3.341 \(\int \frac{1}{x^3 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{\sqrt{a+b x}}{2 a x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0165941, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{\sqrt{a+b x}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a + b*x]),x]

[Out]

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 208

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

Rubi steps

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

Mathematica [C]  time = 0.010191, size = 33, normalized size = 0.49 \[ -\frac{2 b^2 \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{b x}{a}+1\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[a + b*x]),x]

[Out]

(-2*b^2*Sqrt[a + b*x]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x)/a])/a^3

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 66, normalized size = 1. \begin{align*} 2\,{b}^{2} \left ( -1/4\,{\frac{\sqrt{bx+a}}{a{b}^{2}{x}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^2*(-1/4*(b*x+a)^(1/2)/a/b^2/x^2-3/4/a*(-1/2*(b*x+a)^(1/2)/a/b/x+1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(3/2)
))

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.55904, size = 301, normalized size = 4.43 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, a^{3} x^{2}}, \frac{3 \, \sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, a^{3} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 6.00912, size = 102, normalized size = 1.5 \begin{align*} - \frac{1}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{\sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.2173, size = 93, normalized size = 1.37 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3} - 5 \, \sqrt{b x + a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*b^3*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x + a)^(3/2)*b^3 - 5*sqrt(b*x + a)*a*b^3)/(a^
2*b^2*x^2))/b