Optimal. Leaf size=54 \[ 2 e \sqrt {a+b x}-2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2}}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 63, 208} \[ 2 e \sqrt {a+b x}-2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (e+f x)}{x} \, dx &=\frac {2 f (a+b x)^{3/2}}{3 b}+e \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=2 e \sqrt {a+b x}+\frac {2 f (a+b x)^{3/2}}{3 b}+(a e) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=2 e \sqrt {a+b x}+\frac {2 f (a+b x)^{3/2}}{3 b}+\frac {(2 a e) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=2 e \sqrt {a+b x}+\frac {2 f (a+b x)^{3/2}}{3 b}-2 \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 1.02 \[ e \left (2 \sqrt {a+b x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )+\frac {2 f (a+b x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 111, normalized size = 2.06 \[ \left [\frac {3 \, \sqrt {a} b e \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (b f x + 3 \, b e + a f\right )} \sqrt {b x + a}}{3 \, b}, \frac {2 \, {\left (3 \, \sqrt {-a} b e \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (b f x + 3 \, b e + a f\right )} \sqrt {b x + a}\right )}}{3 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 57, normalized size = 1.06 \[ \frac {2 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} b^{2} f + 3 \, \sqrt {b x + a} b^{3} e\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.85 \[ \frac {-2 \sqrt {a}\, b e \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\, b e +\frac {2 \left (b x +a \right )^{\frac {3}{2}} f}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 60, normalized size = 1.11 \[ \sqrt {a} e \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (3 \, \sqrt {b x + a} b e + {\left (b x + a\right )}^{\frac {3}{2}} f\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 45, normalized size = 0.83 \[ 2\,e\,\sqrt {a+b\,x}+\frac {2\,f\,{\left (a+b\,x\right )}^{3/2}}{3\,b}+\sqrt {a}\,e\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.08, size = 54, normalized size = 1.00 \[ \frac {2 a e \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 e \sqrt {a + b x} + \frac {2 f \left (a + b x\right )^{\frac {3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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