\(\int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx\) [608]

Optimal result

Integrand size = 15, antiderivative size = 88 \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=\frac {5 \sqrt {x} \sqrt {2+b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{6 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{3 b}-\frac {5 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}} \]

[Out]

-5*arcsinh(1/2*b^(1/2)*x^(1/2)*2^(1/2))/b^(7/2)-5/6*x^(3/2)*(b*x+2)^(1/2)/b^2+1/3*x^(5/2)*(b*x+2)^(1/2)/b+5/2*
x^(1/2)*(b*x+2)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 88, 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} \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=-\frac {5 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{2 b^3}-\frac {5 x^{3/2} \sqrt {b x+2}}{6 b^2}+\frac {x^{5/2} \sqrt {b x+2}}{3 b} \]

[In]

Int[x^(5/2)/Sqrt[2 + b*x],x]

[Out]

(5*Sqrt[x]*Sqrt[2 + b*x])/(2*b^3) - (5*x^(3/2)*Sqrt[2 + b*x])/(6*b^2) + (x^(5/2)*Sqrt[2 + b*x])/(3*b) - (5*Arc
Sinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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 {x^{5/2} \sqrt {2+b x}}{3 b}-\frac {5 \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx}{3 b} \\ & = -\frac {5 x^{3/2} \sqrt {2+b x}}{6 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{3 b}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{2 b^2} \\ & = \frac {5 \sqrt {x} \sqrt {2+b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{6 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{3 b}-\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b^3} \\ & = \frac {5 \sqrt {x} \sqrt {2+b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{6 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{3 b}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^3} \\ & = \frac {5 \sqrt {x} \sqrt {2+b x}}{2 b^3}-\frac {5 x^{3/2} \sqrt {2+b x}}{6 b^2}+\frac {x^{5/2} \sqrt {2+b x}}{3 b}-\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84 \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=\frac {\sqrt {x} \sqrt {2+b x} \left (15-5 b x+2 b^2 x^2\right )}{6 b^3}+\frac {10 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2+b x}}\right )}{b^{7/2}} \]

[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) + (10*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[2] - Sqrt[2 + b
*x])])/b^(7/2)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72

method	result	size
meijerg	\(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (14 b^{2} x^{2}-35 b x +105\right ) \sqrt {\frac {b x}{2}+1}}{42}-5 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {7}{2}} \sqrt {\pi }}\)	\(63\)
risch	\(\frac {\left (2 b^{2} x^{2}-5 b x +15\right ) \sqrt {x}\, \sqrt {b x +2}}{6 b^{3}}-\frac {5 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{2 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +2}}\)	\(77\)
default	\(\frac {x^{\frac {5}{2}} \sqrt {b x +2}}{3 b}-\frac {5 \left (\frac {x^{\frac {3}{2}} \sqrt {b x +2}}{2 b}-\frac {3 \left (\frac {\sqrt {x}\, \sqrt {b x +2}}{b}-\frac {\ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +2}}\right )}{2 b}\right )}{3 b}\)	\(104\)

[In]

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

[Out]

8/b^(7/2)/Pi^(1/2)*(1/336*Pi^(1/2)*x^(1/2)*2^(1/2)*b^(1/2)*(14*b^2*x^2-35*b*x+105)*(1/2*b*x+1)^(1/2)-5/8*Pi^(1
/2)*arcsinh(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 = 1.41 \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=\left [\frac {{\left (2 \, b^{3} x^{2} - 5 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 15 \, \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{6 \, b^{4}}, \frac {{\left (2 \, b^{3} x^{2} - 5 \, b^{2} x + 15 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 30 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{6 \, b^{4}}\right ] \]

[In]

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

[Out]

[1/6*((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)*sqrt(b)*sqrt(x)
+ 1))/b^4, 1/6*((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]

Sympy [A] (verification not implemented)

Time = 10.96 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08 \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=\frac {x^{\frac {7}{2}}}{3 \sqrt {b x + 2}} - \frac {x^{\frac {5}{2}}}{6 b \sqrt {b x + 2}} + \frac {5 x^{\frac {3}{2}}}{6 b^{2} \sqrt {b x + 2}} + \frac {5 \sqrt {x}}{b^{3} \sqrt {b x + 2}} - \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (63) = 126\).

Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.52 \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=-\frac {\frac {33 \, \sqrt {b x + 2} b^{2}}{\sqrt {x}} - \frac {40 \, {\left (b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {15 \, {\left (b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{6} - \frac {3 \, {\left (b x + 2\right )} b^{5}}{x} + \frac {3 \, {\left (b x + 2\right )}^{2} b^{4}}{x^{2}} - \frac {{\left (b x + 2\right )}^{3} b^{3}}{x^{3}}\right )}} + \frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {7}{2}}} \]

[In]

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

[Out]

-1/3*(33*sqrt(b*x + 2)*b^2/sqrt(x) - 40*(b*x + 2)^(3/2)*b/x^(3/2) + 15*(b*x + 2)^(5/2)/x^(5/2))/(b^6 - 3*(b*x
+ 2)*b^5/x + 3*(b*x + 2)^2*b^4/x^2 - (b*x + 2)^3*b^3/x^3) + 5/2*log(-(sqrt(b) - sqrt(b*x + 2)/sqrt(x))/(sqrt(b
) + sqrt(b*x + 2)/sqrt(x)))/b^(7/2)

Giac [A] (verification not implemented)

none

Time = 6.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=\frac {{\left (\sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} {\left ({\left (b x + 2\right )} {\left (\frac {2 \, {\left (b x + 2\right )}}{b^{2}} - \frac {13}{b^{2}}\right )} + \frac {33}{b^{2}}\right )} + \frac {30 \, \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )}{b^{\frac {3}{2}}}\right )} {\left | b \right |}}{6 \, b^{3}} \]

[In]

integrate(x^(5/2)/(b*x+2)^(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)/b^3

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{5/2}}{\sqrt {2+b x}} \, dx=\int \frac {x^{5/2}}{\sqrt {b\,x+2}} \,d x \]

[In]

int(x^(5/2)/(b*x + 2)^(1/2),x)

[Out]

int(x^(5/2)/(b*x + 2)^(1/2), x)