Optimal. Leaf size=57 \[ \frac {2}{a \sqrt {a+b x}}+\frac {\log \left (\frac {-\sqrt {a}+\sqrt {a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )}{a^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.05, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1973, 53, 65,
214} \begin {gather*} \frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}-\frac {2 \left (\frac {b x}{a}+1\right )^{3/2} \text {arctanh}\left (\sqrt {\frac {b x}{a}+1}\right )}{\sqrt {(a+b x)^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 1973
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \frac {1}{x \left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{\sqrt {(a+b x)^3}}\\ &=\frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}+\frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \frac {1}{x \sqrt {1+\frac {b x}{a}}} \, dx}{\sqrt {(a+b x)^3}}\\ &=\frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}+\frac {\left (2 a \left (1+\frac {b x}{a}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {a x^2}{b}} \, dx,x,\sqrt {1+\frac {b x}{a}}\right )}{b \sqrt {(a+b x)^3}}\\ &=\frac {2 (a+b x)}{a \sqrt {(a+b x)^3}}-\frac {2 \left (1+\frac {b x}{a}\right )^{3/2} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )}{\sqrt {(a+b x)^3}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.98 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {a}-\sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{a^{3/2} \sqrt {(a+b x)^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 47, normalized size = 0.82
method | result | size |
default | \(-\frac {2 \left (b x +a \right ) \left (\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \sqrt {b x +a}-a^{\frac {3}{2}}\right )}{\sqrt {\left (b x +a \right )^{3}}\, a^{\frac {5}{2}}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 125 vs.
\(2 (45) = 90\).
time = 0.60, size = 268, normalized size = 4.70 \begin {gather*} \left [\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a} \log \left (\frac {b^{2} x^{2} + 3 \, a b x + 2 \, a^{2} - 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {a}}{b x^{2} + a x}\right ) + 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} a}{a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}}, \frac {2 \, {\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {-a}}{a b x + a^{2}}\right ) + \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} a\right )}}{a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {\left (a + b x\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 37, normalized size = 0.65 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2}{\sqrt {b x + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {{\left (a+b\,x\right )}^3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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