Integrand size = 28, antiderivative size = 63 \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {394, 65, 223, 212} \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 b \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}-\frac {x \left (\frac {a}{c^2}+\frac {b}{d^2}\right )}{\sqrt {d x-c} \sqrt {c+d x}} \]
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Rule 65
Rule 212
Rule 223
Rule 394
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {b \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{d^3} \\ & = -\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3} \\ & = -\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-\frac {2 \left (b c^2 d+a d^3\right ) x}{c^2 \sqrt {-c+d x} \sqrt {c+d x}}+4 b \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {-c+d x}}\right )}{2 d^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.63
method | result | size |
default | \(\frac {\sqrt {d x -c}\, \left (-\ln \left (\left (\sqrt {-\left (d x +c \right ) \left (-d x +c \right )}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{2} d^{2} x^{2}+\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d^{3} a x +\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d b \,c^{2} x +\ln \left (\left (\sqrt {-\left (d x +c \right ) \left (-d x +c \right )}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{4}\right ) \operatorname {csgn}\left (d \right )}{\left (-d x +c \right ) \sqrt {d^{2} x^{2}-c^{2}}\, c^{2} d^{3} \sqrt {d x +c}}\) | \(166\) |
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (55) = 110\).
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.05 \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {b c^{4} + a c^{2} d^{2} - {\left (b c^{2} d + a d^{3}\right )} \sqrt {d x + c} \sqrt {d x - c} x - {\left (b c^{2} d^{2} + a d^{4}\right )} x^{2} - {\left (b c^{2} d^{2} x^{2} - b c^{4}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{c^{2} d^{5} x^{2} - c^{4} d^{3}} \]
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Timed out. \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \]
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none
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.21 \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {a x}{\sqrt {d^{2} x^{2} - c^{2}} c^{2}} - \frac {b x}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} + \frac {b \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (55) = 110\).
Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.79 \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {b \log \left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}{d^{3}} - \frac {2 \, {\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} c d^{3}} - \frac {{\left (b c^{2} d^{3} + a d^{5}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{2} d^{6}} \]
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Timed out. \[ \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {b\,x^2+a}{{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \]
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