Integrand size = 22, antiderivative size = 164 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}-2 a^{3/2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {103, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=-2 a^{3/2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{3/2}}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{4 d} \]
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Rule 65
Rule 95
Rule 103
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}-\frac {1}{2} \int \frac {\sqrt {a+b x} \left (-2 a c+\frac {1}{2} (-b c-3 a d) x\right )}{x \sqrt {c+d x}} \, dx \\ & = \frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}-\frac {\int \frac {-2 a^2 c d+\frac {1}{4} \left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d} \\ & = \frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}+\left (a^2 c\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d} \\ & = \frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}+\left (2 a^2 c\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {1}{4} \left (-6 a c+\frac {b c^2}{d}-\frac {3 a^2 d}{b}\right ) \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 {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}-2 a^{3/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {1}{4} \left (-6 a c+\frac {b c^2}{d}-\frac {3 a^2 d}{b}\right ) \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 {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {1}{2} (a+b x)^{3/2} \sqrt {c+d x}-2 a^{3/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{3/2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} (5 a d+b (c+2 d x))}{d}-8 a^{3/2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {\left (-b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} d^{3/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(126)=252\).
Time = 0.54 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.03
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} d^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a b c d -\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c^{2}-8 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d +10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d +2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d \sqrt {b d}\, \sqrt {a c}}\) | \(333\) |
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Time = 1.08 (sec) , antiderivative size = 979, normalized size of antiderivative = 5.97 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\left [\frac {8 \, \sqrt {a c} a b d^{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 ) - {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {b d} \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 ) + 4 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d^{2}}, \frac {4 \, \sqrt {a c} a b d^{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 ) + {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-b d} \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 (2 \, b^{2} d^{2} x + b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d^{2}}, \frac {16 \, \sqrt {-a c} a b d^{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 ) - {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {b d} \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 ) + 4 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d^{2}}, \frac {8 \, \sqrt {-a c} a b d^{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 ) + {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-b d} \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 (2 \, b^{2} d^{2} x + b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d^{2}}\right ] \]
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\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}{x}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x} \,d x \]
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