3.2.48 \(\int \frac {1}{x \sqrt {(a+b x)^3}} \, dx\) [148]

Optimal. Leaf size=57 \[ \frac {2}{a \sqrt {a+b x}}+\frac {\log \left (\frac {-\sqrt {a}+\sqrt {a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )}{a^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1973, 53, 65, 214} \begin {gather*} \frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}-\frac {2 \left (\frac {b x}{a}+1\right )^{3/2} \text {arctanh}\left (\sqrt {\frac {b x}{a}+1}\right )}{\sqrt {(a+b x)^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[(a + b*x)^3]),x]

[Out]

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

Rule 53

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 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 214

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

Rule 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \frac {1}{x \left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{\sqrt {(a+b x)^3}}\\ &=\frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}+\frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \frac {1}{x \sqrt {1+\frac {b x}{a}}} \, dx}{\sqrt {(a+b x)^3}}\\ &=\frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}+\frac {\left (2 a \left (1+\frac {b x}{a}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {a x^2}{b}} \, dx,x,\sqrt {1+\frac {b x}{a}}\right )}{b \sqrt {(a+b x)^3}}\\ &=\frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}-\frac {2 \left (1+\frac {b x}{a}\right )^{3/2} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )}{\sqrt {(a+b x)^3}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 56, normalized size = 0.98 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {a}-\sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{a^{3/2} \sqrt {(a+b x)^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[(a + b*x)^3]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 47, normalized size = 0.82

method	result	size
default	\(-\frac {2 \left (b x +a \right ) \left (\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \sqrt {b x +a}-a^{\frac {3}{2}}\right )}{\sqrt {\left (b x +a \right )^{3}}\, a^{\frac {5}{2}}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b*x+a)*(arctanh((b*x+a)^(1/2)/a^(1/2))*a*(b*x+a)^(1/2)-a^(3/2))/((b*x+a)^3)^(1/2)/a^(5/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt((b*x + a)^3)*x), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (45) = 90\).
time = 0.60, size = 268, normalized size = 4.70 \begin {gather*} \left [\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a} \log \left (\frac {b^{2} x^{2} + 3 \, a b x + 2 \, a^{2} - 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {a}}{b x^{2} + a x}\right ) + 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} a}{a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}}, \frac {2 \, {\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {-a}}{a b x + a^{2}}\right ) + \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} a\right )}}{a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {\left (a + b x\right )^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt((a + b*x)**3)), x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 37, normalized size = 0.65 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2}{\sqrt {b x + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {{\left (a+b\,x\right )}^3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((a + b*x)^3)^(1/2)),x)

[Out]

int(1/(x*((a + b*x)^3)^(1/2)), x)

________________________________________________________________________________________