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

Optimal. Leaf size=77 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a+b x}}{4 b^2}+\frac{x^{3/2} \sqrt{a+b x}}{2 b} \]

[Out]

(-3*a*Sqrt[x]*Sqrt[a + b*x])/(4*b^2) + (x^(3/2)*Sqrt[a + b*x])/(2*b) + (3*a^2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a
+ b*x]])/(4*b^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0224716, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a+b x}}{4 b^2}+\frac{x^{3/2} \sqrt{a+b x}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[a + b*x],x]

[Out]

(-3*a*Sqrt[x]*Sqrt[a + b*x])/(4*b^2) + (x^(3/2)*Sqrt[a + b*x])/(2*b) + (3*a^2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a
+ b*x]])/(4*b^(5/2))

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 63

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 217

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]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0591006, size = 85, normalized size = 1.1 \[ \frac{\sqrt{b} \sqrt{x} \left (-3 a^2-a b x+2 b^2 x^2\right )+3 a^{5/2} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{5/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[a + b*x],x]

[Out]

(Sqrt[b]*Sqrt[x]*(-3*a^2 - a*b*x + 2*b^2*x^2) + 3*a^(5/2)*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(4*b^(5/2)*Sqrt[a + b*x])

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 84, normalized size = 1.1 \begin{align*}{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}-{\frac{3\,a}{4\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{3\,{a}^{2}}{8}\sqrt{x \left ( bx+a \right ) }\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.94413, size = 316, normalized size = 4.1 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt{b x + a} \sqrt{x}}{8 \, b^{3}}, -\frac{3 \, a^{2} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt{b x + a} \sqrt{x}}{4 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*a^2*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(2*b^2*x - 3*a*b)*sqrt(b*x + a)*sqrt(
x))/b^3, -1/4*(3*a^2*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (2*b^2*x - 3*a*b)*sqrt(b*x + a)*sqr
t(x))/b^3]

________________________________________________________________________________________

Sympy [A]  time = 4.54513, size = 100, normalized size = 1.3 \begin{align*} - \frac{3 a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a**(3/2)*sqrt(x)/(4*b**2*sqrt(1 + b*x/a)) - sqrt(a)*x**(3/2)/(4*b*sqrt(1 + b*x/a)) + 3*a**2*asinh(sqrt(b)*s
qrt(x)/sqrt(a))/(4*b**(5/2)) + x**(5/2)/(2*sqrt(a)*sqrt(1 + b*x/a))

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out