Optimal. Leaf size=80 \[ -\frac {\sqrt {a+b \sqrt {x}}}{a x}+\frac {3 b \sqrt {a+b \sqrt {x}}}{2 a^2 \sqrt {x}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {272, 44, 65,
214} \begin {gather*} -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {3 b \sqrt {a+b \sqrt {x}}}{2 a^2 \sqrt {x}}-\frac {\sqrt {a+b \sqrt {x}}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {x}} x^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{a x}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{2 a}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{a x}+\frac {3 b \sqrt {a+b \sqrt {x}}}{2 a^2 \sqrt {x}}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{a x}+\frac {3 b \sqrt {a+b \sqrt {x}}}{2 a^2 \sqrt {x}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {x}}\right )}{2 a^2}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{a x}+\frac {3 b \sqrt {a+b \sqrt {x}}}{2 a^2 \sqrt {x}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 68, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+b \sqrt {x}} \left (-2 a+3 b \sqrt {x}\right )}{2 a^2 x}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 72, normalized size = 0.90
method | result | size |
derivativedivides | \(4 b^{2} \left (-\frac {\sqrt {a +b \sqrt {x}}}{4 a \,b^{2} x}-\frac {3 \left (-\frac {\sqrt {a +b \sqrt {x}}}{2 a b \sqrt {x}}+\frac {\arctanh \left (\frac {\sqrt {a +b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\) | \(72\) |
default | \(4 b^{2} \left (-\frac {\sqrt {a +b \sqrt {x}}}{4 a \,b^{2} x}-\frac {3 \left (-\frac {\sqrt {a +b \sqrt {x}}}{2 a b \sqrt {x}}+\frac {\arctanh \left (\frac {\sqrt {a +b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 104, normalized size = 1.30 \begin {gather*} \frac {3 \, b^{2} \log \left (\frac {\sqrt {b \sqrt {x} + a} - \sqrt {a}}{\sqrt {b \sqrt {x} + a} + \sqrt {a}}\right )}{4 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {b \sqrt {x} + a} a b^{2}}{2 \, {\left ({\left (b \sqrt {x} + a\right )}^{2} a^{2} - 2 \, {\left (b \sqrt {x} + a\right )} a^{3} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 137, normalized size = 1.71 \begin {gather*} \left [\frac {3 \, \sqrt {a} b^{2} x \log \left (\frac {b x - 2 \, \sqrt {b \sqrt {x} + a} \sqrt {a} \sqrt {x} + 2 \, a \sqrt {x}}{x}\right ) + 2 \, {\left (3 \, a b \sqrt {x} - 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{4 \, a^{3} x}, \frac {3 \, \sqrt {-a} b^{2} x \arctan \left (\frac {\sqrt {b \sqrt {x} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b \sqrt {x} - 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{2 \, a^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.71, size = 110, normalized size = 1.38 \begin {gather*} - \frac {1}{\sqrt {b} x^{\frac {5}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {\sqrt {b}}{2 a x^{\frac {3}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {3 b^{\frac {3}{2}}}{2 a^{2} \sqrt [4]{x} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt [4]{x}} \right )}}{2 a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 75, normalized size = 0.94 \begin {gather*} \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b \sqrt {x} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{3} - 5 \, \sqrt {b \sqrt {x} + a} a b^{3}}{a^{2} b^{2} x}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.51, size = 57, normalized size = 0.71 \begin {gather*} \frac {3\,{\left (a+b\,\sqrt {x}\right )}^{3/2}}{2\,a^2\,x}-\frac {5\,\sqrt {a+b\,\sqrt {x}}}{2\,a\,x}-\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,\sqrt {x}}}{\sqrt {a}}\right )}{2\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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