Integrand size = 24, antiderivative size = 245 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-12 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}} \]
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Time = 0.24 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {483, 593, 597, 12, 385, 211} \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {b^4 \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}}+\frac {\sqrt {c+d x^2} (b c-2 a d) \left (-8 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{3 a^2 c^4 x (b c-a d)^2}-\frac {\sqrt {c+d x^2} \left (8 a^2 d^2-12 a b c d+b^2 c^2\right )}{3 a c^3 x^3 (b c-a d)^2}-\frac {d (3 b c-2 a d)}{c^2 x^3 \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 483
Rule 593
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {3 (b c-2 a d)-6 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c (b c-a d)} \\ & = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {\int \frac {3 \left (b^2 c^2-12 a b c d+8 a^2 d^2\right )-12 b d (3 b c-2 a d) x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c^2 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}-\frac {\int \frac {3 (b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right )+6 b d \left (b^2 c^2-12 a b c d+8 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{9 a c^3 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {\int \frac {9 b^4 c^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{9 a^2 c^4 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a^2 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a^2 (b c-a d)^2} \\ & = -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {3 b^3 c^3 x^2 \left (c+d x^2\right )^2-a b^2 c^2 \left (c-2 d x^2\right ) \left (c+d x^2\right )^2+a^2 b c d \left (2 c^3-9 c^2 d x^2-36 c d^2 x^4-24 d^3 x^6\right )+a^3 d^2 \left (-c^3+6 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )}{3 a^2 c^4 (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}-\frac {b^4 \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2} (b c-a d)^{5/2}} \]
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Time = 3.13 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(-\frac {-3 b^{4} c^{4} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right ) x^{3} \left (d \,x^{2}+c \right )^{\frac {3}{2}}+\sqrt {\left (a d -b c \right ) a}\, \left (b^{2} \left (-3 b \,x^{2}+a \right ) c^{5}-2 b d \left (3 b^{2} x^{4}+a^{2}\right ) c^{4}+d^{2} \left (-3 b^{3} x^{6}-3 a \,b^{2} x^{4}+9 a^{2} b \,x^{2}+a^{3}\right ) c^{3}-6 \left (\frac {1}{3} b^{2} x^{4}-6 a b \,x^{2}+a^{2}\right ) x^{2} d^{3} a \,c^{2}-24 a^{2} d^{4} x^{4} \left (-b \,x^{2}+a \right ) c -16 a^{3} d^{5} x^{6}\right )}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) a}\, a^{2} x^{3} \left (a d -b c \right )^{2} c^{4}}\) | \(238\) |
risch | \(\text {Expression too large to display}\) | \(1023\) |
default | \(\text {Expression too large to display}\) | \(1552\) |
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Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (221) = 442\).
Time = 0.67 (sec) , antiderivative size = 1128, normalized size of antiderivative = 4.60 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{4} c^{4} d^{2} x^{7} + 2 \, b^{4} c^{5} d x^{5} + b^{4} c^{6} x^{3}\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 (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - {\left (3 \, a b^{4} c^{4} d^{2} - a^{2} b^{3} c^{3} d^{3} - 26 \, a^{3} b^{2} c^{2} d^{4} + 40 \, a^{4} b c d^{5} - 16 \, a^{5} d^{6}\right )} x^{6} - 3 \, {\left (2 \, a b^{4} c^{5} d - a^{2} b^{3} c^{4} d^{2} - 13 \, a^{3} b^{2} c^{3} d^{3} + 20 \, a^{4} b c^{2} d^{4} - 8 \, a^{5} c d^{5}\right )} x^{4} - 3 \, {\left (a b^{4} c^{6} - a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 5 \, a^{4} b c^{3} d^{3} - 2 \, a^{5} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{3} b^{3} c^{7} d^{2} - 3 \, a^{4} b^{2} c^{6} d^{3} + 3 \, a^{5} b c^{5} d^{4} - a^{6} c^{4} d^{5}\right )} x^{7} + 2 \, {\left (a^{3} b^{3} c^{8} d - 3 \, a^{4} b^{2} c^{7} d^{2} + 3 \, a^{5} b c^{6} d^{3} - a^{6} c^{5} d^{4}\right )} x^{5} + {\left (a^{3} b^{3} c^{9} - 3 \, a^{4} b^{2} c^{8} d + 3 \, a^{5} b c^{7} d^{2} - a^{6} c^{6} d^{3}\right )} x^{3}\right )}}, \frac {3 \, {\left (b^{4} c^{4} d^{2} x^{7} + 2 \, b^{4} c^{5} d x^{5} + b^{4} c^{6} x^{3}\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 (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - {\left (3 \, a b^{4} c^{4} d^{2} - a^{2} b^{3} c^{3} d^{3} - 26 \, a^{3} b^{2} c^{2} d^{4} + 40 \, a^{4} b c d^{5} - 16 \, a^{5} d^{6}\right )} x^{6} - 3 \, {\left (2 \, a b^{4} c^{5} d - a^{2} b^{3} c^{4} d^{2} - 13 \, a^{3} b^{2} c^{3} d^{3} + 20 \, a^{4} b c^{2} d^{4} - 8 \, a^{5} c d^{5}\right )} x^{4} - 3 \, {\left (a b^{4} c^{6} - a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 5 \, a^{4} b c^{3} d^{3} - 2 \, a^{5} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{3} b^{3} c^{7} d^{2} - 3 \, a^{4} b^{2} c^{6} d^{3} + 3 \, a^{5} b c^{5} d^{4} - a^{6} c^{4} d^{5}\right )} x^{7} + 2 \, {\left (a^{3} b^{3} c^{8} d - 3 \, a^{4} b^{2} c^{7} d^{2} + 3 \, a^{5} b c^{6} d^{3} - a^{6} c^{5} d^{4}\right )} x^{5} + {\left (a^{3} b^{3} c^{9} - 3 \, a^{4} b^{2} c^{8} d + 3 \, a^{5} b c^{7} d^{2} - a^{6} c^{6} d^{3}\right )} x^{3}\right )}}\right ] \]
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\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (221) = 442\).
Time = 0.97 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {b^{4} \sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\frac {{\left (11 \, b^{3} c^{6} d^{5} - 30 \, a b^{2} c^{5} d^{6} + 27 \, a^{2} b c^{4} d^{7} - 8 \, a^{3} c^{3} d^{8}\right )} x^{2}}{b^{4} c^{11} d - 4 \, a b^{3} c^{10} d^{2} + 6 \, a^{2} b^{2} c^{9} d^{3} - 4 \, a^{3} b c^{8} d^{4} + a^{4} c^{7} d^{5}} + \frac {3 \, {\left (4 \, b^{3} c^{7} d^{4} - 11 \, a b^{2} c^{6} d^{5} + 10 \, a^{2} b c^{5} d^{6} - 3 \, a^{3} c^{4} d^{7}\right )}}{b^{4} c^{11} d - 4 \, a b^{3} c^{10} d^{2} + 6 \, a^{2} b^{2} c^{9} d^{3} - 4 \, a^{3} b c^{8} d^{4} + a^{4} c^{7} d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} - 18 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d} + 8 \, a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} c^{3}} \]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^4\,\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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