\(\int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx\) [493]

Optimal result

Integrand size = 15, antiderivative size = 45 \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {a+b x}}{\sqrt {x}}+2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]

[Out]

2*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))*b^(1/2)-2*(b*x+a)^(1/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {49, 65, 223, 212} \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 \sqrt {a+b x}}{\sqrt {x}} \]

[In]

Int[Sqrt[a + b*x]/x^(3/2),x]

[Out]

(-2*Sqrt[a + b*x])/Sqrt[x] + 2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]]

Rule 49

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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+b x}}{\sqrt {x}}+b \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = -\frac {2 \sqrt {a+b x}}{\sqrt {x}}+(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 \sqrt {a+b x}}{\sqrt {x}}+(2 b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = -\frac {2 \sqrt {a+b x}}{\sqrt {x}}+2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {a+b x}}{\sqrt {x}}+4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right ) \]

[In]

Integrate[Sqrt[a + b*x]/x^(3/2),x]

[Out]

(-2*Sqrt[a + b*x])/Sqrt[x] + 4*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.36

method	result	size
risch	\(-\frac {2 \sqrt {b x +a}}{\sqrt {x}}+\frac {\sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{\sqrt {x}\, \sqrt {b x +a}}\)	\(61\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=\left [\frac {\sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, \sqrt {b x + a} \sqrt {x}}{x}, -\frac {2 \, {\left (\sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + a} \sqrt {x}\right )}}{x}\right ] \]

[In]

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

[Out]

[(sqrt(b)*x*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*sqrt(b*x + a)*sqrt(x))/x, -2*(sqrt(-b)*x*arct
an(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + sqrt(b*x + a)*sqrt(x))/x]

Sympy [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=- \frac {2 \sqrt {a}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 b \sqrt {x}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]

[In]

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

[Out]

-2*sqrt(a)/(sqrt(x)*sqrt(1 + b*x/a)) + 2*sqrt(b)*asinh(sqrt(b)*sqrt(x)/sqrt(a)) - 2*b*sqrt(x)/(sqrt(a)*sqrt(1
+ b*x/a))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=-\sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + a}}{\sqrt {x}} \]

[In]

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

[Out]

-sqrt(b)*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/sqrt(x))) - 2*sqrt(b*x + a)/sqrt(x)

Giac [A] (verification not implemented)

none

Time = 79.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=-\frac {2 \, b^{2} {\left (\frac {\log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{\sqrt {b}} + \frac {\sqrt {b x + a}}{\sqrt {{\left (b x + a\right )} b - a b}}\right )}}{{\left | b \right |}} \]

[In]

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

[Out]

-2*b^2*(log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b)))/sqrt(b) + sqrt(b*x + a)/sqrt((b*x + a)*b -
a*b))/abs(b)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x^{3/2}} \,d x \]

[In]

int((a + b*x)^(1/2)/x^(3/2),x)

[Out]

int((a + b*x)^(1/2)/x^(3/2), x)