Integrand size = 25, antiderivative size = 227 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {158, 152, 52, 65, 214} \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+2 a^3 e \sqrt {c+d x}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]
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Rule 52
Rule 65
Rule 152
Rule 158
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 \int \frac {(a+b x)^2 \sqrt {c+d x} \left (\frac {9 a d e}{2}+\frac {3}{2} (3 b d e-2 b c f+2 a d f) x\right )}{x} \, dx}{9 d} \\ & = \frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {4 \int \frac {(a+b x) \sqrt {c+d x} \left (\frac {63}{4} a^2 d^2 e+\frac {3}{4} \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{x} \, dx}{63 d^2} \\ & = \frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\left (a^3 e\right ) \int \frac {\sqrt {c+d x}}{x} \, dx \\ & = 2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\left (a^3 c e\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx \\ & = 2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\frac {\left (2 a^3 c e\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = 2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}-2 a^3 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \sqrt {c+d x} \left (105 a^3 d^3 (3 d e+c f+d f x)+63 a^2 b d^2 (c+d x) (5 d e-2 c f+3 d f x)+9 a b^2 d (c+d x) \left (8 c^2 f+3 d^2 x (7 e+5 f x)-2 c d (7 e+6 f x)\right )-b^3 (c+d x) \left (16 c^3 f-24 c^2 d (e+f x)+6 c d^2 x (6 e+5 f x)-5 d^3 x^2 (9 e+7 f x)\right )\right )}{315 d^4}-2 a^3 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]
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Time = 1.66 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {-2 a^{3} \sqrt {c}\, d^{4} e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\frac {2 \sqrt {d x +c}\, \left (3 \left (\frac {\left (\frac {7 f x}{9}+e \right ) x^{3} b^{3}}{7}+\frac {3 \left (\frac {5 f x}{7}+e \right ) x^{2} a \,b^{2}}{5}+x \left (\frac {3 f x}{5}+e \right ) a^{2} b +\left (\frac {f x}{3}+e \right ) a^{3}\right ) d^{4}+\left (\frac {3 x^{2} \left (\frac {5 f x}{9}+e \right ) b^{3}}{35}+\frac {3 \left (\frac {3 f x}{7}+e \right ) x a \,b^{2}}{5}+3 \left (\frac {f x}{5}+e \right ) a^{2} b +f \,a^{3}\right ) c \,d^{3}-\frac {6 b \left (\frac {2 \left (\frac {f x}{2}+e \right ) x \,b^{2}}{21}+a \left (\frac {2 f x}{7}+e \right ) b +a^{2} f \right ) c^{2} d^{2}}{5}+\frac {24 \left (\frac {\left (\frac {f x}{3}+e \right ) b}{3}+a f \right ) b^{2} c^{3} d}{35}-\frac {16 b^{3} c^{4} f}{105}\right )}{3}}{d^{4}}\) | \(214\) |
derivativedivides | \(\frac {\frac {2 f \,b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {6 a \,b^{2} d f \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {6 b^{3} c f \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d e \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b \,d^{2} f \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {12 a \,b^{2} c d f \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} d^{2} e \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {6 b^{3} c^{2} f \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {4 b^{3} c d e \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 a^{3} d^{3} f \left (d x +c \right )^{\frac {3}{2}}}{3}-2 a^{2} b c \,d^{2} f \left (d x +c \right )^{\frac {3}{2}}+2 a^{2} b \,d^{3} e \left (d x +c \right )^{\frac {3}{2}}+2 a \,b^{2} c^{2} d f \left (d x +c \right )^{\frac {3}{2}}-2 a \,b^{2} c \,d^{2} e \left (d x +c \right )^{\frac {3}{2}}-\frac {2 b^{3} c^{3} f \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{3} c^{2} d e \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a^{3} d^{4} e \sqrt {d x +c}-2 a^{3} \sqrt {c}\, d^{4} e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d^{4}}\) | \(301\) |
default | \(\frac {\frac {2 f \,b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {6 a \,b^{2} d f \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {6 b^{3} c f \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d e \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b \,d^{2} f \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {12 a \,b^{2} c d f \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} d^{2} e \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {6 b^{3} c^{2} f \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {4 b^{3} c d e \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 a^{3} d^{3} f \left (d x +c \right )^{\frac {3}{2}}}{3}-2 a^{2} b c \,d^{2} f \left (d x +c \right )^{\frac {3}{2}}+2 a^{2} b \,d^{3} e \left (d x +c \right )^{\frac {3}{2}}+2 a \,b^{2} c^{2} d f \left (d x +c \right )^{\frac {3}{2}}-2 a \,b^{2} c \,d^{2} e \left (d x +c \right )^{\frac {3}{2}}-\frac {2 b^{3} c^{3} f \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{3} c^{2} d e \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a^{3} d^{4} e \sqrt {d x +c}-2 a^{3} \sqrt {c}\, d^{4} e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d^{4}}\) | \(301\) |
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Time = 0.27 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\left [\frac {315 \, a^{3} \sqrt {c} d^{4} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}}{315 \, d^{4}}, \frac {2 \, {\left (315 \, a^{3} \sqrt {-c} d^{4} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}\right )}}{315 \, d^{4}}\right ] \]
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Time = 12.08 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\begin {cases} \frac {2 a^{3} c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 a^{3} e \sqrt {c + d x} + \frac {2 b^{3} f \left (c + d x\right )^{\frac {9}{2}}}{9 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {7}{2}} \cdot \left (3 a b^{2} d f - 3 b^{3} c f + b^{3} d e\right )}{7 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} b d^{2} f - 6 a b^{2} c d f + 3 a b^{2} d^{2} e + 3 b^{3} c^{2} f - 2 b^{3} c d e\right )}{5 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a^{2} b d^{3} e + 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e - b^{3} c^{3} f + b^{3} c^{2} d e\right )}{3 d^{4}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (a^{3} e \log {\left (x \right )} + a^{3} f x + 3 a^{2} b e x + \frac {b^{3} f x^{4}}{4} + \frac {x^{3} \cdot \left (3 a b^{2} f + b^{3} e\right )}{3} + \frac {x^{2} \cdot \left (3 a^{2} b f + 3 a b^{2} e\right )}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=a^{3} \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{3} d^{4} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} f + 45 \, {\left (b^{3} d e - 3 \, {\left (b^{3} c - a b^{2} d\right )} f\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 63 \, {\left ({\left (2 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} e - 3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} f\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{315 \, d^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \, a^{3} c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{33} e - 126 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{33} e + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{33} e + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{34} e - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{34} e + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{35} e + 315 \, \sqrt {d x + c} a^{3} d^{36} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} d^{32} f - 135 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} c d^{32} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c^{2} d^{32} f - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{3} d^{32} f + 135 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{2} d^{33} f - 378 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} c d^{33} f + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{33} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b d^{34} f - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b c d^{34} f + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} d^{35} f\right )}}{315 \, d^{36}} \]
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Time = 0.24 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx=\left (c\,\left (c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{d^4}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (c\,f-d\,e\right )}{d^4}\right )\,\sqrt {c+d\,x}+\left (\frac {c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{3\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}+\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{7\,d^4}+\frac {2\,b^3\,c\,f}{7\,d^4}\right )\,{\left (c+d\,x\right )}^{7/2}+\left (\frac {c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )}{5}+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{5\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,b^3\,f\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+a^3\,\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]
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