Integrand size = 21, antiderivative size = 113 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {386, 385, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2} \]
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Rule 214
Rule 385
Rule 386
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {(3 a) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{4 c} \\ & = \frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2} \\ & = \frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2} \\ & = \frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 10.51 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\frac {x \sqrt {a+b x^2} \left (\frac {\sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (5 a c+2 b c x^2+3 a d x^2\right )}{\left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}}}+\frac {3 a \arcsin \left (\frac {\sqrt {\left (-\frac {b}{a}+\frac {d}{c}\right ) x^2}}{\sqrt {1+\frac {d x^2}{c}}}\right )}{\sqrt {\frac {(-b c+a d) x^2}{a c}}}\right )}{8 c^3 \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 2.50 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(-\frac {a^{2} \left (-\frac {\sqrt {b \,x^{2}+a}\, \left (3 a d \,x^{2}+2 c b \,x^{2}+5 a c \right ) x}{a^{2} \left (d \,x^{2}+c \right )^{2}}+\frac {3 \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}}\right )}{8 c^{2}}\) | \(94\) |
default | \(\text {Expression too large to display}\) | \(6921\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (93) = 186\).
Time = 0.33 (sec) , antiderivative size = 526, normalized size of antiderivative = 4.65 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\left [\frac {3 \, {\left (a^{2} d^{2} x^{4} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b c^{6} - a c^{5} d + {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{4} + 2 \, {\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a^{2} d^{2} x^{4} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b c^{6} - a c^{5} d + {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{4} + 2 \, {\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (93) = 186\).
Time = 1.67 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.99 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=-\frac {3 \, a^{2} \sqrt {b} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, \sqrt {-b^{2} c^{2} + a b c d} c^{2}} + \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{3} + 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d - 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{3} + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d + 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{3} + 2 \, a^{4} b^{\frac {3}{2}} c d^{2} + 3 \, a^{5} \sqrt {b} d^{3}}{4 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2} c^{2} d^{2}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^3} \,d x \]
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