Integrand size = 24, antiderivative size = 83 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac {\arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {677, 223, 209} \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\frac {\arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)} \]
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Rule 209
Rule 223
Rule 677
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}-\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx \\ & = \frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\int \frac {1}{\sqrt {a^2-b^2 x^2}} \, dx \\ & = \frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\text {Subst}\left (\int \frac {1}{1+b^2 x^2} \, dx,x,\frac {x}{\sqrt {a^2-b^2 x^2}}\right ) \\ & = \frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac {\tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\frac {4 (a+2 b x) \sqrt {a^2-b^2 x^2}}{3 b (a+b x)^2}-\frac {2 \arctan \left (\frac {b x}{\sqrt {a^2}-\sqrt {a^2-b^2 x^2}}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs. \(2(75)=150\).
Time = 2.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.51
method | result | size |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{3 a b \left (x +\frac {a}{b}\right )^{4}}-\frac {b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{a b \left (x +\frac {a}{b}\right )^{3}}-\frac {2 b \left (\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{a b \left (x +\frac {a}{b}\right )^{2}}+\frac {3 b \left (\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{3}+a b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a}{b}\right )+2 a b \right ) \sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}{4 b^{2}}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )\right )}{a}\right )}{a}\right )}{3 a}}{b^{4}}\) | \(291\) |
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Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\frac {2 \, {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + 2 \, \sqrt {-b^{2} x^{2} + a^{2}} {\left (2 \, b x + a\right )}\right )}}{3 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]
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\[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=-\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{3 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac {\arcsin \left (\frac {b x}{a}\right )}{b} + \frac {7 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3 \, {\left (b^{2} x + a b\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\frac {\arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{{\left | b \right |}} - \frac {8 \, {\left (\frac {3 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + 1\right )}}{3 \, {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx=\int \frac {{\left (a^2-b^2\,x^2\right )}^{3/2}}{{\left (a+b\,x\right )}^4} \,d x \]
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