3.5.59 \(\int \frac {1}{\sqrt {x} (a+b x)^2} \, dx\) [459]

Optimal. Leaf size=45 \[ \frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {44, 65, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x)^2),x]

[Out]

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

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 211

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*x)^2),x]

[Out]

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

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Maple [A]
time = 0.10, size = 36, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {\sqrt {x}}{a \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\)	\(36\)
default	\(\frac {\sqrt {x}}{a \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)/a/(b*x+a)+1/a/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2))

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Maxima [A]
time = 0.57, size = 35, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x}}{a b x + a^{2}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.47, size = 116, normalized size = 2.58 \begin {gather*} \left [\frac {2 \, a b \sqrt {x} - \sqrt {-a b} {\left (b x + a\right )} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac {a b \sqrt {x} - \sqrt {a b} {\left (b x + a\right )} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right )}{a^{2} b^{2} x + a^{3} b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (37) = 74\).
time = 2.65, size = 277, normalized size = 6.16 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} + \frac {2 b \sqrt {x} \sqrt {- \frac {a}{b}}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} + \frac {b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} - \frac {b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/a**2, Eq(b, 0)), (-2/(3*b**2*x**(3/2)), Eq(a, 0)), (
a*log(sqrt(x) - sqrt(-a/b))/(2*a**2*b*sqrt(-a/b) + 2*a*b**2*x*sqrt(-a/b)) - a*log(sqrt(x) + sqrt(-a/b))/(2*a**
2*b*sqrt(-a/b) + 2*a*b**2*x*sqrt(-a/b)) + 2*b*sqrt(x)*sqrt(-a/b)/(2*a**2*b*sqrt(-a/b) + 2*a*b**2*x*sqrt(-a/b))
+ b*x*log(sqrt(x) - sqrt(-a/b))/(2*a**2*b*sqrt(-a/b) + 2*a*b**2*x*sqrt(-a/b)) - b*x*log(sqrt(x) + sqrt(-a/b))
/(2*a**2*b*sqrt(-a/b) + 2*a*b**2*x*sqrt(-a/b)), True))

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Giac [A]
time = 0.58, size = 35, normalized size = 0.78 \begin {gather*} \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {\sqrt {x}}{{\left (b x + a\right )} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.09, size = 33, normalized size = 0.73 \begin {gather*} \frac {\sqrt {x}}{a\,\left (a+b\,x\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b*x)^2),x)

[Out]

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

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