Integrand size = 22, antiderivative size = 209 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=\frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}+3 \sqrt {b} \sqrt {d} (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {99, 154, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=-\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}+3 \sqrt {b} \sqrt {d} (a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x}+\frac {3 d \sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c} \]
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Rule 65
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {1}{2} \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{2} (b c+a d)+3 b d x\right )}{x^2} \, dx \\ & = -\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{4} \left (b^2 c^2+6 a b c d+a^2 d^2\right )+\frac {3}{2} b d (3 b c+a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 c} \\ & = \frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {\int \frac {\frac {3}{4} b c \left (b^2 c^2+6 a b c d+a^2 d^2\right )+3 b^2 c d (b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b c} \\ & = \frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {1}{2} (3 b d (b c+a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{8} \left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx \\ & = \frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+(3 d (b c+a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{4} \left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right ) \\ & = \frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}+(3 d (b c+a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right ) \\ & = \frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}+3 \sqrt {b} \sqrt {d} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} (b x (5 c-4 d x)+a (2 c+5 d x))}{4 x^2}-\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 \sqrt {a} \sqrt {c}}+3 \sqrt {b} \sqrt {d} (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(165)=330\).
Time = 1.54 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.04
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,d^{2} x^{2} \sqrt {a c}+12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c d \,x^{2} \sqrt {a c}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2} \sqrt {b d}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2} \sqrt {b d}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2} \sqrt {b d}+8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d \,x^{2}-10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x -10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x -4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {b d}\, \sqrt {a c}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(427\) |
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Time = 0.77 (sec) , antiderivative size = 1089, normalized size of antiderivative = 5.21 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=\left [\frac {12 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {b d} x^{2} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a c x^{2}}, -\frac {24 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {-b d} x^{2} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a c x^{2}}, \frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 6 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {b d} x^{2} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 2 \, {\left (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a c x^{2}}, \frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 12 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {-b d} x^{2} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a c x^{2}}\right ] \]
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\[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1168 vs. \(2 (165) = 330\).
Time = 0.98 (sec) , antiderivative size = 1168, normalized size of antiderivative = 5.59 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^3} \,d x \]
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