Integrand size = 21, antiderivative size = 149 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {c (3 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 149, 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 = {390, 386, 385, 211} \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\frac {c (3 b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac {x \sqrt {c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
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Rule 211
Rule 385
Rule 386
Rule 390
Rubi steps \begin{align*} \text {integral}& = \frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {(3 b c-4 a d) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx}{4 a (b c-a d)} \\ & = \frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {(c (3 b c-4 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 a^2 (b c-a d)} \\ & = \frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {(c (3 b c-4 a d)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 a^2 (b c-a d)} \\ & = \frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {c (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\frac {x \sqrt {c+d x^2} \left (-5 a b c+4 a^2 d-3 b^2 c x^2+2 a b d x^2\right )}{8 a^2 (-b c+a d) \left (a+b x^2\right )^2}-\frac {c (3 b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b \sqrt {d} x^2-b x \sqrt {c+d x^2}}{\sqrt {a} \sqrt {b c-a d}}\right )}{8 a^{5/2} (b c-a d)^{3/2}} \]
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Time = 2.53 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(-\frac {c \left (-\frac {\sqrt {d \,x^{2}+c}\, \left (2 x^{2} a b d -3 b^{2} c \,x^{2}+4 a^{2} d -5 a b c \right ) x}{c \left (b \,x^{2}+a \right )^{2}}-\frac {\left (4 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{8 \left (a d -b c \right ) a^{2}}\) | \(121\) |
default | \(\text {Expression too large to display}\) | \(4155\) |
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Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (129) = 258\).
Time = 0.38 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.68 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\left [-\frac {{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d + {\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \, {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{3} + {\left (5 \, a^{2} b^{2} c^{2} - 9 \, a^{3} b c d + 4 \, a^{4} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{32 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )} x^{2}\right )}}, \frac {{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d + {\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \, {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{3} + {\left (5 \, a^{2} b^{2} c^{2} - 9 \, a^{3} b c d + 4 \, a^{4} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (129) = 258\).
Time = 1.66 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.27 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (3 \, b c^{2} \sqrt {d} - 4 \, a c d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{8 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{3} c^{2} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b^{2} c d^{\frac {3}{2}} - 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{3} c^{3} \sqrt {d} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{2} c^{2} d^{\frac {3}{2}} - 40 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b c d^{\frac {5}{2}} + 16 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{3} d^{\frac {7}{2}} + 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{4} \sqrt {d} - 28 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{3} d^{\frac {3}{2}} + 16 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c^{2} d^{\frac {5}{2}} - 3 \, b^{3} c^{5} \sqrt {d} + 2 \, a b^{2} c^{4} d^{\frac {3}{2}}}{4 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}^{2} {\left (a^{2} b^{2} c - a^{3} b d\right )}} \]
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Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^3} \,d x \]
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