Optimal. Leaf size=77 \[ -\frac {2 (a+b x)^{3/2} (2 a d f-5 b (c f+d e)-3 b d f x)}{15 b^2}+2 c e \sqrt {a+b x}-2 \sqrt {a} c e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 (a+b x)^{3/2} (2 a d f-5 b (c f+d e)-3 b d f x)}{15 b^2}+2 c e \sqrt {a+b x}-2 \sqrt {a} c e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 147
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x) (e+f x)}{x} \, dx &=-\frac {2 (a+b x)^{3/2} (2 a d f-5 b (d e+c f)-3 b d f x)}{15 b^2}+(c e) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=2 c e \sqrt {a+b x}-\frac {2 (a+b x)^{3/2} (2 a d f-5 b (d e+c f)-3 b d f x)}{15 b^2}+(a c e) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=2 c e \sqrt {a+b x}-\frac {2 (a+b x)^{3/2} (2 a d f-5 b (d e+c f)-3 b d f x)}{15 b^2}+\frac {(2 a c e) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=2 c e \sqrt {a+b x}-\frac {2 (a+b x)^{3/2} (2 a d f-5 b (d e+c f)-3 b d f x)}{15 b^2}-2 \sqrt {a} c e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 87, normalized size = 1.13 \[ \frac {2 (a+b x)^{3/2} (-a d f+b c f+b d e)}{3 b^2}+\frac {2 d f (a+b x)^{5/2}}{5 b^2}+2 c e \sqrt {a+b x}-2 \sqrt {a} c e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 217, normalized size = 2.82 \[ \left [\frac {15 \, \sqrt {a} b^{2} c e \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, b^{2} d f x^{2} + 5 \, {\left (3 \, b^{2} c + a b d\right )} e + {\left (5 \, a b c - 2 \, a^{2} d\right )} f + {\left (5 \, b^{2} d e + {\left (5 \, b^{2} c + a b d\right )} f\right )} x\right )} \sqrt {b x + a}}{15 \, b^{2}}, \frac {2 \, {\left (15 \, \sqrt {-a} b^{2} c e \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, b^{2} d f x^{2} + 5 \, {\left (3 \, b^{2} c + a b d\right )} e + {\left (5 \, a b c - 2 \, a^{2} d\right )} f + {\left (5 \, b^{2} d e + {\left (5 \, b^{2} c + a b d\right )} f\right )} x\right )} \sqrt {b x + a}\right )}}{15 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 105, normalized size = 1.36 \[ \frac {2 \, a c \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a}} + \frac {2 \, {\left (5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{9} c f + 3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{8} d f - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{8} d f + 15 \, \sqrt {b x + a} b^{10} c e + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{9} d e\right )}}{15 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 1.16 \[ \frac {-2 \sqrt {a}\, b^{2} c e \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\, b^{2} c e -\frac {2 \left (b x +a \right )^{\frac {3}{2}} a d f}{3}+\frac {2 \left (b x +a \right )^{\frac {3}{2}} b c f}{3}+\frac {2 \left (b x +a \right )^{\frac {3}{2}} b d e}{3}+\frac {2 \left (b x +a \right )^{\frac {5}{2}} d f}{5}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 90, normalized size = 1.17 \[ \sqrt {a} c e \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (15 \, \sqrt {b x + a} b^{2} c e + 3 \, {\left (b x + a\right )}^{\frac {5}{2}} d f + 5 \, {\left (b d e + {\left (b c - a d\right )} f\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{15 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.49, size = 136, normalized size = 1.77 \[ \left (a\,\left (\frac {2\,b\,c\,f-4\,a\,d\,f+2\,b\,d\,e}{b^2}+\frac {2\,a\,d\,f}{b^2}\right )+\frac {2\,\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )}{b^2}\right )\,\sqrt {a+b\,x}+\left (\frac {2\,b\,c\,f-4\,a\,d\,f+2\,b\,d\,e}{3\,b^2}+\frac {2\,a\,d\,f}{3\,b^2}\right )\,{\left (a+b\,x\right )}^{3/2}+\frac {2\,d\,f\,{\left (a+b\,x\right )}^{5/2}}{5\,b^2}+\sqrt {a}\,c\,e\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.97, size = 92, normalized size = 1.19 \[ \frac {2 a c e \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 c e \sqrt {a + b x} + \frac {2 d f \left (a + b x\right )^{\frac {5}{2}}}{5 b^{2}} + \frac {2 \left (a + b x\right )^{\frac {3}{2}} \left (- a d f + b c f + b d e\right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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