Integrand size = 21, antiderivative size = 175 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\frac {b (2 b c-a d) x \sqrt {a+b x^2}}{2 c d^2}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac {b^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^3}+\frac {(b c-a d)^{3/2} (4 b c+a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} d^3} \]
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Time = 0.15 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {424, 542, 537, 223, 212, 385, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=-\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (4 b c-5 a d)}{2 d^3}+\frac {(b c-a d)^{3/2} (a d+4 b c) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} d^3}+\frac {b x \sqrt {a+b x^2} (2 b c-a d)}{2 c d^2}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]
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Rule 212
Rule 214
Rule 223
Rule 385
Rule 424
Rule 537
Rule 542
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}+\frac {\int \frac {\sqrt {a+b x^2} \left (a (b c+a d)+2 b (2 b c-a d) x^2\right )}{c+d x^2} \, dx}{2 c d} \\ & = \frac {b (2 b c-a d) x \sqrt {a+b x^2}}{2 c d^2}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}+\frac {\int \frac {-2 a \left (2 b^2 c^2-2 a b c d-a^2 d^2\right )-2 b^2 c (4 b c-5 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{4 c d^2} \\ & = \frac {b (2 b c-a d) x \sqrt {a+b x^2}}{2 c d^2}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac {\left (b^2 (4 b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 d^3}+\frac {\left ((b c-a d)^2 (4 b c+a d)\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 c d^3} \\ & = \frac {b (2 b c-a d) x \sqrt {a+b x^2}}{2 c d^2}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac {\left (b^2 (4 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 d^3}+\frac {\left ((b c-a d)^2 (4 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 c d^3} \\ & = \frac {b (2 b c-a d) x \sqrt {a+b x^2}}{2 c d^2}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^3}+\frac {(b c-a d)^{3/2} (4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} d^3} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\frac {\frac {d x \sqrt {a+b x^2} \left (-2 a b c d+a^2 d^2+b^2 c \left (2 c+d x^2\right )\right )}{c \left (c+d x^2\right )}+\frac {\sqrt {-b c+a d} \left (4 b^2 c^2-3 a b c d-a^2 d^2\right ) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{c^{3/2}}+b^{3/2} (4 b c-5 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^3} \]
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Time = 2.59 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\frac {\left (d \,x^{2}+c \right ) \left (a d +4 b c \right ) \left (a d -b c \right )^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{4}+\left (\left (d \,x^{2}+c \right ) \left (b^{\frac {5}{2}} c -\frac {5 a d \,b^{\frac {3}{2}}}{4}\right ) c \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\frac {x \sqrt {b \,x^{2}+a}\, d \left (2 b^{2} c^{2}-2 \left (-\frac {b \,x^{2}}{2}+a \right ) b d c +a^{2} d^{2}\right )}{4}\right ) \sqrt {\left (a d -b c \right ) c}\right )}{\sqrt {\left (a d -b c \right ) c}\, d^{3} c \left (d \,x^{2}+c \right )}\) | \(180\) |
risch | \(\text {Expression too large to display}\) | \(1017\) |
default | \(\text {Expression too large to display}\) | \(5230\) |
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Time = 0.66 (sec) , antiderivative size = 1236, normalized size of antiderivative = 7.06 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{\left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (147) = 294\).
Time = 0.32 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\frac {\sqrt {b x^{2} + a} b^{2} x}{2 \, d^{2}} + \frac {{\left (4 \, b^{\frac {5}{2}} c - 5 \, a b^{\frac {3}{2}} d\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{4 \, d^{3}} - \frac {{\left (4 \, b^{\frac {7}{2}} c^{3} - 7 \, a b^{\frac {5}{2}} c^{2} d + 2 \, a^{2} b^{\frac {3}{2}} c d^{2} + a^{3} \sqrt {b} d^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, \sqrt {-b^{2} c^{2} + a b c d} c d^{3}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {7}{2}} c^{3} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {5}{2}} c^{2} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {3}{2}} c d^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} \sqrt {b} d^{3} + a^{2} b^{\frac {5}{2}} c^{2} d - 2 \, a^{3} b^{\frac {3}{2}} c d^{2} + a^{4} \sqrt {b} d^{3}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )} c d^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^2} \,d x \]
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