Integrand size = 21, antiderivative size = 105 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {424, 393, 223, 212} \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x (b c-a d) (3 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}}+\frac {x \left (c+d x^2\right ) (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rule 212
Rule 223
Rule 393
Rule 424
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {c (2 b c+a d)+3 a d^2 x^2}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b} \\ & = \frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b^2} \\ & = \frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b^2} \\ & = \frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {(-b c+a d) x \left (3 a b c+3 a^2 d+2 b^2 c x^2+4 a b d x^2\right )}{3 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac {d^2 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}} \]
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Time = 2.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(\frac {a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} d^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (-c \left (\frac {2 d \,x^{2}}{3}+c \right ) a \,b^{\frac {5}{2}}+\frac {4 b^{\frac {3}{2}} a^{2} d^{2} x^{2}}{3}+\sqrt {b}\, a^{3} d^{2}-\frac {2 b^{\frac {7}{2}} c^{2} x^{2}}{3}\right )}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{\frac {5}{2}} a^{2}}\) | \(105\) |
default | \(c^{2} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+2 c d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(156\) |
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Time = 0.27 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.03 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.40 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {2 \, c^{2} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {2 \, c d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, c d x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {d^{2} x}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}}{a^{2} b^{3}} + \frac {3 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )}}{a^{2} b^{3}}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {d^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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