Integrand size = 21, antiderivative size = 194 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac {(b c-a d) (4 b c+3 a d) x \sqrt {a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {\sqrt {b c-a d} \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} d^3} \]
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Time = 0.14 (sec) , antiderivative size = 194, 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, 540, 537, 223, 212, 385, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=-\frac {\sqrt {b c-a d} \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right ) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} d^3}+\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {x \sqrt {a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left (c+d x^2\right )}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
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Rule 212
Rule 214
Rule 223
Rule 385
Rule 424
Rule 537
Rule 540
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}+\frac {\int \frac {\sqrt {a+b x^2} \left (a (b c+3 a d)+4 b^2 c x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac {(b c-a d) (4 b c+3 a d) x \sqrt {a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}-\frac {\int \frac {-a \left (4 b^2 c^2+a d (b c+3 a d)\right )-8 b^3 c^2 x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 d^2} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac {(b c-a d) (4 b c+3 a d) x \sqrt {a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac {b^3 \int \frac {1}{\sqrt {a+b x^2}} \, dx}{d^3}-\frac {\left ((b c-a d) \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 d^3} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac {(b c-a d) (4 b c+3 a d) x \sqrt {a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac {b^3 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {\left ((b c-a d) \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right )\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 d^3} \\ & = -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac {(b c-a d) (4 b c+3 a d) x \sqrt {a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {\sqrt {b c-a d} \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} d^3} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=-\frac {\frac {d (b c-a d) x \sqrt {a+b x^2} \left (2 b c \left (2 c+3 d x^2\right )+a d \left (5 c+3 d x^2\right )\right )}{c^2 \left (c+d x^2\right )^2}+\frac {3 (-4 b c+a d)^2 \sqrt {-b c+a d} \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{c^{5/2}}-\frac {4 b (10 b c-7 a d) \sqrt {b c-a d} \text {arctanh}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {b c-a d}}\right )}{c^{3/2}}+8 b^{5/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 d^3} \]
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Time = 2.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\left (d \,x^{2}+c \right )^{2} \left (a^{2} d^{2}+\frac {4}{3} a b c d +\frac {8}{3} b^{2} c^{2}\right ) \left (a d -b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )-\frac {5 \sqrt {\left (a d -b c \right ) c}\, \left (\frac {8 c^{2} b^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{5}+x d \left (\frac {4 b \,c^{2}}{5}+d \left (\frac {6 b \,x^{2}}{5}+a \right ) c +\frac {3 a \,d^{2} x^{2}}{5}\right ) \sqrt {b \,x^{2}+a}\, \left (a d -b c \right )\right )}{3}\right )}{8 \sqrt {\left (a d -b c \right ) c}\, d^{3} \left (d \,x^{2}+c \right )^{2} c^{2}}\) | \(194\) |
default | \(\text {Expression too large to display}\) | \(10683\) |
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (168) = 336\).
Time = 0.50 (sec) , antiderivative size = 1517, normalized size of antiderivative = 7.82 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{\left (c + d x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (168) = 336\).
Time = 0.32 (sec) , antiderivative size = 659, normalized size of antiderivative = 3.40 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=-\frac {b^{\frac {5}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, d^{3}} + \frac {{\left (8 \, b^{\frac {7}{2}} c^{3} - 4 \, a b^{\frac {5}{2}} c^{2} d - a^{2} b^{\frac {3}{2}} c d^{2} - 3 \, 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 )}{8 \, \sqrt {-b^{2} c^{2} + a b c d} c^{2} d^{3}} - \frac {16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {7}{2}} c^{3} d - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {5}{2}} c^{2} d^{2} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {3}{2}} c d^{3} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} \sqrt {b} d^{4} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {9}{2}} c^{4} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {7}{2}} c^{3} d + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} c^{2} d^{2} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {3}{2}} c d^{3} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} \sqrt {b} d^{4} + 32 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {7}{2}} c^{3} d - 28 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {5}{2}} c^{2} d^{2} - 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {3}{2}} c d^{3} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} \sqrt {b} d^{4} + 6 \, a^{4} b^{\frac {5}{2}} c^{2} d^{2} - 3 \, a^{5} b^{\frac {3}{2}} c d^{3} - 3 \, a^{6} \sqrt {b} d^{4}}{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^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^3} \,d x \]
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