Integrand size = 21, antiderivative size = 164 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=-3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x+3 \sqrt {a} c^2 (b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
[Out]
Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {382, 99, 158, 152, 52, 65, 214} \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=3 \sqrt {a} c^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (2 a d+b c)-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{35 b^2}-3 c^2 \sqrt {a+\frac {b}{x}} (2 a d+b c)+x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \]
[In]
[Out]
Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b x)^{3/2} (c+d x)^3}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\text {Subst}\left (\int \frac {\sqrt {a+b x} (c+d x)^2 \left (\frac {3}{2} (b c+2 a d)+\frac {9 b d x}{2}\right )}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {2 \text {Subst}\left (\int \frac {\sqrt {a+b x} (c+d x) \left (\frac {21}{4} b c (b c+2 a d)+\frac {3}{4} b d (19 b c+2 a d) x\right )}{x} \, dx,x,\frac {1}{x}\right )}{7 b} \\ & = -\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {1}{2} \left (3 c^2 (b c+2 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {1}{2} \left (3 a c^2 (b c+2 a d)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {\left (3 a c^2 (b c+2 a d)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b} \\ & = -3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x+3 \sqrt {a} c^2 (b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.97 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (4 a^3 d^3 x^3-2 a^2 b d^2 x^2 (d+21 c x)+a b^2 x \left (-16 d^3-84 c d^2 x-280 c^2 d x^2+35 c^3 x^3\right )-2 b^3 \left (5 d^3+21 c d^2 x+35 c^2 d x^2+35 c^3 x^3\right )\right )}{35 b^2 x^3}+3 \sqrt {a} c^2 (b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\frac {\left (35 a \,b^{2} c^{3} x^{4}+4 x^{3} a^{3} d^{3}-42 x^{3} a^{2} b c \,d^{2}-280 x^{3} a \,b^{2} c^{2} d -70 x^{3} b^{3} c^{3}-2 x^{2} a^{2} b \,d^{3}-84 x^{2} a \,b^{2} c \,d^{2}-70 x^{2} b^{3} c^{2} d -16 x a \,b^{2} d^{3}-42 x \,b^{3} c \,d^{2}-10 b^{3} d^{3}\right ) \sqrt {\frac {a x +b}{x}}}{35 x^{3} b^{2}}+\frac {3 \left (2 a d +b c \right ) \sqrt {a}\, c^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 \left (a x +b \right )}\) | \(214\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, \left (420 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x}\, b \,c^{2} d \,x^{5}+210 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, b^{2} c^{3} x^{5}+210 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c^{2} d \,x^{5}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c^{3} x^{5}-420 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,c^{2} d \,x^{3}-140 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} c^{3} x^{3}+8 a^{\frac {5}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} d^{3} x^{2}-84 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b c \,d^{2} x^{2}-140 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} c^{2} d \,x^{2}-12 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,d^{3} x -84 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} c \,d^{2} x -20 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} d^{3}\right )}{70 x^{4} b^{2} \sqrt {x \left (a x +b \right )}\, \sqrt {a}}\) | \(353\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.32 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\left [\frac {105 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt {a} x^{3} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \, {\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \, {\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{70 \, b^{2} x^{3}}, -\frac {105 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \, {\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \, {\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{35 \, b^{2} x^{3}}\right ] \]
[In]
[Out]
Time = 33.36 (sec) , antiderivative size = 1828, normalized size of antiderivative = 11.15 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.16 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=-\frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} c d^{2}}{5 \, b} + \frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c^{3} - {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} c^{2} d - \frac {2}{35} \, {\left (\frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{b^{2}} - \frac {7 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{b^{2}}\right )} d^{3} \]
[In]
[Out]
Exception generated. \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 8.03 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.99 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx={\left (a+\frac {b}{x}\right )}^{5/2}\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{5\,b^2}-\frac {4\,a\,d^3}{5\,b^2}\right )+\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,{\left (a\,d-b\,c\right )}^3}{b^2}+2\,a\,\left (2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )-\frac {6\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{b^2}\right )-a^2\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )\right )+{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (\frac {2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )}{3}-\frac {2\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{3\,b^2}\right )-\frac {2\,d^3\,{\left (a+\frac {b}{x}\right )}^{7/2}}{7\,b^2}+a\,c^3\,x\,\sqrt {a+\frac {b}{x}}-2\,c^2\,\mathrm {atan}\left (\frac {2\,c^2\,\sqrt {a+\frac {b}{x}}\,\left (2\,a\,d+b\,c\right )\,\sqrt {-\frac {9\,a}{4}}}{6\,d\,a^2\,c^2+3\,b\,a\,c^3}\right )\,\left (2\,a\,d+b\,c\right )\,\sqrt {-\frac {9\,a}{4}} \]
[In]
[Out]