3.6.57 \(\int x^{5/2} (2+b x)^{5/2} \, dx\)
Optimal. Leaf size=144 \[ -\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{16 b^3}-\frac {5 x^{3/2} \sqrt {b x+2}}{48 b^2}+\frac {1}{6} x^{7/2} (b x+2)^{5/2}+\frac {1}{6} x^{7/2} (b x+2)^{3/2}+\frac {1}{8} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{24 b} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used =
{50, 54, 215} \begin {gather*} -\frac {5 x^{3/2} \sqrt {b x+2}}{48 b^2}+\frac {5 \sqrt {x} \sqrt {b x+2}}{16 b^3}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}}+\frac {1}{6} x^{7/2} (b x+2)^{5/2}+\frac {1}{6} x^{7/2} (b x+2)^{3/2}+\frac {1}{8} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{24 b} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^(5/2)*(2 + b*x)^(5/2),x]
[Out]
(5*Sqrt[x]*Sqrt[2 + b*x])/(16*b^3) - (5*x^(3/2)*Sqrt[2 + b*x])/(48*b^2) + (x^(5/2)*Sqrt[2 + b*x])/(24*b) + (x^
(7/2)*Sqrt[2 + b*x])/8 + (x^(7/2)*(2 + b*x)^(3/2))/6 + (x^(7/2)*(2 + b*x)^(5/2))/6 - (5*ArcSinh[(Sqrt[b]*Sqrt[
x])/Sqrt[2]])/(8*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)^{5/2} \, dx &=\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {5}{6} \int x^{5/2} (2+b x)^{3/2} \, dx\\ &=\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {1}{2} \int x^{5/2} \sqrt {2+b x} \, dx\\ &=\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx\\ &=\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx}{24 b}\\ &=-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{16 b^2}\\ &=\frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{16 b^3}\\ &=\frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3}\\ &=\frac {5 \sqrt {x} \sqrt {2+b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{48 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2+b x}+\frac {1}{6} x^{7/2} (2+b x)^{3/2}+\frac {1}{6} x^{7/2} (2+b x)^{5/2}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 0.60 \begin {gather*} \frac {\sqrt {x} \sqrt {b x+2} \left (8 b^5 x^5+40 b^4 x^4+54 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{48 b^3}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^(5/2)*(2 + b*x)^(5/2),x]
[Out]
(Sqrt[x]*Sqrt[2 + b*x]*(15 - 5*b*x + 2*b^2*x^2 + 54*b^3*x^3 + 40*b^4*x^4 + 8*b^5*x^5))/(48*b^3) - (5*ArcSinh[(
Sqrt[b]*Sqrt[x])/Sqrt[2]])/(8*b^(7/2))
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IntegrateAlgebraic [A] time = 0.11, size = 105, normalized size = 0.73 \begin {gather*} \frac {5 \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{8 b^{7/2}}+\frac {\sqrt {b x+2} \left (8 b^5 x^{11/2}+40 b^4 x^{9/2}+54 b^3 x^{7/2}+2 b^2 x^{5/2}-5 b x^{3/2}+15 \sqrt {x}\right )}{48 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^(5/2)*(2 + b*x)^(5/2),x]
[Out]
(Sqrt[2 + b*x]*(15*Sqrt[x] - 5*b*x^(3/2) + 2*b^2*x^(5/2) + 54*b^3*x^(7/2) + 40*b^4*x^(9/2) + 8*b^5*x^(11/2)))/
(48*b^3) + (5*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[2 + b*x]])/(8*b^(7/2))
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fricas [A] time = 1.29, size = 172, normalized size = 1.19 \begin {gather*} \left [\frac {{\left (8 \, b^{6} x^{5} + 40 \, b^{5} x^{4} + 54 \, 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 )}{48 \, b^{4}}, \frac {{\left (8 \, b^{6} x^{5} + 40 \, b^{5} x^{4} + 54 \, 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 )}{48 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(5/2)*(b*x+2)^(5/2),x, algorithm="fricas")
[Out]
[1/48*((8*b^6*x^5 + 40*b^5*x^4 + 54*b^4*x^3 + 2*b^3*x^2 - 5*b^2*x + 15*b)*sqrt(b*x + 2)*sqrt(x) + 15*sqrt(b)*l
og(b*x - sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1))/b^4, 1/48*((8*b^6*x^5 + 40*b^5*x^4 + 54*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 \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)^(5/2),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 [83.4865739918,53.112478131]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 [38.6973876911,89.629912049]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 [6.82230772497,55.0343274642]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 [53.4880634798,16.0204098616]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 [46.2456374937,66.0382199469]Wa
rning, 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 [94.9264369817,51.84415
26662]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 [98.7121795234,
4.66774101928]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 [90.210
2860468,38.2197840363]1/b*(2*b^3*abs(b)/b^2*(2*(((((113400*b^29/2721600/b^34*sqrt(b*x+2)*sqrt(b*x+2)-1383480*b
^29/2721600/b^34)*sqrt(b*x+2)*sqrt(b*x+2)+7093170*b^29/2721600/b^34)*sqrt(b*x+2)*sqrt(b*x+2)-19737270*b^29/272
1600/b^34)*sqrt(b*x+2)*sqrt(b*x+2)+32304825*b^29/2721600/b^34)*sqrt(b*x+2)*sqrt(b*x+2)-33722325*b^29/2721600/b
^34)*sqrt(b*x+2)*sqrt(b*(b*x+2)-2*b)-231/16/b^4/sqrt(b)*ln(abs(sqrt(b*(b*x+2)-2*b)-sqrt(b)*sqrt(b*x+2))))+12*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)*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))))+24*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))))+16*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 = 0.00, size = 138, normalized size = 0.96 \begin {gather*} \frac {\left (b x +2\right )^{\frac {7}{2}} x^{\frac {5}{2}}}{6 b}-\frac {\left (b x +2\right )^{\frac {7}{2}} x^{\frac {3}{2}}}{6 b^{2}}+\frac {\left (b x +2\right )^{\frac {7}{2}} \sqrt {x}}{8 b^{3}}-\frac {\left (b x +2\right )^{\frac {5}{2}} \sqrt {x}}{24 b^{3}}-\frac {5 \left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{48 b^{3}}-\frac {5 \sqrt {b x +2}\, \sqrt {x}}{16 b^{3}}-\frac {5 \sqrt {\left (b x +2\right ) x}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{16 \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)^(5/2),x)
[Out]
1/6/b*x^(5/2)*(b*x+2)^(7/2)-1/6/b^2*x^(3/2)*(b*x+2)^(7/2)+1/8/b^3*x^(1/2)*(b*x+2)^(7/2)-1/24*(b*x+2)^(5/2)/b^3
*x^(1/2)-5/48*(b*x+2)^(3/2)/b^3*x^(1/2)-5/16*(b*x+2)^(1/2)/b^3*x^(1/2)-5/16*((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 = 2.98, size = 223, normalized size = 1.55 \begin {gather*} \frac {\frac {15 \, \sqrt {b x + 2} b^{5}}{\sqrt {x}} - \frac {85 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}} + \frac {198 \, {\left (b x + 2\right )}^{\frac {7}{2}} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (b x + 2\right )}^{\frac {9}{2}} b}{x^{\frac {9}{2}}} + \frac {15 \, {\left (b x + 2\right )}^{\frac {11}{2}}}{x^{\frac {11}{2}}}}{24 \, {\left (b^{9} - \frac {6 \, {\left (b x + 2\right )} b^{8}}{x} + \frac {15 \, {\left (b x + 2\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x + 2\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x + 2\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x + 2\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x + 2\right )}^{6} b^{3}}{x^{6}}\right )}} + \frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{16 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(5/2)*(b*x+2)^(5/2),x, algorithm="maxima")
[Out]
1/24*(15*sqrt(b*x + 2)*b^5/sqrt(x) - 85*(b*x + 2)^(3/2)*b^4/x^(3/2) + 198*(b*x + 2)^(5/2)*b^3/x^(5/2) + 198*(b
*x + 2)^(7/2)*b^2/x^(7/2) - 85*(b*x + 2)^(9/2)*b/x^(9/2) + 15*(b*x + 2)^(11/2)/x^(11/2))/(b^9 - 6*(b*x + 2)*b^
8/x + 15*(b*x + 2)^2*b^7/x^2 - 20*(b*x + 2)^3*b^6/x^3 + 15*(b*x + 2)^4*b^5/x^4 - 6*(b*x + 2)^5*b^4/x^5 + (b*x
+ 2)^6*b^3/x^6) + 5/16*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 \begin {gather*} \int x^{5/2}\,{\left (b\,x+2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(5/2)*(b*x + 2)^(5/2),x)
[Out]
int(x^(5/2)*(b*x + 2)^(5/2), x)
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sympy [A] time = 22.96, size = 158, normalized size = 1.10 \begin {gather*} \frac {b^{3} x^{\frac {13}{2}}}{6 \sqrt {b x + 2}} + \frac {7 b^{2} x^{\frac {11}{2}}}{6 \sqrt {b x + 2}} + \frac {67 b x^{\frac {9}{2}}}{24 \sqrt {b x + 2}} + \frac {55 x^{\frac {7}{2}}}{24 \sqrt {b x + 2}} - \frac {x^{\frac {5}{2}}}{48 b \sqrt {b x + 2}} + \frac {5 x^{\frac {3}{2}}}{48 b^{2} \sqrt {b x + 2}} + \frac {5 \sqrt {x}}{8 b^{3} \sqrt {b x + 2}} - \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{8 b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**(5/2)*(b*x+2)**(5/2),x)
[Out]
b**3*x**(13/2)/(6*sqrt(b*x + 2)) + 7*b**2*x**(11/2)/(6*sqrt(b*x + 2)) + 67*b*x**(9/2)/(24*sqrt(b*x + 2)) + 55*
x**(7/2)/(24*sqrt(b*x + 2)) - x**(5/2)/(48*b*sqrt(b*x + 2)) + 5*x**(3/2)/(48*b**2*sqrt(b*x + 2)) + 5*sqrt(x)/(
8*b**3*sqrt(b*x + 2)) - 5*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(8*b**(7/2))
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