Optimal. Leaf size=51 \[ -\frac {(a+b x)^{3/2}}{x}+3 b \sqrt {a+b x}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \begin {gather*} -\frac {(a+b x)^{3/2}}{x}+3 b \sqrt {a+b x}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x^2} \, dx &=-\frac {(a+b x)^{3/2}}{x}+\frac {1}{2} (3 b) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=3 b \sqrt {a+b x}-\frac {(a+b x)^{3/2}}{x}+\frac {1}{2} (3 a b) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=3 b \sqrt {a+b x}-\frac {(a+b x)^{3/2}}{x}+(3 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=3 b \sqrt {a+b x}-\frac {(a+b x)^{3/2}}{x}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.65 \begin {gather*} \frac {2 b (a+b x)^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x}{a}+1\right )}{5 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 49, normalized size = 0.96 \begin {gather*} \frac {\sqrt {a+b x} (2 (a+b x)-3 a)}{x}-3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 102, normalized size = 2.00 \begin {gather*} \left [\frac {3 \, \sqrt {a} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, b x - a\right )} \sqrt {b x + a}}{2 \, x}, \frac {3 \, \sqrt {-a} b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, b x - a\right )} \sqrt {b x + a}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 56, normalized size = 1.10 \begin {gather*} \frac {\frac {3 \, a b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {b x + a} b^{2} - \frac {\sqrt {b x + a} a b}{x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.92 \begin {gather*} 2 \left (\left (-\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {b x +a}}{2 b x}\right ) a +\sqrt {b x +a}\right ) b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.87, size = 58, normalized size = 1.14 \begin {gather*} \frac {3}{2} \, \sqrt {a} b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x + a} b - \frac {\sqrt {b x + a} a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 42, normalized size = 0.82 \begin {gather*} 2\,b\,\sqrt {a+b\,x}-3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )-\frac {a\,\sqrt {a+b\,x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.66, size = 92, normalized size = 1.80 \begin {gather*} - 3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {a^{2}}{\sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {a \sqrt {b}}{\sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {2 b^{\frac {3}{2}} \sqrt {x}}{\sqrt {\frac {a}{b x} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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