Integrand size = 23, antiderivative size = 77 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=2 a e \sqrt {c+d x}-\frac {2 (c+d x)^{3/2} (2 b c f-5 d (b e+a f)-3 b d f x)}{15 d^2}-2 a \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {152, 52, 65, 214} \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=-2 a \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )-\frac {2 (c+d x)^{3/2} (-5 d (a f+b e)+2 b c f-3 b d f x)}{15 d^2}+2 a e \sqrt {c+d x} \]
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Rule 52
Rule 65
Rule 152
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\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) \text {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*}
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \sqrt {c+d x} (-b (c+d x) (-5 d e+2 c f-3 d f x)+5 a d (3 d e+c f+d f x))}{15 d^2}-2 a \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]
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Time = 1.56 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {-2 a \sqrt {c}\, d^{2} e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\frac {2 \sqrt {d x +c}\, \left (\left (x \left (\frac {3 f x}{5}+e \right ) b +3 \left (\frac {f x}{3}+e \right ) a \right ) d^{2}+\left (\left (\frac {f x}{5}+e \right ) b +a f \right ) c d -\frac {2 c^{2} b f}{5}\right )}{3}}{d^{2}}\) | \(83\) |
derivativedivides | \(\frac {\frac {2 f b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 a d f \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {2 b c f \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b d e \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a \,d^{2} e \sqrt {d x +c}-2 a \sqrt {c}\, d^{2} e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d^{2}}\) | \(89\) |
default | \(\frac {\frac {2 f b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 a d f \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {2 b c f \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b d e \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a \,d^{2} e \sqrt {d x +c}-2 a \sqrt {c}\, d^{2} e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d^{2}}\) | \(89\) |
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Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.84 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=\left [\frac {15 \, a \sqrt {c} d^{2} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (3 \, b d^{2} f x^{2} + 5 \, {\left (b c d + 3 \, a d^{2}\right )} e - {\left (2 \, b c^{2} - 5 \, a c d\right )} f + {\left (5 \, b d^{2} e + {\left (b c d + 5 \, a d^{2}\right )} f\right )} x\right )} \sqrt {d x + c}}{15 \, d^{2}}, \frac {2 \, {\left (15 \, a \sqrt {-c} d^{2} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (3 \, b d^{2} f x^{2} + 5 \, {\left (b c d + 3 \, a d^{2}\right )} e - {\left (2 \, b c^{2} - 5 \, a c d\right )} f + {\left (5 \, b d^{2} e + {\left (b c d + 5 \, a d^{2}\right )} f\right )} x\right )} \sqrt {d x + c}\right )}}{15 \, d^{2}}\right ] \]
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Time = 10.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=\begin {cases} \frac {2 a c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 a e \sqrt {c + d x} + \frac {2 b f \left (c + d x\right )^{\frac {5}{2}}}{5 d^{2}} + \frac {2 \left (c + d x\right )^{\frac {3}{2}} \left (a d f - b c f + b d e\right )}{3 d^{2}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (a e \log {\left (x \right )} + a f x + b e x + \frac {b f x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=a \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (15 \, \sqrt {d x + c} a d^{2} e + 3 \, {\left (d x + c\right )}^{\frac {5}{2}} b f + 5 \, {\left (b d e - {\left (b c - a d\right )} f\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{15 \, d^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \, a c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (5 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{9} e + 15 \, \sqrt {d x + c} a d^{10} e + 3 \, {\left (d x + c\right )}^{\frac {5}{2}} b d^{8} f - 5 \, {\left (d x + c\right )}^{\frac {3}{2}} b c d^{8} f + 5 \, {\left (d x + c\right )}^{\frac {3}{2}} a d^{9} f\right )}}{15 \, d^{10}} \]
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Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \sqrt {c+d x} (e+f x)}{x} \, dx=\left (c\,\left (\frac {2\,a\,d\,f-4\,b\,c\,f+2\,b\,d\,e}{d^2}+\frac {2\,b\,c\,f}{d^2}\right )-\frac {2\,\left (a\,d-b\,c\right )\,\left (c\,f-d\,e\right )}{d^2}\right )\,\sqrt {c+d\,x}+\left (\frac {2\,a\,d\,f-4\,b\,c\,f+2\,b\,d\,e}{3\,d^2}+\frac {2\,b\,c\,f}{3\,d^2}\right )\,{\left (c+d\,x\right )}^{3/2}+\frac {2\,b\,f\,{\left (c+d\,x\right )}^{5/2}}{5\,d^2}+a\,\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]
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