Integrand size = 21, antiderivative size = 126 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=-c (3 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {c (3 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}+\sqrt {a} c (3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {382, 91, 81, 52, 65, 214} \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (4 a d+3 b c)+\frac {c^2 x \left (a+\frac {b}{x}\right )^{5/2}}{a}-\frac {c \left (a+\frac {b}{x}\right )^{3/2} (4 a d+3 b c)}{3 a}-c \sqrt {a+\frac {b}{x}} (4 a d+3 b c)-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b x)^{3/2} (c+d x)^2}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2} \left (\frac {1}{2} c (3 b c+4 a d)+a d^2 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{a} \\ & = -\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {(c (3 b c+4 a d)) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\frac {c (3 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {1}{2} (c (3 b c+4 a d)) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -c (3 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {c (3 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {1}{2} (a c (3 b c+4 a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -c (3 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {c (3 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}-\frac {(a c (3 b c+4 a d)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b} \\ & = -c (3 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {c (3 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}}{3 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{5/2} x}{a}+\sqrt {a} c (3 b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (-6 a^2 d^2 x^2+a b x \left (-12 d^2-80 c d x+15 c^2 x^2\right )-2 b^2 \left (3 d^2+10 c d x+15 c^2 x^2\right )\right )}{15 b x^2}+\sqrt {a} c (3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {\left (-15 a b \,c^{2} x^{3}+6 a^{2} d^{2} x^{2}+80 a b c d \,x^{2}+30 b^{2} c^{2} x^{2}+12 x a b \,d^{2}+20 x \,b^{2} c d +6 b^{2} d^{2}\right ) \sqrt {\frac {a x +b}{x}}}{15 x^{2} b}+\frac {\left (4 a d +3 b c \right ) \sqrt {a}\, c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 a x +2 b}\) | \(153\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, \left (120 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x}\, c d \,x^{4}+90 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, b \,c^{2} x^{4}-120 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} c d \,x^{2}+60 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b c d \,x^{4}+45 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} c^{2} x^{4}-12 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} d^{2} x -60 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,c^{2} x^{2}-40 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b c d x -12 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,d^{2}\right )}{30 x^{3} b \sqrt {x \left (a x +b \right )}\, \sqrt {a}}\) | \(260\) |
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Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.13 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\left [\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {a} x^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{30 \, b x^{2}}, -\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{15 \, b x^{2}}\right ] \]
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Time = 22.39 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.33 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {4 a^{\frac {11}{2}} b^{\frac {5}{2}} d^{2} x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {7}{2}} d^{2} x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {9}{2}} d^{2} x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {11}{2}} d^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \sqrt {a} b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} - \frac {4 a^{6} b^{2} d^{2} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {4 a^{5} b^{3} d^{2} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + a \sqrt {b} c^{2} \sqrt {x} \sqrt {\frac {a x}{b} + 1} - 2 a c 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 ) + a d^{2} \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 ) - b c^{2} \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 ) + 2 b c 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 = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.21 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} d^{2}}{5 \, b} + \frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c^{2} - \frac {2}{3} \, {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} c d \]
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Exception generated. \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\text {Exception raised: TypeError} \]
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Time = 6.71 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.56 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \, dx=\sqrt {a+\frac {b}{x}}\,\left (2\,a\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b}+\frac {2\,a^2\,d^2}{b}\right )+\left (\frac {4\,a\,d^2-4\,b\,c\,d}{3\,b}-\frac {4\,a\,d^2}{3\,b}\right )\,{\left (a+\frac {b}{x}\right )}^{3/2}-\frac {2\,d^2\,{\left (a+\frac {b}{x}\right )}^{5/2}}{5\,b}+a\,c^2\,x\,\sqrt {a+\frac {b}{x}}-2\,c\,\mathrm {atan}\left (\frac {2\,c\,\sqrt {a+\frac {b}{x}}\,\left (4\,a\,d+3\,b\,c\right )\,\sqrt {-\frac {a}{4}}}{4\,d\,a^2\,c+3\,b\,a\,c^2}\right )\,\left (4\,a\,d+3\,b\,c\right )\,\sqrt {-\frac {a}{4}} \]
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