Integrand size = 22, antiderivative size = 144 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}-\frac {\sqrt {c} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
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Time = 0.08 (sec) , antiderivative size = 144, 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 = {99, 159, 163, 65, 223, 212, 95, 214} \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=-\frac {\sqrt {c} (3 a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}+2 d \sqrt {a+b x} \sqrt {c+d x} \]
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Rule 65
Rule 95
Rule 99
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}+\int \frac {\sqrt {c+d x} \left (\frac {1}{2} (b c+3 a d)+2 b d x\right )}{x \sqrt {a+b x}} \, dx \\ & = 2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}+\frac {\int \frac {\frac {1}{2} b c (b c+3 a d)+\frac {1}{2} b d (3 b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{b} \\ & = 2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}+\frac {1}{2} (d (3 b c+a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{2} (c (b c+3 a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx \\ & = 2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}+\frac {(d (3 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 )}{b}+(c (b c+3 a d)) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right ) \\ & = 2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}-\frac {\sqrt {c} (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {(d (3 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 )}{b} \\ & = 2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}-\frac {\sqrt {c} (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \left (-c^2+d^2 x^2\right )}{x \sqrt {c+d x}}-\frac {\sqrt {c} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(112)=224\).
Time = 1.47 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a c d x \sqrt {b d}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b \,c^{2} x \sqrt {b d}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,d^{2} x \sqrt {a c}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b c d x \sqrt {a c}-2 d x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 c \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\) | \(298\) |
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none
Time = 0.59 (sec) , antiderivative size = 893, normalized size of antiderivative = 6.20 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\left [\frac {{\left (3 \, b c + a d\right )} x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (b c + 3 \, a d\right )} x \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} {\left (d x - c\right )}}{4 \, x}, -\frac {2 \, {\left (3 \, b c + a d\right )} x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - {\left (b c + 3 \, a d\right )} x \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, \sqrt {b x + a} \sqrt {d x + c} {\left (d x - c\right )}}{4 \, x}, \frac {2 \, {\left (b c + 3 \, a d\right )} x \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + {\left (3 \, b c + a d\right )} x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} {\left (d x - c\right )}}{4 \, x}, \frac {{\left (b c + 3 \, a d\right )} x \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - {\left (3 \, b c + a d\right )} x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} \sqrt {d x + c} {\left (d x - c\right )}}{2 \, x}\right ] \]
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\[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (112) = 224\).
Time = 0.48 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.67 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\frac {\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} d {\left | b \right |}}{b} - \frac {{\left (3 \, \sqrt {b d} b c {\left | b \right |} + \sqrt {b d} a d {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b} - \frac {2 \, {\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} + 3 \, \sqrt {b d} a b c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {4 \, {\left (\sqrt {b d} b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}}}{2 \, b} \]
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Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \]
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