Integrand size = 22, antiderivative size = 270 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {8 b^2 c^2-55 a b c d+35 a^2 d^2}{24 a c^3 (b c-a d)^2 x^3}+\frac {8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3}{8 a^2 c^4 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \left (63 b^2 c^2-90 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^3} \]
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Time = 0.32 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {483, 593, 597, 536, 211} \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \left (35 a^2 d^2-90 a b c d+63 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^3}-\frac {35 a^2 d^2-55 a b c d+8 b^2 c^2}{24 a c^3 x^3 (b c-a d)^2}+\frac {35 a^3 d^3-55 a^2 b c d^2+8 a b^2 c^2 d+8 b^3 c^3}{8 a^2 c^4 x (b c-a d)^2}-\frac {d (11 b c-7 a d)}{8 c^2 x^3 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)} \]
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Rule 211
Rule 483
Rule 536
Rule 593
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-7 a d-7 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)} \\ & = -\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {\int \frac {8 b^2 c^2-55 a b c d+35 a^2 d^2-5 b d (11 b c-7 a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2} \\ & = -\frac {\frac {8 b^2 c}{a}-55 b d+\frac {35 a d^2}{c}}{24 c^2 (b c-a d)^2 x^3}-\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}-\frac {\int \frac {3 \left (8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3\right )+3 b d \left (8 b^2 c^2-55 a b c d+35 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{24 a c^3 (b c-a d)^2} \\ & = -\frac {\frac {8 b^2 c}{a}-55 b d+\frac {35 a d^2}{c}}{24 c^2 (b c-a d)^2 x^3}+\frac {8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3}{8 a^2 c^4 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {\int \frac {3 \left (8 b^4 c^4+8 a b^3 c^3 d+8 a^2 b^2 c^2 d^2-55 a^3 b c d^3+35 a^4 d^4\right )+3 b d \left (8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{24 a^2 c^4 (b c-a d)^2} \\ & = -\frac {\frac {8 b^2 c}{a}-55 b d+\frac {35 a d^2}{c}}{24 c^2 (b c-a d)^2 x^3}+\frac {8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3}{8 a^2 c^4 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b^5 \int \frac {1}{a+b x^2} \, dx}{a^2 (b c-a d)^3}-\frac {\left (d^3 \left (63 b^2 c^2-90 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^4 (b c-a d)^3} \\ & = -\frac {\frac {8 b^2 c}{a}-55 b d+\frac {35 a d^2}{c}}{24 c^2 (b c-a d)^2 x^3}+\frac {8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3}{8 a^2 c^4 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \left (63 b^2 c^2-90 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^3} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {1}{3 a c^3 x^3}+\frac {b c+3 a d}{a^2 c^4 x}-\frac {d^3 x}{4 c^3 (b c-a d) \left (c+d x^2\right )^2}-\frac {d^3 (15 b c-11 a d) x}{8 c^4 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (-b c+a d)^3}-\frac {d^{5/2} \left (63 b^2 c^2-90 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^3} \]
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Time = 2.79 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {1}{3 a \,c^{3} x^{3}}-\frac {-3 a d -b c}{x \,a^{2} c^{4}}-\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{2} \left (a d -b c \right )^{3} \sqrt {a b}}+\frac {d^{3} \left (\frac {\left (\frac {11}{8} a^{2} d^{3}-\frac {13}{4} a b c \,d^{2}+\frac {15}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (13 a^{2} d^{2}-30 a b c d +17 b^{2} c^{2}\right ) x}{8}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (35 a^{2} d^{2}-90 a b c d +63 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}}\right )}{c^{4} \left (a d -b c \right )^{3}}\) | \(190\) |
risch | \(\text {Expression too large to display}\) | \(1924\) |
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Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (244) = 488\).
Time = 3.32 (sec) , antiderivative size = 2397, normalized size of antiderivative = 8.88 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.63 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (63 \, b^{2} c^{2} d^{3} - 90 \, a b c d^{4} + 35 \, a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} \sqrt {c d}} - \frac {8 \, a b^{2} c^{5} - 16 \, a^{2} b c^{4} d + 8 \, a^{3} c^{3} d^{2} - 3 \, {\left (8 \, b^{3} c^{3} d^{2} + 8 \, a b^{2} c^{2} d^{3} - 55 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{6} - {\left (48 \, b^{3} c^{4} d + 40 \, a b^{2} c^{3} d^{2} - 275 \, a^{2} b c^{2} d^{3} + 175 \, a^{3} c d^{4}\right )} x^{4} - 8 \, {\left (3 \, b^{3} c^{5} + a b^{2} c^{4} d - 11 \, a^{2} b c^{3} d^{2} + 7 \, a^{3} c^{2} d^{3}\right )} x^{2}}{24 \, {\left ({\left (a^{2} b^{2} c^{6} d^{2} - 2 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} x^{7} + 2 \, {\left (a^{2} b^{2} c^{7} d - 2 \, a^{3} b c^{6} d^{2} + a^{4} c^{5} d^{3}\right )} x^{5} + {\left (a^{2} b^{2} c^{8} - 2 \, a^{3} b c^{7} d + a^{4} c^{6} d^{2}\right )} x^{3}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (63 \, b^{2} c^{2} d^{3} - 90 \, a b c d^{4} + 35 \, a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} \sqrt {c d}} - \frac {15 \, b c d^{4} x^{3} - 11 \, a d^{5} x^{3} + 17 \, b c^{2} d^{3} x - 13 \, a c d^{4} x}{8 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {3 \, b c x^{2} + 9 \, a d x^{2} - a c}{3 \, a^{2} c^{4} x^{3}} \]
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Time = 6.51 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.91 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {\frac {x^2\,\left (7\,a\,d+3\,b\,c\right )}{3\,a^2\,c^2}-\frac {1}{3\,a\,c}+\frac {x^4\,\left (175\,a^3\,d^4-275\,a^2\,b\,c\,d^3+40\,a\,b^2\,c^2\,d^2+48\,b^3\,c^3\,d\right )}{24\,a^2\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^6\,\left (35\,a^3\,d^5-55\,a^2\,b\,c\,d^4+8\,a\,b^2\,c^2\,d^3+8\,b^3\,c^3\,d^2\right )}{8\,a^2\,c^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2\,x^3+2\,c\,d\,x^5+d^2\,x^7}+\frac {\mathrm {atan}\left (\frac {b\,c^9\,x\,{\left (-a^5\,b^9\right )}^{3/2}\,64{}\mathrm {i}+a^{14}\,b\,d^9\,x\,\sqrt {-a^5\,b^9}\,1225{}\mathrm {i}+a^{10}\,b^5\,c^4\,d^5\,x\,\sqrt {-a^5\,b^9}\,3969{}\mathrm {i}-a^{11}\,b^4\,c^3\,d^6\,x\,\sqrt {-a^5\,b^9}\,11340{}\mathrm {i}+a^{12}\,b^3\,c^2\,d^7\,x\,\sqrt {-a^5\,b^9}\,12510{}\mathrm {i}-a^{13}\,b^2\,c\,d^8\,x\,\sqrt {-a^5\,b^9}\,6300{}\mathrm {i}}{-1225\,a^{17}\,b^5\,d^9+6300\,a^{16}\,b^6\,c\,d^8-12510\,a^{15}\,b^7\,c^2\,d^7+11340\,a^{14}\,b^8\,c^3\,d^6-3969\,a^{13}\,b^9\,c^4\,d^5+64\,a^8\,b^{14}\,c^9}\right )\,\sqrt {-a^5\,b^9}\,1{}\mathrm {i}}{a^8\,d^3-3\,a^7\,b\,c\,d^2+3\,a^6\,b^2\,c^2\,d-a^5\,b^3\,c^3}+\frac {\mathrm {atan}\left (\frac {a^9\,d^5\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,1225{}\mathrm {i}+b^9\,c^{18}\,d\,x\,\sqrt {-c^9\,d^5}\,64{}\mathrm {i}-a^6\,b^3\,c^3\,d^2\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,11340{}\mathrm {i}+a^7\,b^2\,c^2\,d^3\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,12510{}\mathrm {i}-a^8\,b\,c\,d^4\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,6300{}\mathrm {i}+a^5\,b^4\,c^4\,d\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,3969{}\mathrm {i}}{1225\,a^9\,c^{14}\,d^{12}-6300\,a^8\,b\,c^{15}\,d^{11}+12510\,a^7\,b^2\,c^{16}\,d^{10}-11340\,a^6\,b^3\,c^{17}\,d^9+3969\,a^5\,b^4\,c^{18}\,d^8-64\,b^9\,c^{23}\,d^3}\right )\,\sqrt {-c^9\,d^5}\,\left (35\,a^2\,d^2-90\,a\,b\,c\,d+63\,b^2\,c^2\right )\,1{}\mathrm {i}}{8\,\left (-a^3\,c^9\,d^3+3\,a^2\,b\,c^{10}\,d^2-3\,a\,b^2\,c^{11}\,d+b^3\,c^{12}\right )} \]
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