∫ √__x (2 + bx)5∕2 dx

3.6.59 \(\int \sqrt {x} (2+b x)^{5/2} \, dx\)

Optimal. Leaf size=102 \[ -\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}}+\frac {1}{4} x^{3/2} (b x+2)^{5/2}+\frac {5}{12} x^{3/2} (b x+2)^{3/2}+\frac {5}{8} x^{3/2} \sqrt {b x+2}+\frac {5 \sqrt {x} \sqrt {b x+2}}{8 b} \]

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Rubi [A] time = 0.02, antiderivative size = 102, 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 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}}+\frac {1}{4} x^{3/2} (b x+2)^{5/2}+\frac {5}{12} x^{3/2} (b x+2)^{3/2}+\frac {5}{8} x^{3/2} \sqrt {b x+2}+\frac {5 \sqrt {x} \sqrt {b x+2}}{8 b} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[x]*(2 + b*x)^(5/2),x]

[Out]

(5*Sqrt[x]*Sqrt[2 + b*x])/(8*b) + (5*x^(3/2)*Sqrt[2 + b*x])/8 + (5*x^(3/2)*(2 + b*x)^(3/2))/12 + (x^(3/2)*(2 +

b*x)^(5/2))/4 - (5*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(3/2))

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

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Mathematica [A] time = 0.04, size = 70, normalized size = 0.69 \begin {gather*} \frac {\sqrt {x} \sqrt {b x+2} \left (6 b^3 x^3+34 b^2 x^2+59 b x+15\right )}{24 b}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{3/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[x]*(2 + b*x)^(5/2),x]

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(15 + 59*b*x + 34*b^2*x^2 + 6*b^3*x^3))/(24*b) - (5*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])

/(4*b^(3/2))

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IntegrateAlgebraic [A] time = 0.11, size = 85, normalized size = 0.83 \begin {gather*} \frac {5 \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{4 b^{3/2}}+\frac {\sqrt {b x+2} \left (6 b^3 x^{7/2}+34 b^2 x^{5/2}+59 b x^{3/2}+15 \sqrt {x}\right )}{24 b} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[2 + b*x]*(15*Sqrt[x] + 59*b*x^(3/2) + 34*b^2*x^(5/2) + 6*b^3*x^(7/2)))/(24*b) + (5*Log[-(Sqrt[b]*Sqrt[x]

) + Sqrt[2 + b*x]])/(4*b^(3/2))

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fricas [A] time = 1.43, size = 140, normalized size = 1.37 \begin {gather*} \left [\frac {{\left (6 \, b^{4} x^{3} + 34 \, b^{3} x^{2} + 59 \, 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^{2}}, \frac {{\left (6 \, b^{4} x^{3} + 34 \, b^{3} x^{2} + 59 \, 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^{2}}\right ] \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*((6*b^4*x^3 + 34*b^3*x^2 + 59*b^2*x + 15*b)*sqrt(b*x + 2)*sqrt(x) + 15*sqrt(b)*log(b*x - sqrt(b*x + 2)*s

qrt(b)*sqrt(x) + 1))/b^2, 1/24*((6*b^4*x^3 + 34*b^3*x^2 + 59*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^2]

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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((b*x+2)^(5/2)*x^(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 [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*(((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))))+12*b^2*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))))+24*b*abs(b)/b^2/b*(2*(1/8*sqrt(b*x

+2)*sqrt(b*x+2)-5/8)*sqrt(b*x+2)*sqrt(b*(b*x+2)-2*b)-6*b/4/sqrt(b)*ln(abs(sqrt(b*(b*x+2)-2*b)-sqrt(b)*sqrt(b*x

+2))))+16*abs(b)/b^2*(1/2*sqrt(b*x+2)*sqrt(b*(b*x+2)-2*b)+2*b/2/sqrt(b)*ln(abs(sqrt(b*(b*x+2)-2*b)-sqrt(b)*sqr

t(b*x+2)))))

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maple [A] time = 0.01, size = 99, normalized size = 0.97 \begin {gather*} \frac {\left (b x +2\right )^{\frac {5}{2}} x^{\frac {3}{2}}}{4}+\frac {5 \left (b x +2\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{12}+\frac {5 \sqrt {b x +2}\, x^{\frac {3}{2}}}{8}+\frac {5 \sqrt {b x +2}\, \sqrt {x}}{8 b}-\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 {3}{2}} \sqrt {x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

*((b*x+2)*x)^(1/2)/(b*x+2)^(1/2)/b^(3/2)/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 = 161, normalized size = 1.58 \begin {gather*} \frac {\frac {15 \, \sqrt {b x + 2} b^{3}}{\sqrt {x}} - \frac {55 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {73 \, {\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^{5} - \frac {4 \, {\left (b x + 2\right )} b^{4}}{x} + \frac {6 \, {\left (b x + 2\right )}^{2} b^{3}}{x^{2}} - \frac {4 \, {\left (b x + 2\right )}^{3} b^{2}}{x^{3}} + \frac {{\left (b x + 2\right )}^{4} b}{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 {3}{2}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

2)^(7/2)/x^(7/2))/(b^5 - 4*(b*x + 2)*b^4/x + 6*(b*x + 2)^2*b^3/x^2 - 4*(b*x + 2)^3*b^2/x^3 + (b*x + 2)^4*b/x^

4) + 5/8*log(-(sqrt(b) - sqrt(b*x + 2)/sqrt(x))/(sqrt(b) + sqrt(b*x + 2)/sqrt(x)))/b^(3/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A] time = 8.61, size = 119, normalized size = 1.17 \begin {gather*} \frac {b^{3} x^{\frac {9}{2}}}{4 \sqrt {b x + 2}} + \frac {23 b^{2} x^{\frac {7}{2}}}{12 \sqrt {b x + 2}} + \frac {127 b x^{\frac {5}{2}}}{24 \sqrt {b x + 2}} + \frac {133 x^{\frac {3}{2}}}{24 \sqrt {b x + 2}} + \frac {5 \sqrt {x}}{4 b \sqrt {b x + 2}} - \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {3}{2}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x**(9/2)/(4*sqrt(b*x + 2)) + 23*b**2*x**(7/2)/(12*sqrt(b*x + 2)) + 127*b*x**(5/2)/(24*sqrt(b*x + 2)) + 13

3*x**(3/2)/(24*sqrt(b*x + 2)) + 5*sqrt(x)/(4*b*sqrt(b*x + 2)) - 5*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(3/2)

)

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