Optimal. Leaf size=77 \[ -\frac{2 (c+d x)^{3/2} (-5 d (a f+b e)+2 b c f-3 b d f x)}{15 d^2}+2 a e \sqrt{c+d x}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]
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Rubi [A] time = 0.0244644, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {147, 50, 63, 208} \[ -\frac{2 (c+d x)^{3/2} (-5 d (a f+b e)+2 b c f-3 b d f x)}{15 d^2}+2 a e \sqrt{c+d x}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) \sqrt{c+d x} (e+f x)}{x} \, dx &=-\frac{2 (c+d x)^{3/2} (2 b c f-5 d (b e+a f)-3 b d f x)}{15 d^2}+(a e) \int \frac{\sqrt{c+d x}}{x} \, dx\\ &=2 a e \sqrt{c+d x}-\frac{2 (c+d x)^{3/2} (2 b c f-5 d (b e+a f)-3 b d f x)}{15 d^2}+(a c e) \int \frac{1}{x \sqrt{c+d x}} \, dx\\ &=2 a e \sqrt{c+d x}-\frac{2 (c+d x)^{3/2} (2 b c f-5 d (b e+a f)-3 b d f x)}{15 d^2}+\frac{(2 a c e) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=2 a e \sqrt{c+d x}-\frac{2 (c+d x)^{3/2} (2 b c f-5 d (b e+a f)-3 b d f x)}{15 d^2}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.150035, size = 81, normalized size = 1.05 \[ \frac{2 \sqrt{c+d x} (5 a d (c f+3 d e+d f x)-b (c+d x) (2 c f-5 d e-3 d f x))}{15 d^2}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 89, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{{d}^{2}} \left ( 1/5\,fb \left ( dx+c \right ) ^{5/2}+1/3\, \left ( dx+c \right ) ^{3/2}adf-1/3\, \left ( dx+c \right ) ^{3/2}bcf+1/3\, \left ( dx+c \right ) ^{3/2}bde+a{d}^{2}e\sqrt{dx+c}-a\sqrt{c}{d}^{2}e{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.349, size = 517, normalized size = 6.71 \begin{align*} \left [\frac{15 \, a \sqrt{c} d^{2} e \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (3 \, b d^{2} f x^{2} + 5 \,{\left (b c d + 3 \, a d^{2}\right )} e -{\left (2 \, b c^{2} - 5 \, a c d\right )} f +{\left (5 \, b d^{2} e +{\left (b c d + 5 \, a d^{2}\right )} f\right )} x\right )} \sqrt{d x + c}}{15 \, d^{2}}, \frac{2 \,{\left (15 \, a \sqrt{-c} d^{2} e \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) +{\left (3 \, b d^{2} f x^{2} + 5 \,{\left (b c d + 3 \, a d^{2}\right )} e -{\left (2 \, b c^{2} - 5 \, a c d\right )} f +{\left (5 \, b d^{2} e +{\left (b c d + 5 \, a d^{2}\right )} f\right )} x\right )} \sqrt{d x + c}\right )}}{15 \, d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.4181, size = 92, normalized size = 1.19 \begin{align*} \frac{2 a c e \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} + 2 a e \sqrt{c + d x} + \frac{2 b f \left (c + d x\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a d f - b c f + b d e\right )}{3 d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47718, size = 142, normalized size = 1.84 \begin{align*} \frac{2 \, a c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{\sqrt{-c}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b d^{8} f - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b c d^{8} f + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a d^{9} f + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{9} e + 15 \, \sqrt{d x + c} a d^{10} e\right )}}{15 \, d^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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