3.537 \(\int \frac {(2+b x)^{3/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=58 \[ -\frac {2 (b x+2)^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {b x+2}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]

[Out]

6*arcsinh(1/2*b^(1/2)*x^(1/2)*2^(1/2))*b^(1/2)-2*(b*x+2)^(3/2)/x^(1/2)+3*b*x^(1/2)*(b*x+2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {47, 50, 54, 215} \[ -\frac {2 (b x+2)^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {b x+2}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 + b*x)^(3/2)/x^(3/2),x]

[Out]

3*b*Sqrt[x]*Sqrt[2 + b*x] - (2*(2 + b*x)^(3/2))/Sqrt[x] + 6*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 \frac {(2+b x)^{3/2}}{x^{3/2}} \, dx &=-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+(3 b) \int \frac {\sqrt {2+b x}}{\sqrt {x}} \, dx\\ &=3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+(3 b) \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+(6 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=3 b \sqrt {x} \sqrt {2+b x}-\frac {2 (2+b x)^{3/2}}{\sqrt {x}}+6 \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 28, normalized size = 0.48 \[ -\frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x}{2}\right )}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + b*x)^(3/2)/x^(3/2),x]

[Out]

(-4*Sqrt[2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -1/2*(b*x)])/Sqrt[x]

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fricas [A]  time = 0.48, size = 99, normalized size = 1.71 \[ \left [\frac {3 \, \sqrt {b} x \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) + \sqrt {b x + 2} {\left (b x - 4\right )} \sqrt {x}}{x}, -\frac {6 \, \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + 2} {\left (b x - 4\right )} \sqrt {x}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(3*sqrt(b)*x*log(b*x + sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1) + sqrt(b*x + 2)*(b*x - 4)*sqrt(x))/x, -(6*sqrt(-b)*
x*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))) - sqrt(b*x + 2)*(b*x - 4)*sqrt(x))/x]

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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((b*x+2)^(3/2)/x^(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 [85.3561567818,61.7937478349]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 [71.707969239,78.6493344628]b/abs(b)*b^2/b*(2*(1/2*sq
rt(b*x+2)*sqrt(b*x+2)-3)*sqrt(b*x+2)*sqrt(b*(b*x+2)-2*b)/(b*(b*x+2)-2*b)-6/sqrt(b)*ln(abs(sqrt(b*(b*x+2)-2*b)-
sqrt(b)*sqrt(b*x+2))))

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maple [A]  time = 0.02, size = 72, normalized size = 1.24 \[ \frac {3 \sqrt {\left (b x +2\right ) x}\, \sqrt {b}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{\sqrt {b x +2}\, \sqrt {x}}+\frac {b^{2} x^{2}-2 b x -8}{\sqrt {b x +2}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+2)^(3/2)/x^(3/2),x)

[Out]

(b^2*x^2-2*b*x-8)/(b*x+2)^(1/2)/x^(1/2)+3*((b*x+2)*x)^(1/2)/(b*x+2)^(1/2)*b^(1/2)/x^(1/2)*ln((b*x+1)/b^(1/2)+(
b*x^2+2*x)^(1/2))

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maxima [A]  time = 2.91, size = 81, normalized size = 1.40 \[ -3 \, \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right ) - \frac {4 \, \sqrt {b x + 2}}{\sqrt {x}} - \frac {2 \, \sqrt {b x + 2} b}{{\left (b - \frac {b x + 2}{x}\right )} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)^(3/2)/x^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x+2\right )}^{3/2}}{x^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + 2)^(3/2)/x^(3/2),x)

[Out]

int((b*x + 2)^(3/2)/x^(3/2), x)

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sympy [A]  time = 2.44, size = 73, normalized size = 1.26 \[ 6 \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {b x + 2}} - \frac {2 b \sqrt {x}}{\sqrt {b x + 2}} - \frac {8}{\sqrt {x} \sqrt {b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*sqrt(b)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2) + b**2*x**(3/2)/sqrt(b*x + 2) - 2*b*sqrt(x)/sqrt(b*x + 2) - 8/(sqrt
(x)*sqrt(b*x + 2))

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