\(\int \sqrt {x} (2+b x)^{3/2} \, dx\) [535]

Optimal result

Integrand size = 15, antiderivative size = 82 \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\frac {\sqrt {x} \sqrt {2+b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2+b x}+\frac {1}{3} x^{3/2} (2+b x)^{3/2}-\frac {\text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221} \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=-\frac {\text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}+\frac {1}{3} x^{3/2} (b x+2)^{3/2}+\frac {1}{2} x^{3/2} \sqrt {b x+2}+\frac {\sqrt {x} \sqrt {b x+2}}{2 b} \]

[In]

Int[Sqrt[x]*(2 + b*x)^(3/2),x]

[Out]

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

Rule 52

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 56

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 221

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\frac {\sqrt {x} \sqrt {2+b x} \left (3+7 b x+2 b^2 x^2\right )}{6 b}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2+b x}}\right )}{b^{3/2}} \]

[In]

Integrate[Sqrt[x]*(2 + b*x)^(3/2),x]

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(3 + 7*b*x + 2*b^2*x^2))/(6*b) + (2*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[2] - Sqrt[2 + b*x])
])/b^(3/2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77

method	result	size
meijerg	\(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (2 b^{2} x^{2}+7 b x +3\right ) \sqrt {\frac {b x}{2}+1}}{6}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {3}{2}} \sqrt {\pi }}\)	\(63\)
risch	\(\frac {\left (2 b^{2} x^{2}+7 b x +3\right ) \sqrt {x}\, \sqrt {b x +2}}{6 b}-\frac {\ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +2}}\)	\(77\)
default	\(\frac {x^{\frac {3}{2}} \left (b x +2\right )^{\frac {3}{2}}}{3}+\frac {x^{\frac {3}{2}} \sqrt {b x +2}}{2}+\frac {\sqrt {x}\, \sqrt {b x +2}}{2 b}-\frac {\ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +2}}\)	\(87\)

[In]

int((b*x+2)^(3/2)*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

6/b^(3/2)/Pi^(1/2)*(1/36*Pi^(1/2)*x^(1/2)*2^(1/2)*b^(1/2)*(2*b^2*x^2+7*b*x+3)*(1/2*b*x+1)^(1/2)-1/6*Pi^(1/2)*a
rcsinh(1/2*b^(1/2)*x^(1/2)*2^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.51 \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\left [\frac {{\left (2 \, b^{3} x^{2} + 7 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 3 \, \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{6 \, b^{2}}, \frac {{\left (2 \, b^{3} x^{2} + 7 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 6 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{6 \, b^{2}}\right ] \]

[In]

integrate((b*x+2)^(3/2)*x^(1/2),x, algorithm="fricas")

[Out]

[1/6*((2*b^3*x^2 + 7*b^2*x + 3*b)*sqrt(b*x + 2)*sqrt(x) + 3*sqrt(b)*log(b*x - sqrt(b*x + 2)*sqrt(b)*sqrt(x) +
1))/b^2, 1/6*((2*b^3*x^2 + 7*b^2*x + 3*b)*sqrt(b*x + 2)*sqrt(x) + 6*sqrt(-b)*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*
sqrt(x))))/b^2]

Sympy [A] (verification not implemented)

Time = 5.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12 \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\frac {b^{2} x^{\frac {7}{2}}}{3 \sqrt {b x + 2}} + \frac {11 b x^{\frac {5}{2}}}{6 \sqrt {b x + 2}} + \frac {17 x^{\frac {3}{2}}}{6 \sqrt {b x + 2}} + \frac {\sqrt {x}}{b \sqrt {b x + 2}} - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} \]

[In]

integrate((b*x+2)**(3/2)*x**(1/2),x)

[Out]

b**2*x**(7/2)/(3*sqrt(b*x + 2)) + 11*b*x**(5/2)/(6*sqrt(b*x + 2)) + 17*x**(3/2)/(6*sqrt(b*x + 2)) + sqrt(x)/(b
*sqrt(b*x + 2)) - asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (57) = 114\).

Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.61 \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\frac {\frac {3 \, \sqrt {b x + 2} b^{2}}{\sqrt {x}} - \frac {8 \, {\left (b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{4} - \frac {3 \, {\left (b x + 2\right )} b^{3}}{x} + \frac {3 \, {\left (b x + 2\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x + 2\right )}^{3} b}{x^{3}}\right )}} + \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {3}{2}}} \]

[In]

integrate((b*x+2)^(3/2)*x^(1/2),x, algorithm="maxima")

[Out]

1/3*(3*sqrt(b*x + 2)*b^2/sqrt(x) - 8*(b*x + 2)^(3/2)*b/x^(3/2) - 3*(b*x + 2)^(5/2)/x^(5/2))/(b^4 - 3*(b*x + 2)
*b^3/x + 3*(b*x + 2)^2*b^2/x^2 - (b*x + 2)^3*b/x^3) + 1/2*log(-(sqrt(b) - sqrt(b*x + 2)/sqrt(x))/(sqrt(b) + sq
rt(b*x + 2)/sqrt(x)))/b^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (57) = 114\).

Time = 16.77 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.70 \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\frac {{\left (\sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} {\left ({\left (b x + 2\right )} {\left (\frac {2 \, {\left (b x + 2\right )}}{b^{2}} - \frac {13}{b^{2}}\right )} + \frac {33}{b^{2}}\right )} + \frac {30 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{b^{\frac {3}{2}}}\right )} {\left | b \right |} + \frac {12 \, {\left (\sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} {\left (b x - 3\right )} - 6 \, \sqrt {b} \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )\right )} {\left | b \right |}}{b^{2}} + \frac {24 \, {\left (2 \, \sqrt {b} \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right ) + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2}\right )} {\left | b \right |}}{b^{2}}}{6 \, b} \]

[In]

integrate((b*x+2)^(3/2)*x^(1/2),x, algorithm="giac")

[Out]

1/6*((sqrt((b*x + 2)*b - 2*b)*sqrt(b*x + 2)*((b*x + 2)*(2*(b*x + 2)/b^2 - 13/b^2) + 33/b^2) + 30*log(abs(-sqrt
(b*x + 2)*sqrt(b) + sqrt((b*x + 2)*b - 2*b)))/b^(3/2))*abs(b) + 12*(sqrt((b*x + 2)*b - 2*b)*sqrt(b*x + 2)*(b*x
- 3) - 6*sqrt(b)*log(abs(-sqrt(b*x + 2)*sqrt(b) + sqrt((b*x + 2)*b - 2*b))))*abs(b)/b^2 + 24*(2*sqrt(b)*log(a
bs(-sqrt(b*x + 2)*sqrt(b) + sqrt((b*x + 2)*b - 2*b))) + sqrt((b*x + 2)*b - 2*b)*sqrt(b*x + 2))*abs(b)/b^2)/b

Mupad [F(-1)]

Timed out. \[ \int \sqrt {x} (2+b x)^{3/2} \, dx=\int \sqrt {x}\,{\left (b\,x+2\right )}^{3/2} \,d x \]

[In]

int(x^(1/2)*(b*x + 2)^(3/2),x)

[Out]

int(x^(1/2)*(b*x + 2)^(3/2), x)