Optimal. Leaf size=70 \[ -\frac {5 b \sqrt {x}}{a^3}+\frac {5 x^{3/2}}{3 a^2}-\frac {x^{5/2}}{a (b+a x)}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \]
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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.
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Rule 43
Rule 52
Rule 65
Rule 211
Rule 269
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.
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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.
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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.
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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.
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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.
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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.
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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.
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