Integrand size = 21, antiderivative size = 90 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 x}{a b^2 \sqrt {a+b x^2}}+\frac {d^2 x \sqrt {a+b x^2}}{2 b^2}+\frac {d (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {424, 396, 223, 212} \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (4 b c-3 a d)}{2 b^{5/2}}-\frac {d x \sqrt {a+b x^2} (2 b c-3 a d)}{2 a b^2}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}} \]
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Rule 212
Rule 223
Rule 396
Rule 424
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {\int \frac {a c d-d (2 b c-3 a d) x^2}{\sqrt {a+b x^2}} \, dx}{a b} \\ & = -\frac {d (2 b c-3 a d) x \sqrt {a+b x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {(d (4 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^2} \\ & = -\frac {d (2 b c-3 a d) x \sqrt {a+b x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {(d (4 b c-3 a d)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^2} \\ & = -\frac {d (2 b c-3 a d) x \sqrt {a+b x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (2 b^2 c^2-4 a b c d+3 a^2 d^2+a b d^2 x^2\right )}{2 a b^2 \sqrt {a+b x^2}}-\frac {d (4 b c-3 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{5/2}} \]
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Time = 2.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\sqrt {b \,x^{2}+a}\, a d \left (a d -\frac {4 b c}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (-\frac {4 \left (-\frac {d \,x^{2}}{4}+c \right ) d a \,b^{\frac {3}{2}}}{3}+\sqrt {b}\, a^{2} d^{2}+\frac {2 b^{\frac {5}{2}} c^{2}}{3}\right )\right )}{2 \sqrt {b \,x^{2}+a}\, b^{\frac {5}{2}} a}\) | \(93\) |
risch | \(\frac {d^{2} x \sqrt {b \,x^{2}+a}}{2 b^{2}}-\frac {\frac {a \,d^{2} x}{\sqrt {b \,x^{2}+a}}-\frac {2 b^{2} c^{2} x}{a \sqrt {b \,x^{2}+a}}+\left (3 a b \,d^{2}-4 b^{2} c d \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b^{2}}\) | \(114\) |
default | \(\frac {c^{2} x}{a \sqrt {b \,x^{2}+a}}+d^{2} \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+2 c d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(123\) |
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Time = 0.27 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.07 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (a b^{2} d^{2} x^{3} + {\left (2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, -\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (a b^{2} d^{2} x^{3} + {\left (2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d^{2} x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {c^{2} x}{\sqrt {b x^{2} + a} a} - \frac {2 \, c d x}{\sqrt {b x^{2} + a} b} + \frac {3 \, a d^{2} x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {2 \, c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {3 \, a d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {d^{2} x^{2}}{b} + \frac {2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}}{a b^{3}}\right )} x}{2 \, \sqrt {b x^{2} + a}} - \frac {{\left (4 \, b c d - 3 \, a d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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