3.6.39 \(\int x^{5/2} (2-b x)^{3/2} \, dx\)

Optimal. Leaf size=131 \[ \frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{20} x^{7/2} \sqrt {2-b x}-\frac {x^{5/2} \sqrt {2-b x}}{20 b} \]

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Rubi [A]  time = 0.04, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{20} x^{7/2} \sqrt {2-b x}-\frac {x^{5/2} \sqrt {2-b x}}{20 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(2 - b*x)^(3/2),x]

[Out]

(-3*Sqrt[x]*Sqrt[2 - b*x])/(8*b^3) - (x^(3/2)*Sqrt[2 - b*x])/(8*b^2) - (x^(5/2)*Sqrt[2 - b*x])/(20*b) + (3*x^(
7/2)*Sqrt[2 - b*x])/20 + (x^(7/2)*(2 - b*x)^(3/2))/5 + (3*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} (2-b x)^{3/2} \, dx &=\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{5} \int x^{5/2} \sqrt {2-b x} \, dx\\ &=\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{20} \int \frac {x^{5/2}}{\sqrt {2-b x}} \, dx\\ &=-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {\int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{4 b}\\ &=-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{8 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{8 b^3}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 79, normalized size = 0.60 \begin {gather*} \frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}-\frac {\sqrt {x} \sqrt {2-b x} \left (8 b^4 x^4-22 b^3 x^3+2 b^2 x^2+5 b x+15\right )}{40 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(2 - b*x)^(3/2),x]

[Out]

-1/40*(Sqrt[x]*Sqrt[2 - b*x]*(15 + 5*b*x + 2*b^2*x^2 - 22*b^3*x^3 + 8*b^4*x^4))/b^3 + (3*ArcSin[(Sqrt[b]*Sqrt[
x])/Sqrt[2]])/(4*b^(7/2))

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IntegrateAlgebraic [A]  time = 0.16, size = 104, normalized size = 0.79 \begin {gather*} \frac {3 \sqrt {-b} \log \left (\sqrt {2-b x}-\sqrt {-b} \sqrt {x}\right )}{4 b^4}+\frac {\sqrt {2-b x} \left (-8 b^4 x^{9/2}+22 b^3 x^{7/2}-2 b^2 x^{5/2}-5 b x^{3/2}-15 \sqrt {x}\right )}{40 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/2)*(2 - b*x)^(3/2),x]

[Out]

(Sqrt[2 - b*x]*(-15*Sqrt[x] - 5*b*x^(3/2) - 2*b^2*x^(5/2) + 22*b^3*x^(7/2) - 8*b^4*x^(9/2)))/(40*b^3) + (3*Sqr
t[-b]*Log[-(Sqrt[-b]*Sqrt[x]) + Sqrt[2 - b*x]])/(4*b^4)

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fricas [A]  time = 0.91, size = 157, normalized size = 1.20 \begin {gather*} \left [-\frac {{\left (8 \, b^{5} x^{4} - 22 \, 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 )}{40 \, b^{4}}, -\frac {{\left (8 \, b^{5} x^{4} - 22 \, 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 )}{40 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-b*x+2)^(3/2),x, algorithm="fricas")

[Out]

[-1/40*((8*b^5*x^4 - 22*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, -1/40*((8*b^5*x^4 - 22*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)^(3/2),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 [-18.2719481629,8.05231268331]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 [-36.6004387327,28.4266860783]Warning, choosing ro
ot 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 [-61.8196598012,96.7771189027]Warning, c
hoosing 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 [-5.40303052077,66.1769613782]
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 [-61.0171700171,94.1
262030317]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 [-12.71137
13701,17.6881634681]1/b*(-2*b^2*abs(b)/b^2*(2*((((5040*b^19/100800/b^23*sqrt(-b*x+2)*sqrt(-b*x+2)-51660*b^19/1
00800/b^23)*sqrt(-b*x+2)*sqrt(-b*x+2)+215460*b^19/100800/b^23)*sqrt(-b*x+2)*sqrt(-b*x+2)-469350*b^19/100800/b^
23)*sqrt(-b*x+2)*sqrt(-b*x+2)+607950*b^19/100800/b^23)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-63/8/b^3/sqrt(-b)*ln
(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))+8*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)-35/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b
*x+2))))-8*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 = 132, normalized size = 1.01 \begin {gather*} -\frac {\left (-b x +2\right )^{\frac {5}{2}} x^{\frac {5}{2}}}{5 b}-\frac {\left (-b x +2\right )^{\frac {5}{2}} x^{\frac {3}{2}}}{4 b^{2}}-\frac {\left (-b x +2\right )^{\frac {5}{2}} \sqrt {x}}{4 b^{3}}+\frac {\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{8 b^{3}}+\frac {3 \sqrt {-b x +2}\, \sqrt {x}}{8 b^{3}}+\frac {3 \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)^(3/2),x)

[Out]

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

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maxima [A]  time = 2.94, size = 179, normalized size = 1.37 \begin {gather*} \frac {\frac {15 \, \sqrt {-b x + 2} b^{4}}{\sqrt {x}} + \frac {70 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{3}}{x^{\frac {3}{2}}} - \frac {128 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b^{2}}{x^{\frac {5}{2}}} - \frac {70 \, {\left (-b x + 2\right )}^{\frac {7}{2}} b}{x^{\frac {7}{2}}} - \frac {15 \, {\left (-b x + 2\right )}^{\frac {9}{2}}}{x^{\frac {9}{2}}}}{20 \, {\left (b^{8} - \frac {5 \, {\left (b x - 2\right )} b^{7}}{x} + \frac {10 \, {\left (b x - 2\right )}^{2} b^{6}}{x^{2}} - \frac {10 \, {\left (b x - 2\right )}^{3} b^{5}}{x^{3}} + \frac {5 \, {\left (b x - 2\right )}^{4} b^{4}}{x^{4}} - \frac {{\left (b x - 2\right )}^{5} b^{3}}{x^{5}}\right )}} - \frac {3 \, \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)^(3/2),x, algorithm="maxima")

[Out]

1/20*(15*sqrt(-b*x + 2)*b^4/sqrt(x) + 70*(-b*x + 2)^(3/2)*b^3/x^(3/2) - 128*(-b*x + 2)^(5/2)*b^2/x^(5/2) - 70*
(-b*x + 2)^(7/2)*b/x^(7/2) - 15*(-b*x + 2)^(9/2)/x^(9/2))/(b^8 - 5*(b*x - 2)*b^7/x + 10*(b*x - 2)^2*b^6/x^2 -
10*(b*x - 2)^3*b^5/x^3 + 5*(b*x - 2)^4*b^4/x^4 - (b*x - 2)^5*b^3/x^5) - 3/4*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqr
t(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}\,{\left (2-b\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(2 - b*x)^(3/2),x)

[Out]

int(x^(5/2)*(2 - b*x)^(3/2), x)

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sympy [A]  time = 15.28, size = 291, normalized size = 2.22 \begin {gather*} \begin {cases} - \frac {i b^{2} x^{\frac {11}{2}}}{5 \sqrt {b x - 2}} + \frac {19 i b x^{\frac {9}{2}}}{20 \sqrt {b x - 2}} - \frac {23 i x^{\frac {7}{2}}}{20 \sqrt {b x - 2}} - \frac {i x^{\frac {5}{2}}}{40 b \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{8 b^{2} \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{4 b^{3} \sqrt {b x - 2}} - \frac {3 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^{2} x^{\frac {11}{2}}}{5 \sqrt {- b x + 2}} - \frac {19 b x^{\frac {9}{2}}}{20 \sqrt {- b x + 2}} + \frac {23 x^{\frac {7}{2}}}{20 \sqrt {- b x + 2}} + \frac {x^{\frac {5}{2}}}{40 b \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{8 b^{2} \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{4 b^{3} \sqrt {- b x + 2}} + \frac {3 \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)**(3/2),x)

[Out]

Piecewise((-I*b**2*x**(11/2)/(5*sqrt(b*x - 2)) + 19*I*b*x**(9/2)/(20*sqrt(b*x - 2)) - 23*I*x**(7/2)/(20*sqrt(b
*x - 2)) - I*x**(5/2)/(40*b*sqrt(b*x - 2)) - I*x**(3/2)/(8*b**2*sqrt(b*x - 2)) + 3*I*sqrt(x)/(4*b**3*sqrt(b*x
- 2)) - 3*I*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/2)), Abs(b*x)/2 > 1), (b**2*x**(11/2)/(5*sqrt(-b*x + 2))
- 19*b*x**(9/2)/(20*sqrt(-b*x + 2)) + 23*x**(7/2)/(20*sqrt(-b*x + 2)) + x**(5/2)/(40*b*sqrt(-b*x + 2)) + x**(
3/2)/(8*b**2*sqrt(-b*x + 2)) - 3*sqrt(x)/(4*b**3*sqrt(-b*x + 2)) + 3*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/
2)), True))

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