3.6.5 \(\int x^{5/2} \sqrt {2+b x} \, dx\)
Optimal. Leaf size=108 \[ -\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{8 b^3}-\frac {5 x^{3/2} \sqrt {b x+2}}{24 b^2}+\frac {1}{4} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{12 b} \]
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Rubi [A] time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00,
number of steps used = 6, 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}}{24 b^2}+\frac {5 \sqrt {x} \sqrt {b x+2}}{8 b^3}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}+\frac {1}{4} x^{7/2} \sqrt {b x+2}+\frac {x^{5/2} \sqrt {b x+2}}{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*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} \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 \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 = 70, normalized size = 0.65 \begin {gather*} \frac {\sqrt {x} \sqrt {b x+2} \left (6 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{24 b^3}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}} \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*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])
/(4*b^(7/2))
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IntegrateAlgebraic [A] time = 0.10, size = 85, normalized size = 0.79 \begin {gather*} \frac {5 \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{4 b^{7/2}}+\frac {\sqrt {b x+2} \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*Log[-(Sqrt[b]*Sqrt[x]
) + Sqrt[2 + b*x]])/(4*b^(7/2))
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fricas [A] time = 0.92, size = 140, normalized size = 1.30 \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}}{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)*sqr
t(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)*arc
tan(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)^(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 [59.8656459874,25.8388736797]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 [33.9285577983,15.451549686]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 [54.7579903365,81.9516051291]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 [18.4052062202,51.6443148847]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)-35/8/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 = 108, normalized size = 1.00 \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}\, \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}} \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)*(x*(b*x+2))^(1/2)/(b*x+2)^(1/2)/x^(1/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*x)^(1/2))
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maxima [B] time = 3.03, size = 163, normalized size = 1.51 \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 \, \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}}} \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) + 5/8*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 \begin {gather*} \int x^{5/2}\,\sqrt {b\,x+2} \,d 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]
int(x^(5/2)*(b*x + 2)^(1/2), x)
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sympy [A] time = 10.12, size = 117, normalized size = 1.08 \begin {gather*} \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 {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \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)
[Out]
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*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/2))
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