Integrand size = 21, antiderivative size = 143 \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=-\frac {7}{5} d \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2-\frac {d \sqrt {a+\frac {b}{x}} \left (2 \left (57 b^2 c^2+15 a b c d-2 a^2 d^2\right )+\frac {b d (33 b c+2 a d)}{x}\right )}{15 b^2}+\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 x+\frac {c^2 (b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {382, 99, 158, 152, 65, 214} \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=-\frac {d \sqrt {a+\frac {b}{x}} \left (2 \left (-2 a^2 d^2+15 a b c d+57 b^2 c^2\right )+\frac {b d (2 a d+33 b c)}{x}\right )}{15 b^2}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (6 a d+b c)}{\sqrt {a}}+x \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3-\frac {7}{5} d \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 \]
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Rule 65
Rule 99
Rule 152
Rule 158
Rule 214
Rule 382
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x} (c+d x)^3}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 x-\text {Subst}\left (\int \frac {(c+d x)^2 \left (\frac {1}{2} (b c+6 a d)+\frac {7 b d x}{2}\right )}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {7}{5} d \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2+\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 x-\frac {2 \text {Subst}\left (\int \frac {(c+d x) \left (\frac {5}{4} b c (b c+6 a d)+\frac {1}{4} b d (33 b c+2 a d) x\right )}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{5 b} \\ & = -\frac {7}{5} d \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2-\frac {d \sqrt {a+\frac {b}{x}} \left (2 \left (57 b^2 c^2+15 a b c d-2 a^2 d^2\right )+\frac {b d (33 b c+2 a d)}{x}\right )}{15 b^2}+\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 x-\frac {1}{2} \left (c^2 (b c+6 a d)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {7}{5} d \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2-\frac {d \sqrt {a+\frac {b}{x}} \left (2 \left (57 b^2 c^2+15 a b c d-2 a^2 d^2\right )+\frac {b d (33 b c+2 a d)}{x}\right )}{15 b^2}+\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 x-\frac {\left (c^2 (b c+6 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} \\ & = -\frac {7}{5} d \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2-\frac {d \sqrt {a+\frac {b}{x}} \left (2 \left (57 b^2 c^2+15 a b c d-2 a^2 d^2\right )+\frac {b d (33 b c+2 a d)}{x}\right )}{15 b^2}+\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 x+\frac {c^2 (b c+6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (4 a^2 d^3 x^2-2 a b d^2 x (d+15 c x)-3 b^2 \left (2 d^3+10 c d^2 x+30 c^2 d x^2-5 c^3 x^3\right )\right )}{15 b^2 x^2}+\frac {c^2 (b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (15 b^{2} c^{3} x^{3}+4 a^{2} d^{3} x^{2}-30 a b c \,d^{2} x^{2}-90 b^{2} c^{2} d \,x^{2}-2 a \,d^{3} x b -30 c \,d^{2} x \,b^{2}-6 b^{2} d^{3}\right ) \sqrt {\frac {a x +b}{x}}}{15 x^{2} b^{2}}+\frac {\left (6 a d +b c \right ) c^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 \sqrt {a}\, \left (a x +b \right )}\) | \(160\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, \left (180 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, b \,c^{2} d \,x^{4}+30 \sqrt {a}\, \sqrt {a \,x^{2}+b x}\, b^{2} c^{3} x^{4}+90 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} c^{2} d \,x^{4}+15 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3} c^{3} x^{4}-180 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,c^{2} d \,x^{2}+8 a^{\frac {3}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} d^{3} x -60 c \,d^{2} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b x -12 \sqrt {a}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,d^{3}\right )}{30 x^{3} \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{2}}\) | \(248\) |
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Time = 0.32 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.14 \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=\left [\frac {15 \, {\left (b^{3} c^{3} + 6 \, a b^{2} c^{2} d\right )} \sqrt {a} x^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (15 \, a b^{2} c^{3} x^{3} - 6 \, a b^{2} d^{3} - 2 \, {\left (45 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{2} - 2 \, {\left (15 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{30 \, a b^{2} x^{2}}, -\frac {15 \, {\left (b^{3} c^{3} + 6 \, a b^{2} c^{2} d\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (15 \, a b^{2} c^{3} x^{3} - 6 \, a b^{2} d^{3} - 2 \, {\left (45 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{2} - 2 \, {\left (15 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{15 \, a b^{2} x^{2}}\right ] \]
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Time = 14.65 (sec) , antiderivative size = 461, normalized size of antiderivative = 3.22 \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=\frac {4 a^{\frac {11}{2}} b^{\frac {3}{2}} d^{3} x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {5}{2}} d^{3} x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {7}{2}} d^{3} x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {9}{2}} d^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {4 a^{6} b d^{3} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {4 a^{5} b^{2} d^{3} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \sqrt {b} c^{3} \sqrt {x} \sqrt {\frac {a x}{b} + 1} - 3 c^{2} d \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 3 c d^{2} \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + \frac {b c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{\sqrt {a}} \]
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Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.15 \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=\frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} x - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{\sqrt {a}}\right )} c^{3} - 3 \, {\left (\sqrt {a} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, \sqrt {a + \frac {b}{x}}\right )} c^{2} d - \frac {2}{15} \, d^{3} {\left (\frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a}{b^{2}}\right )} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} c d^{2}}{b} \]
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Exception generated. \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx=\text {Exception raised: TypeError} \]
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Time = 6.79 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.21 \[ \int \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3 \, dx={\left (a+\frac {b}{x}\right )}^{3/2}\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{3\,b^2}-\frac {4\,a\,d^3}{3\,b^2}\right )+\sqrt {a+\frac {b}{x}}\,\left (2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )-\frac {6\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{b^2}\right )+c^3\,x\,\sqrt {a+\frac {b}{x}}-\frac {2\,d^3\,{\left (a+\frac {b}{x}\right )}^{5/2}}{5\,b^2}-\frac {c^2\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\left (6\,a\,d+b\,c\right )\,1{}\mathrm {i}}{\sqrt {a}} \]
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