3.533 \(\int x^{5/2} (2+b x)^{3/2} \, dx\)

Optimal. Leaf size=126 \[ -\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {3 \sqrt {x} \sqrt {b x+2}}{8 b^3}-\frac {x^{3/2} \sqrt {b x+2}}{8 b^2}+\frac {1}{5} x^{7/2} (b x+2)^{3/2}+\frac {3}{20} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{20 b} \]

[Out]

1/5*x^(7/2)*(b*x+2)^(3/2)-3/4*arcsinh(1/2*b^(1/2)*x^(1/2)*2^(1/2))/b^(7/2)-1/8*x^(3/2)*(b*x+2)^(1/2)/b^2+1/20*
x^(5/2)*(b*x+2)^(1/2)/b+3/20*x^(7/2)*(b*x+2)^(1/2)+3/8*x^(1/2)*(b*x+2)^(1/2)/b^3

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Rubi [A]  time = 0.03, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {50, 54, 215} \[ -\frac {x^{3/2} \sqrt {b x+2}}{8 b^2}+\frac {3 \sqrt {x} \sqrt {b x+2}}{8 b^3}-\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {1}{5} x^{7/2} (b x+2)^{3/2}+\frac {3}{20} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{20 b} \]

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*ArcSinh[(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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 \sinh ^{-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 = 78, normalized size = 0.62 \[ \frac {\sqrt {x} \sqrt {b x+2} \left (8 b^4 x^4+22 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{40 b^3}-\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(2 + b*x)^(3/2),x]

[Out]

(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))/(40*b^3) - (3*ArcSinh[(Sqrt[b]*Sqrt[
x])/Sqrt[2]])/(4*b^(7/2))

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fricas [A]  time = 0.44, size = 156, normalized size = 1.24 \[ \left [\frac {{\left (8 \, b^{5} x^{4} + 22 \, b^{4} x^{3} + 2 \, b^{3} x^{2} - 5 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 15 \, \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{40 \, b^{4}}, \frac {{\left (8 \, b^{5} x^{4} + 22 \, b^{4} x^{3} + 2 \, b^{3} x^{2} - 5 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 30 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{40 \, b^{4}}\right ] \]

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)/(b*sqrt(x))))/b^4]

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

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 [82.7280518371,8.05231268331]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 [64.3995612673,28.4266860783]Warning, choosing root o
f [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]%%%}+%%%{2
0,[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 [39.1803401988,96.7771189027]Warning, choosin
g 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 [95.5969694792,66.1769613782]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 [39.9828299829,94.1262030317]
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]%%%}+%%%{2
0,[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 [88.2886286299,17.688
1634681]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^19/100800/b^23)*sqr
t(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)*sqr
t(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*sq
rt(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 = 123, normalized size = 0.98 \[ \frac {\left (b x +2\right )^{\frac {5}{2}} x^{\frac {5}{2}}}{5 b}-\frac {\left (b x +2\right )^{\frac {5}{2}} x^{\frac {3}{2}}}{4 b^{2}}+\frac {\left (b x +2\right )^{\frac {5}{2}} \sqrt {x}}{4 b^{3}}-\frac {\left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{8 b^{3}}-\frac {3 \sqrt {b x +2}\, \sqrt {x}}{8 b^{3}}-\frac {3 \sqrt {\left (b x +2\right ) x}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{8 \sqrt {b x +2}\, b^{\frac {7}{2}} \sqrt {x}} \]

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)*ln((b*x+1)/b^(1/2)+(
b*x^2+2*x)^(1/2))

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maxima [B]  time = 2.99, size = 194, normalized size = 1.54 \[ \frac {\frac {15 \, \sqrt {b x + 2} b^{4}}{\sqrt {x}} - \frac {70 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{3}}{x^{\frac {3}{2}}} - \frac {128 \, {\left (b x + 2\right )}^{\frac {5}{2}} b^{2}}{x^{\frac {5}{2}}} + \frac {70 \, {\left (b x + 2\right )}^{\frac {7}{2}} b}{x^{\frac {7}{2}}} - \frac {15 \, {\left (b x + 2\right )}^{\frac {9}{2}}}{x^{\frac {9}{2}}}}{20 \, {\left (b^{8} - \frac {5 \, {\left (b x + 2\right )} b^{7}}{x} + \frac {10 \, {\left (b x + 2\right )}^{2} b^{6}}{x^{2}} - \frac {10 \, {\left (b x + 2\right )}^{3} b^{5}}{x^{3}} + \frac {5 \, {\left (b x + 2\right )}^{4} b^{4}}{x^{4}} - \frac {{\left (b x + 2\right )}^{5} b^{3}}{x^{5}}\right )}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{8 \, b^{\frac {7}{2}}} \]

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) + 3/8*log(-(sqrt(b) - sqrt(b*x + 2)/sqrt(x))/
(sqrt(b) + sqrt(b*x + 2)/sqrt(x)))/b^(7/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{5/2}\,{\left (b\,x+2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x + 2)^(3/2),x)

[Out]

int(x^(5/2)*(b*x + 2)^(3/2), x)

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sympy [A]  time = 15.67, size = 136, normalized size = 1.08 \[ \frac {b^{2} x^{\frac {11}{2}}}{5 \sqrt {b x + 2}} + \frac {19 b x^{\frac {9}{2}}}{20 \sqrt {b x + 2}} + \frac {23 x^{\frac {7}{2}}}{20 \sqrt {b x + 2}} - \frac {x^{\frac {5}{2}}}{40 b \sqrt {b x + 2}} + \frac {x^{\frac {3}{2}}}{8 b^{2} \sqrt {b x + 2}} + \frac {3 \sqrt {x}}{4 b^{3} \sqrt {b x + 2}} - \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x+2)**(3/2),x)

[Out]

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*asinh(sqrt(2)*
sqrt(b)*sqrt(x)/2)/(4*b**(7/2))

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