[1,4]%%%}+%%%{-20
,[1,3]%%%}+%%%{40,[1,2]%%%}+%%%{-48,[1,1]%%%}+%%%{32,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-8,[0,3]%%%}+%%%{24,[0,2]%%
%}+%%%{-32,[0,1]%%%}+%%%{16,[0,0]%%%}] at parameters values [-67.0714422017,15.451549686]Warning, choosing roo
t of [1,0,%%%{4,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+
%%%{6,[2,0]%%%}+%%%{-4,[1,2]%%%}+%%%{-28,[1,1]%%%}+%%%{-8,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{8,[0,1]%%%}+%%%{24,[0,
0]%%%},0,%%%{4,[3,3]%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-64,[2,2]%%%}+%
%%{20,[2,1]%%%}+%%%{8,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-20,[1,2]%%%}+%%%{128,[1,1]%%%}+%%%{-16,[1,0]%%%}+%%%{-4,
[0,3]%%%}+%%%{8,[0,2]%%%}+%%%{16,[0,1]%%%}+%%%{-32,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%
}+%%%{-4,[4,1]%%%}+%%%{1,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-12,[3,3]%%%}+%%%{20,[3,2]%%%}+%%%{-20,[3,1]%%%}+%%%{8,
[3,0]%%%}+%%%{6,[2,4]%%%}+%%%{-20,[2,3]%%%}+%%%{46,[2,2]%%%}+%%%{-40,[2,1]%%%}+%%%{24,[2,0]%%%}+%%%{4,[1,4]%%%
}+%%%{-20,[1,3]%%%}+%%%{40,[1,2]%%%}+%%%{-48,[1,1]%%%}+%%%{32,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-8,[0,3]%%%}+%%%{2
4,[0,2]%%%}+%%%{-32,[0,1]%%%}+%%%{16,[0,0]%%%}] at parameters values [-46.2420096635,81.9516051291]Warning, ch
oosing root of [1,0,%%%{4,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,
[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{-4,[1,2]%%%}+%%%{-28,[1,1]%%%}+%%%{-8,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{8,[0,1]%%%}+
%%%{24,[0,0]%%%},0,%%%{4,[3,3]%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-64,[
2,2]%%%}+%%%{20,[2,1]%%%}+%%%{8,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-20,[1,2]%%%}+%%%{128,[1,1]%%%}+%%%{-16,[1,0]%%
%}+%%%{-4,[0,3]%%%}+%%%{8,[0,2]%%%}+%%%{16,[0,1]%%%}+%%%{-32,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{
6,[4,2]%%%}+%%%{-4,[4,1]%%%}+%%%{1,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-12,[3,3]%%%}+%%%{20,[3,2]%%%}+%%%{-20,[3,1]%
%%}+%%%{8,[3,0]%%%}+%%%{6,[2,4]%%%}+%%%{-20,[2,3]%%%}+%%%{46,[2,2]%%%}+%%%{-40,[2,1]%%%}+%%%{24,[2,0]%%%}+%%%{
4,[1,4]%%%}+%%%{-20,[1,3]%%%}+%%%{40,[1,2]%%%}+%%%{-48,[1,1]%%%}+%%%{32,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-8,[0,3]
%%%}+%%%{24,[0,2]%%%}+%%%{-32,[0,1]%%%}+%%%{16,[0,0]%%%}] at parameters values [-82.5947937798,51.6443148847]1
/b*(2*b*abs(b)/b^2*(2*(((-90*b^11/1440/b^14*sqrt(-b*x+2)*sqrt(-b*x+2)+750*b^11/1440/b^14)*sqrt(-b*x+2)*sqrt(-b
*x+2)-2445*b^11/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)+4185*b^11/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-3
5/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-4*abs(b)/b^2*(2*((12*b^5/144/b^7*sqrt(-
b*x+2)*sqrt(-b*x+2)-78*b^5/144/b^7)*sqrt(-b*x+2)*sqrt(-b*x+2)+198*b^5/144/b^7)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2
*b)-5/2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2)))))

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maple [A]  time = 0.01, size = 116, normalized size = 1.04 \begin {gather*} -\frac {\left (-b x +2\right )^{\frac {3}{2}} x^{\frac {5}{2}}}{4 b}-\frac {5 \left (-b x +2\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{12 b^{2}}-\frac {5 \left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{8 b^{3}}+\frac {5 \sqrt {-b x +2}\, \sqrt {x}}{8 b^{3}}+\frac {5 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{8 \sqrt {-b x +2}\, b^{\frac {7}{2}} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(-b*x+2)^(1/2),x)

[Out]

-1/4/b*x^(5/2)*(-b*x+2)^(3/2)-5/12/b^2*x^(3/2)*(-b*x+2)^(3/2)-5/8/b^3*x^(1/2)*(-b*x+2)^(3/2)+5/8*x^(1/2)*(-b*x
+2)^(1/2)/b^3+5/8/b^(7/2)*((-b*x+2)*x)^(1/2)/(-b*x+2)^(1/2)/x^(1/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))

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maxima [A]  time = 2.93, size = 147, normalized size = 1.31 \begin {gather*} \frac {\frac {15 \, \sqrt {-b x + 2} b^{3}}{\sqrt {x}} - \frac {73 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} - \frac {55 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} - \frac {15 \, {\left (-b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{12 \, {\left (b^{7} - \frac {4 \, {\left (b x - 2\right )} b^{6}}{x} + \frac {6 \, {\left (b x - 2\right )}^{2} b^{5}}{x^{2}} - \frac {4 \, {\left (b x - 2\right )}^{3} b^{4}}{x^{3}} + \frac {{\left (b x - 2\right )}^{4} b^{3}}{x^{4}}\right )}} - \frac {5 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-b*x+2)^(1/2),x, algorithm="maxima")

[Out]

1/12*(15*sqrt(-b*x + 2)*b^3/sqrt(x) - 73*(-b*x + 2)^(3/2)*b^2/x^(3/2) - 55*(-b*x + 2)^(5/2)*b/x^(5/2) - 15*(-b
*x + 2)^(7/2)/x^(7/2))/(b^7 - 4*(b*x - 2)*b^6/x + 6*(b*x - 2)^2*b^5/x^2 - 4*(b*x - 2)^3*b^4/x^3 + (b*x - 2)^4*
b^3/x^4) - 5/4*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x)))/b^(7/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{5/2}\,\sqrt {2-b\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(2 - b*x)^(1/2),x)

[Out]

int(x^(5/2)*(2 - b*x)^(1/2), x)

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sympy [A]  time = 9.92, size = 252, normalized size = 2.25 \begin {gather*} \begin {cases} \frac {i b x^{\frac {9}{2}}}{4 \sqrt {b x - 2}} - \frac {7 i x^{\frac {7}{2}}}{12 \sqrt {b x - 2}} - \frac {i x^{\frac {5}{2}}}{24 b \sqrt {b x - 2}} - \frac {5 i x^{\frac {3}{2}}}{24 b^{2} \sqrt {b x - 2}} + \frac {5 i \sqrt {x}}{4 b^{3} \sqrt {b x - 2}} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {7}{2}}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {b x^{\frac {9}{2}}}{4 \sqrt {- b x + 2}} + \frac {7 x^{\frac {7}{2}}}{12 \sqrt {- b x + 2}} + \frac {x^{\frac {5}{2}}}{24 b \sqrt {- b x + 2}} + \frac {5 x^{\frac {3}{2}}}{24 b^{2} \sqrt {- b x + 2}} - \frac {5 \sqrt {x}}{4 b^{3} \sqrt {- b x + 2}} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(-b*x+2)**(1/2),x)

[Out]

Piecewise((I*b*x**(9/2)/(4*sqrt(b*x - 2)) - 7*I*x**(7/2)/(12*sqrt(b*x - 2)) - I*x**(5/2)/(24*b*sqrt(b*x - 2))
- 5*I*x**(3/2)/(24*b**2*sqrt(b*x - 2)) + 5*I*sqrt(x)/(4*b**3*sqrt(b*x - 2)) - 5*I*acosh(sqrt(2)*sqrt(b)*sqrt(x
)/2)/(4*b**(7/2)), Abs(b*x)/2 > 1), (-b*x**(9/2)/(4*sqrt(-b*x + 2)) + 7*x**(7/2)/(12*sqrt(-b*x + 2)) + x**(5/2
)/(24*b*sqrt(-b*x + 2)) + 5*x**(3/2)/(24*b**2*sqrt(-b*x + 2)) - 5*sqrt(x)/(4*b**3*sqrt(-b*x + 2)) + 5*asin(sqr
t(2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/2)), True))

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