Optimal. Leaf size=49 \[ 2 a \sqrt {a+b x}+\frac {2}{3} (a+b x)^{3/2}-2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {52, 65, 214}
\begin {gather*} -2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+2 a \sqrt {a+b x}+\frac {2}{3} (a+b x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x} \, dx &=\frac {2}{3} (a+b x)^{3/2}+a \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=2 a \sqrt {a+b x}+\frac {2}{3} (a+b x)^{3/2}+a^2 \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=2 a \sqrt {a+b x}+\frac {2}{3} (a+b x)^{3/2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=2 a \sqrt {a+b x}+\frac {2}{3} (a+b x)^{3/2}-2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 44, normalized size = 0.90 \begin {gather*} \frac {2}{3} \sqrt {a+b x} (4 a+b x)-2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.08, size = 62, normalized size = 1.27 \begin {gather*} \frac {\sqrt {a} \left (-6 a \text {Log}\left [1+\sqrt {\frac {a+b x}{a}}\right ]+3 a \text {Log}\left [\frac {b x}{a}\right ]+8 a \sqrt {\frac {a+b x}{a}}+2 b x \sqrt {\frac {a+b x}{a}}\right )}{3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 38, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 a \sqrt {b x +a}\) | \(38\) |
default | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 a \sqrt {b x +a}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 52, normalized size = 1.06 \begin {gather*} a^{\frac {3}{2}} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} + 2 \, \sqrt {b x + a} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 88, normalized size = 1.80 \begin {gather*} \left [a^{\frac {3}{2}} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \frac {2}{3} \, {\left (b x + 4 \, a\right )} \sqrt {b x + a}, 2 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \frac {2}{3} \, {\left (b x + 4 \, a\right )} \sqrt {b x + a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.13, size = 71, normalized size = 1.45 \begin {gather*} \frac {8 a^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{3} + a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} - 2 a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )} + \frac {2 \sqrt {a} b x \sqrt {1 + \frac {b x}{a}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 63, normalized size = 1.29 \begin {gather*} \frac {2}{3} \sqrt {a+b x} \left (a+b x\right )+2 \sqrt {a+b x} a+\frac {4 a^{2} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a}}\right )}{2 \sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 37, normalized size = 0.76 \begin {gather*} 2\,a\,\sqrt {a+b\,x}+\frac {2\,{\left (a+b\,x\right )}^{3/2}}{3}-2\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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