3.5.65 \(\int \frac {\sqrt {x}}{(a+b x)^3} \, dx\) [465]

Optimal. Leaf size=73 \[ -\frac {\sqrt {x}}{2 b (a+b x)^2}+\frac {\sqrt {x}}{4 a b (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 44, 65, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a+b x)}-\frac {\sqrt {x}}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b*x)^3,x]

[Out]

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

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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

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Mathematica [A]
time = 0.11, size = 60, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {x} (a-b x)}{4 a b (a+b x)^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b*x)^3,x]

[Out]

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

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Maple [A]
time = 0.13, size = 52, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {\frac {x^{\frac {3}{2}}}{4 a}-\frac {\sqrt {x}}{4 b}}{\left (b x +a \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a b \sqrt {a b}}\)	\(52\)
default	\(\frac {\frac {x^{\frac {3}{2}}}{4 a}-\frac {\sqrt {x}}{4 b}}{\left (b x +a \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a b \sqrt {a b}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 64, normalized size = 0.88 \begin {gather*} \frac {b x^{\frac {3}{2}} - a \sqrt {x}}{4 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.93, size = 186, normalized size = 2.55 \begin {gather*} \left [-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (a b^{2} x - a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) - {\left (a b^{2} x - a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (58) = 116\).
time = 7.04, size = 627, normalized size = 8.59 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{3} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {2 a b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} + \frac {2 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {2 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} + \frac {2 b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} + \frac {b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a**3), Eq(b, 0)), (-2/(3*b**3*x**(3/2)), Eq(a, 0
)), (a**2*log(sqrt(x) - sqrt(-a/b))/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8*a*b**4*x**2*sqrt(-
a/b)) - a**2*log(sqrt(x) + sqrt(-a/b))/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8*a*b**4*x**2*sqr
t(-a/b)) - 2*a*b*sqrt(x)*sqrt(-a/b)/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8*a*b**4*x**2*sqrt(-
a/b)) + 2*a*b*x*log(sqrt(x) - sqrt(-a/b))/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8*a*b**4*x**2*
sqrt(-a/b)) - 2*a*b*x*log(sqrt(x) + sqrt(-a/b))/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8*a*b**4
*x**2*sqrt(-a/b)) + 2*b**2*x**(3/2)*sqrt(-a/b)/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8*a*b**4*
x**2*sqrt(-a/b)) + b**2*x**2*log(sqrt(x) - sqrt(-a/b))/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-a/b) + 8
*a*b**4*x**2*sqrt(-a/b)) - b**2*x**2*log(sqrt(x) + sqrt(-a/b))/(8*a**3*b**2*sqrt(-a/b) + 16*a**2*b**3*x*sqrt(-
a/b) + 8*a*b**4*x**2*sqrt(-a/b)), True))

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Giac [A]
time = 1.21, size = 52, normalized size = 0.71 \begin {gather*} \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b} + \frac {b x^{\frac {3}{2}} - a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.13, size = 56, normalized size = 0.77 \begin {gather*} \frac {\frac {x^{3/2}}{4\,a}-\frac {\sqrt {x}}{4\,b}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{3/2}\,b^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a + b*x)^3,x)

[Out]

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

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