Integrand size = 22, antiderivative size = 119 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {100, 163, 65, 223, 212, 95, 214} \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}} \]
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Rule 65
Rule 95
Rule 100
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {2 \int \frac {\frac {a^2 d}{2}+\frac {1}{2} b^2 c x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c d} \\ & = -\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {a^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c}+\frac {b^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d} \\ & = -\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d} \\ & = -\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d} \\ & = -\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.61 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {2 i a^{3/2} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b c^{5/2}}-\frac {2 a^{3/2} \left (\sqrt {a} \sqrt {d}+i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b c^{5/2}}-\frac {4 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{d^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs. \(2(91)=182\).
Time = 0.58 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.57
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x \sqrt {b d}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c d x \sqrt {a c}-\sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c^{2}+2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d -2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \right ) \sqrt {b x +a}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {d x +c}\, c d}\) | \(306\) |
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (91) = 182\).
Time = 0.51 (sec) , antiderivative size = 956, normalized size of antiderivative = 8.03 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\left [\frac {{\left (b c d x + b c^{2}\right )} \sqrt {\frac {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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (a d^{2} x + a c d\right )} \sqrt {\frac {a}{c}} \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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, -\frac {2 \, {\left (b c d x + b c^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (a d^{2} x + a c d\right )} \sqrt {\frac {a}{c}} \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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, \frac {2 \, {\left (a d^{2} x + a c d\right )} \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + {\left (b c d x + b c^{2}\right )} \sqrt {\frac {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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, \frac {{\left (a d^{2} x + a c d\right )} \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - {\left (b c d x + b c^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - 2 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{c d^{2} x + c^{2} d}\right ] \]
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\[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (91) = 182\).
Time = 0.35 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b d} a^{2} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c {\left | b \right |}} - \frac {\sqrt {b d} b^{2} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2} {\left | b \right |}} - \frac {2 \, {\left (b^{3} c {\left | b \right |} - a b^{2} d {\left | b \right |}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} b^{2} c d} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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