Integrand size = 19, antiderivative size = 100 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=-\left ((3 b c+2 a d) \sqrt {a+\frac {b}{x}}\right )-\frac {(3 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}+\frac {c \left (a+\frac {b}{x}\right )^{5/2} x}{a}+\sqrt {a} (3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {382, 79, 52, 65, 214} \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (2 a d+3 b c)-\frac {\left (a+\frac {b}{x}\right )^{3/2} (2 a d+3 b c)}{3 a}-\sqrt {a+\frac {b}{x}} (2 a d+3 b c)+\frac {c x \left (a+\frac {b}{x}\right )^{5/2}}{a} \]
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Rule 52
Rule 65
Rule 79
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b x)^{3/2} (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {\left (\frac {3 b c}{2}+a d\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x}\right )}{a} \\ & = -\frac {(3 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}+\frac {c \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {1}{2} (3 b c+2 a d) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\left ((3 b c+2 a d) \sqrt {a+\frac {b}{x}}\right )-\frac {(3 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}+\frac {c \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {1}{2} (a (3 b c+2 a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -\left ((3 b c+2 a d) \sqrt {a+\frac {b}{x}}\right )-\frac {(3 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}+\frac {c \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {(a (3 b c+2 a d)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b} \\ & = -\left ((3 b c+2 a d) \sqrt {a+\frac {b}{x}}\right )-\frac {(3 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}+\frac {c \left (a+\frac {b}{x}\right )^{5/2} x}{a}+\sqrt {a} (3 b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.73 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {\sqrt {a+\frac {b}{x}} (a x (-8 d+3 c x)-2 b (d+3 c x))}{3 x}+\sqrt {a} (3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {\left (3 a c \,x^{2}-8 a d x -6 b c x -2 b d \right ) \sqrt {\frac {a x +b}{x}}}{3 x}+\frac {\left (2 a d +3 b c \right ) \sqrt {a}\, \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 a x +2 b}\) | \(105\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, \left (12 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x}\, d \,x^{3}+18 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, b c \,x^{3}-12 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} d x +6 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b d \,x^{3}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} c \,x^{3}-12 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b c x -4 d \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b \right )}{6 x^{2} \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b}\) | \(205\) |
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Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.64 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\left [\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {a} x \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (3 \, a c x^{2} - 2 \, b d - 2 \, {\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, x}, -\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (3 \, a c x^{2} - 2 \, b d - 2 \, {\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, x}\right ] \]
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Time = 14.38 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.77 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\sqrt {a} b c \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} + a \sqrt {b} c \sqrt {x} \sqrt {\frac {a x}{b} + 1} - a d \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c - \frac {1}{3} \, {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} d \]
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Exception generated. \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\text {Exception raised: TypeError} \]
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Time = 6.70 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=2\,a^{3/2}\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )-\frac {2\,d\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3}-2\,a\,d\,\sqrt {a+\frac {b}{x}}-\frac {2\,c\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {a\,x}{b}\right )}{{\left (\frac {a\,x}{b}+1\right )}^{3/2}} \]
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