Integrand size = 19, antiderivative size = 98 \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=-\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}}+\frac {3 b \sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {3 \sqrt {b} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {49, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=-\frac {3 \sqrt {b} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}+\frac {3 b \sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}} \]
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}}+\frac {(3 b) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d} \\ & = -\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}}+\frac {3 b \sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {(3 b (b c-a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2} \\ & = -\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}}+\frac {3 b \sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {(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 )}{d^2} \\ & = -\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}}+\frac {3 b \sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {(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 )}{d^2} \\ & = -\frac {2 (a+b x)^{3/2}}{d \sqrt {c+d x}}+\frac {3 b \sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {3 \sqrt {b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} (3 b c-2 a d+b d x)}{d^2 \sqrt {c+d x}}-\frac {3 \sqrt {b} (b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{5/2}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (d x +c \right )^{\frac {3}{2}}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.17 \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\left [-\frac {3 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (b d x + 3 \, b c - 2 \, a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (d^{3} x + c d^{2}\right )}}, \frac {3 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (b d x + 3 \, b c - 2 \, a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (d^{3} x + c d^{2}\right )}}\right ] \]
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\[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} b^{2}}{d {\left | b \right |}} + \frac {3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}{d^{3} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {3 \, {\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2} {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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