3.17.75 \(\int \frac {\sqrt {x}}{(a+\frac {b}{x})^2} \, dx\) [1675]

Optimal. Leaf size=70 \[ -\frac {5 b \sqrt {x}}{a^3}+\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (b+a x)}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \]

[Out]

5/3*x^(3/2)/a^2-x^(5/2)/a/(a*x+b)+5*b^(3/2)*arctan(a^(1/2)*x^(1/2)/b^(1/2))/a^(7/2)-5*b*x^(1/2)/a^3

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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 43, 52, 65, 211} \begin {gather*} \frac {5 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}}-\frac {5 b \sqrt {x}}{a^3}+\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (a x+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b/x)^2,x]

[Out]

(-5*b*Sqrt[x])/a^3 + (5*x^(3/2))/(3*a^2) - x^(5/2)/(a*(b + a*x)) + (5*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]
])/a^(7/2)

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 52

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 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]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^{5/2}}{(b+a x)^2} \, dx\\ &=-\frac {x^{5/2}}{a (b+a x)}+\frac {5 \int \frac {x^{3/2}}{b+a x} \, dx}{2 a}\\ &=\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (b+a x)}-\frac {(5 b) \int \frac {\sqrt {x}}{b+a x} \, dx}{2 a^2}\\ &=-\frac {5 b \sqrt {x}}{a^3}+\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (b+a x)}+\frac {\left (5 b^2\right ) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 a^3}\\ &=-\frac {5 b \sqrt {x}}{a^3}+\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (b+a x)}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=-\frac {5 b \sqrt {x}}{a^3}+\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (b+a x)}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 68, normalized size = 0.97 \begin {gather*} \frac {\sqrt {x} \left (-15 b^2-10 a b x+2 a^2 x^2\right )}{3 a^3 (b+a x)}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b/x)^2,x]

[Out]

(Sqrt[x]*(-15*b^2 - 10*a*b*x + 2*a^2*x^2))/(3*a^3*(b + a*x)) + (5*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a
^(7/2)

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Maple [A]
time = 0.06, size = 59, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {2 a \,x^{\frac {3}{2}}}{3}-4 b \sqrt {x}}{a^{3}}+\frac {2 b^{2} \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}\)	\(59\)
default	\(\frac {\frac {2 a \,x^{\frac {3}{2}}}{3}-4 b \sqrt {x}}{a^{3}}+\frac {2 b^{2} \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}\)	\(59\)
risch	\(\frac {2 \left (a x -6 b \right ) \sqrt {x}}{3 a^{3}}-\frac {b^{2} \sqrt {x}}{a^{3} \left (a x +b \right )}+\frac {5 b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 66, normalized size = 0.94 \begin {gather*} \frac {2 \, a^{2} - \frac {10 \, a b}{x} - \frac {15 \, b^{2}}{x^{2}}}{3 \, {\left (\frac {a^{4}}{x^{\frac {3}{2}}} + \frac {a^{3} b}{x^{\frac {5}{2}}}\right )}} - \frac {5 \, b^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*a^2 - 10*a*b/x - 15*b^2/x^2)/(a^4/x^(3/2) + a^3*b/x^(5/2)) - 5*b^2*arctan(b/(sqrt(a*b)*sqrt(x)))/(sqrt(
a*b)*a^3)

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Fricas [A]
time = 0.38, size = 161, normalized size = 2.30 \begin {gather*} \left [\frac {15 \, {\left (a b x + b^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (2 \, a^{2} x^{2} - 10 \, a b x - 15 \, b^{2}\right )} \sqrt {x}}{6 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {15 \, {\left (a b x + b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (2 \, a^{2} x^{2} - 10 \, a b x - 15 \, b^{2}\right )} \sqrt {x}}{3 \, {\left (a^{4} x + a^{3} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*(a*b*x + b^2)*sqrt(-b/a)*log((a*x + 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x + b)) + 2*(2*a^2*x^2 - 10*a*b*x
- 15*b^2)*sqrt(x))/(a^4*x + a^3*b), 1/3*(15*(a*b*x + b^2)*sqrt(b/a)*arctan(a*sqrt(x)*sqrt(b/a)/b) + (2*a^2*x^2
- 10*a*b*x - 15*b^2)*sqrt(x))/(a^4*x + a^3*b)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (63) = 126\).
time = 2.32, size = 389, normalized size = 5.56 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {7}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{2}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 b^{2}} & \text {for}\: a = 0 \\\frac {4 a^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} - \frac {20 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} - \frac {30 a b^{2} \sqrt {x} \sqrt {- \frac {b}{a}}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} + \frac {15 a b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} - \frac {15 a b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} + \frac {15 b^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} - \frac {15 b^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a^{5} x \sqrt {- \frac {b}{a}} + 6 a^{4} b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a**2), Eq(b, 0)), (2*x**(7/2)/(7*b**2), Eq(a, 0)
), (4*a**3*x**(5/2)*sqrt(-b/a)/(6*a**5*x*sqrt(-b/a) + 6*a**4*b*sqrt(-b/a)) - 20*a**2*b*x**(3/2)*sqrt(-b/a)/(6*
a**5*x*sqrt(-b/a) + 6*a**4*b*sqrt(-b/a)) - 30*a*b**2*sqrt(x)*sqrt(-b/a)/(6*a**5*x*sqrt(-b/a) + 6*a**4*b*sqrt(-
b/a)) + 15*a*b**2*x*log(sqrt(x) - sqrt(-b/a))/(6*a**5*x*sqrt(-b/a) + 6*a**4*b*sqrt(-b/a)) - 15*a*b**2*x*log(sq
rt(x) + sqrt(-b/a))/(6*a**5*x*sqrt(-b/a) + 6*a**4*b*sqrt(-b/a)) + 15*b**3*log(sqrt(x) - sqrt(-b/a))/(6*a**5*x*
sqrt(-b/a) + 6*a**4*b*sqrt(-b/a)) - 15*b**3*log(sqrt(x) + sqrt(-b/a))/(6*a**5*x*sqrt(-b/a) + 6*a**4*b*sqrt(-b/
a)), True))

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Giac [A]
time = 0.51, size = 65, normalized size = 0.93 \begin {gather*} \frac {5 \, b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {b^{2} \sqrt {x}}{{\left (a x + b\right )} a^{3}} + \frac {2 \, {\left (a^{4} x^{\frac {3}{2}} - 6 \, a^{3} b \sqrt {x}\right )}}{3 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*b^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) - b^2*sqrt(x)/((a*x + b)*a^3) + 2/3*(a^4*x^(3/2) - 6*a^3*b*s
qrt(x))/a^6

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Mupad [B]
time = 0.06, size = 58, normalized size = 0.83 \begin {gather*} \frac {2\,x^{3/2}}{3\,a^2}-\frac {4\,b\,\sqrt {x}}{a^3}-\frac {b^2\,\sqrt {x}}{x\,a^4+b\,a^3}+\frac {5\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a + b/x)^2,x)

[Out]

(2*x^(3/2))/(3*a^2) - (4*b*x^(1/2))/a^3 - (b^2*x^(1/2))/(a^3*b + a^4*x) + (5*b^(3/2)*atan((a^(1/2)*x^(1/2))/b^
(1/2)))/a^(7/2)

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