Integrand size = 21, antiderivative size = 144 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} \sqrt {b c-a d}} \]
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Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {386, 385, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\frac {5 a^3 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} \sqrt {b c-a d}}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3} \]
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Rule 214
Rule 385
Rule 386
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {(5 a) \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx}{6 c} \\ & = \frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {\left (5 a^2\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{8 c^2} \\ & = \frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{16 c^3} \\ & = \frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 c^3} \\ & = \frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 10.60 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\frac {x \sqrt {a+b x^2} \left (\frac {\sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (8 b^2 c^2 x^4+2 a b c x^2 \left (13 c+5 d x^2\right )+a^2 \left (33 c^2+40 c d x^2+15 d^2 x^4\right )\right )}{\left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}}}+\frac {15 a^2 \arcsin \left (\frac {\sqrt {\left (-\frac {b}{a}+\frac {d}{c}\right ) x^2}}{\sqrt {1+\frac {d x^2}{c}}}\right )}{\sqrt {\frac {(-b c+a d) x^2}{a c}}}\right )}{48 c^4 \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 2.67 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(-\frac {-33 x \sqrt {b \,x^{2}+a}\, \left (\left (\frac {8}{33} b^{2} x^{4}+\frac {26}{33} a b \,x^{2}+a^{2}\right ) c^{2}+\frac {40 x^{2} \left (\frac {b \,x^{2}}{4}+a \right ) d a c}{33}+\frac {5 a^{2} d^{2} x^{4}}{11}\right ) \sqrt {\left (a d -b c \right ) c}+15 a^{3} \left (d \,x^{2}+c \right )^{3} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{48 \sqrt {\left (a d -b c \right ) c}\, c^{3} \left (d \,x^{2}+c \right )^{3}}\) | \(144\) |
default | \(\text {Expression too large to display}\) | \(19519\) |
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (120) = 240\).
Time = 0.37 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.90 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\left [\frac {15 \, {\left (a^{3} d^{3} x^{6} + 3 \, a^{3} c d^{2} x^{4} + 3 \, a^{3} c^{2} d x^{2} + a^{3} c^{3}\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 (8 \, b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{5} + 2 \, {\left (13 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d - 20 \, a^{3} c^{2} d^{2}\right )} x^{3} + 33 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{192 \, {\left (b c^{8} - a c^{7} d + {\left (b c^{5} d^{3} - a c^{4} d^{4}\right )} x^{6} + 3 \, {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 3 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{2}\right )}}, -\frac {15 \, {\left (a^{3} d^{3} x^{6} + 3 \, a^{3} c d^{2} x^{4} + 3 \, a^{3} c^{2} d x^{2} + a^{3} c^{3}\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 (8 \, b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{5} + 2 \, {\left (13 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d - 20 \, a^{3} c^{2} d^{2}\right )} x^{3} + 33 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b c^{8} - a c^{7} d + {\left (b c^{5} d^{3} - a c^{4} d^{4}\right )} x^{6} + 3 \, {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 3 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (120) = 240\).
Time = 1.31 (sec) , antiderivative size = 846, normalized size of antiderivative = 5.88 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=-\frac {5 \, a^{3} \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 )}{16 \, \sqrt {-b^{2} c^{2} + a b c d} c^{3}} + \frac {48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {7}{2}} c^{3} d^{2} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} \sqrt {b} d^{5} + 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c^{4} d + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} c^{3} d^{2} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} c d^{4} + 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} \sqrt {b} d^{5} + 256 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {11}{2}} c^{5} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c^{4} d + 288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} c^{3} d^{2} - 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c^{2} d^{3} + 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} c d^{4} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} \sqrt {b} d^{5} + 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c^{4} d + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} c^{3} d^{2} + 360 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c^{2} d^{3} - 420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} c d^{4} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} \sqrt {b} d^{5} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} c^{3} d^{2} + 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c^{2} d^{3} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} c d^{4} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} \sqrt {b} d^{5} + 8 \, a^{6} b^{\frac {5}{2}} c^{2} d^{3} + 10 \, a^{7} b^{\frac {3}{2}} c d^{4} + 15 \, a^{8} \sqrt {b} d^{5}}{24 \, {\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 )}^{3} c^{3} d^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^4} \,d x \]
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