Integrand size = 17, antiderivative size = 85 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\frac {3 d \sqrt {c+d x}}{b^2}-\frac {(c+d x)^{3/2}}{b (a+b x)}-\frac {3 d \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 52, 65, 214} \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=-\frac {3 d \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}}-\frac {(c+d x)^{3/2}}{b (a+b x)}+\frac {3 d \sqrt {c+d x}}{b^2} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{3/2}}{b (a+b x)}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b} \\ & = \frac {3 d \sqrt {c+d x}}{b^2}-\frac {(c+d x)^{3/2}}{b (a+b x)}+\frac {(3 d (b c-a d)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^2} \\ & = \frac {3 d \sqrt {c+d x}}{b^2}-\frac {(c+d x)^{3/2}}{b (a+b x)}+\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^2} \\ & = \frac {3 d \sqrt {c+d x}}{b^2}-\frac {(c+d x)^{3/2}}{b (a+b x)}-\frac {3 d \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\frac {\sqrt {c+d x} (-b c+3 a d+2 b d x)}{b^2 (a+b x)}-\frac {3 d \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{5/2}} \]
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Time = 0.55 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {2 d \sqrt {d x +c}}{b^{2}}-\frac {\left (2 a d -2 b c \right ) d \left (-\frac {\sqrt {d x +c}}{2 \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{2}}\) | \(93\) |
pseudoelliptic | \(\frac {-3 d \left (b x +a \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+3 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}\, \left (\frac {\left (2 d x -c \right ) b}{3}+a d \right )}{b^{2} \left (b x +a \right ) \sqrt {\left (a d -b c \right ) b}}\) | \(99\) |
derivativedivides | \(2 d \left (\frac {\sqrt {d x +c}}{b^{2}}-\frac {\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {3 \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{2}}\right )\) | \(100\) |
default | \(2 d \left (\frac {\sqrt {d x +c}}{b^{2}}-\frac {\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {3 \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{2}}\right )\) | \(100\) |
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Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.47 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\left [\frac {3 \, {\left (b d x + a d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt {d x + c}}{2 \, {\left (b^{3} x + a b^{2}\right )}}, -\frac {3 \, {\left (b d x + a d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt {d x + c}}{b^{3} x + a b^{2}}\right ] \]
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\[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\frac {2 \, \sqrt {d x + c} d}{b^{2}} + \frac {3 \, {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{2}} - \frac {\sqrt {d x + c} b c d - \sqrt {d x + c} a d^{2}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx=\frac {\left (a\,d^2-b\,c\,d\right )\,\sqrt {c+d\,x}}{b^3\,\left (c+d\,x\right )-b^3\,c+a\,b^2\,d}+\frac {2\,d\,\sqrt {c+d\,x}}{b^2}-\frac {3\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,d\,\sqrt {a\,d-b\,c}\,\sqrt {c+d\,x}}{a\,d^2-b\,c\,d}\right )\,\sqrt {a\,d-b\,c}}{b^{5/2}} \]
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