Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}} \]
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Time = 0.06 (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} \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}} \]
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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 {c+d x}}{a b \sqrt {a+b x}}+\frac {2 \int \frac {\frac {b c^2}{2}+\frac {1}{2} a d^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a b} \\ & = \frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}+\frac {c^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a}+\frac {d^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b} \\ & = \frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a}+\frac {\left (2 d^2\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 )}{b^2} \\ & = \frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {\left (2 d^2\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 )}{b^2} \\ & = \frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.67 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.62 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}+\frac {2 i \sqrt {c} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{a^{3/2} b}-\frac {2 i \sqrt {c} \left (-i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{a^{3/2} b}-\frac {4 d^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs. \(2(91)=182\).
Time = 0.62 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.57
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x \sqrt {b d}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,d^{2} x \sqrt {a c}-\sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} d^{2}-2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d +2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \right ) \sqrt {d x +c}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {b x +a}\, a b}\) | \(306\) |
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (91) = 182\).
Time = 0.50 (sec) , antiderivative size = 956, normalized size of antiderivative = 8.03 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\left [\frac {{\left (a b d x + a^{2} d\right )} \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^{2} c x + a b c\right )} \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 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b^{2} x + a^{2} b\right )}}, -\frac {2 \, {\left (a b d x + a^{2} d\right )} \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^{2} c x + a b c\right )} \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 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b^{2} x + a^{2} b\right )}}, \frac {2 \, {\left (b^{2} c x + a b c\right )} \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 (a b d x + a^{2} d\right )} \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 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b^{2} x + a^{2} b\right )}}, \frac {{\left (b^{2} c x + a b c\right )} \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 (a b d x + a^{2} d\right )} \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 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{a b^{2} x + a^{2} b}\right ] \]
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\[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x \left (a + b x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
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