Integrand size = 31, antiderivative size = 166 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {3 d^2 \left (4 b c^2+5 a d^2\right )}{8 c^6 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {3 d^2 \left (4 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{8 c^7} \]
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Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {465, 105, 12, 106, 21, 94, 211} \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {3 d^2 \left (5 a d^2+4 b c^2\right ) \arctan \left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{8 c^7}-\frac {3 d^2 \left (5 a d^2+4 b c^2\right )}{8 c^6 \sqrt {d x-c} \sqrt {c+d x}}+\frac {5 a d^2+4 b c^2}{8 c^4 x^2 \sqrt {d x-c} \sqrt {c+d x}}+\frac {a}{4 c^2 x^4 \sqrt {d x-c} \sqrt {c+d x}} \]
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Rule 12
Rule 21
Rule 94
Rule 105
Rule 106
Rule 211
Rule 465
Rubi steps \begin{align*} \text {integral}& = \frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {1}{4} \left (4 b+\frac {5 a d^2}{c^2}\right ) \int \frac {1}{x^3 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx \\ & = \frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {\left (4 b c^2+5 a d^2\right ) \int \frac {3 d^2}{x (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx}{8 c^4} \\ & = \frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {\left (3 d^2 \left (4 b c^2+5 a d^2\right )\right ) \int \frac {1}{x (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx}{8 c^4} \\ & = -\frac {3 d^2 \left (4 b c^2+5 a d^2\right )}{8 c^6 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 d \left (4 b c^2+5 a d^2\right )\right ) \int \frac {c d+d^2 x}{x \sqrt {-c+d x} (c+d x)^{3/2}} \, dx}{8 c^6} \\ & = -\frac {3 d^2 \left (4 b c^2+5 a d^2\right )}{8 c^6 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 d^2 \left (4 b c^2+5 a d^2\right )\right ) \int \frac {1}{x \sqrt {-c+d x} \sqrt {c+d x}} \, dx}{8 c^6} \\ & = -\frac {3 d^2 \left (4 b c^2+5 a d^2\right )}{8 c^6 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 d^3 \left (4 b c^2+5 a d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2 d+d x^2} \, dx,x,\sqrt {-c+d x} \sqrt {c+d x}\right )}{8 c^6} \\ & = -\frac {3 d^2 \left (4 b c^2+5 a d^2\right )}{8 c^6 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {a}{4 c^2 x^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {4 b c^2+5 a d^2}{8 c^4 x^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {3 d^2 \left (4 b c^2+5 a d^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{8 c^7} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {\frac {4 b c^3 x^2 \left (c^2-3 d^2 x^2\right )+a \left (2 c^5+5 c^3 d^2 x^2-15 c d^4 x^4\right )}{x^4 \sqrt {-c+d x} \sqrt {c+d x}}+6 d^2 \left (4 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {c+d x}}{\sqrt {-c+d x}}\right )}{8 c^7} \]
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Time = 4.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.61
method | result | size |
risch | \(\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (7 a \,d^{2} x^{2}+4 b \,c^{2} x^{2}+2 c^{2} a \right )}{8 c^{6} x^{4} \sqrt {d x -c}}-\frac {d^{2} \left (-\frac {\left (15 a \,d^{2}+12 b \,c^{2}\right ) \ln \left (\frac {-2 c^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}}{x}\right )}{\sqrt {-c^{2}}}+\frac {4 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d^{2} \left (x -\frac {c}{d}\right )^{2}+2 c d \left (x -\frac {c}{d}\right )}}{d c \left (x -\frac {c}{d}\right )}-\frac {4 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d^{2} \left (x +\frac {c}{d}\right )^{2}-2 c d \left (x +\frac {c}{d}\right )}}{d c \left (x +\frac {c}{d}\right )}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{8 c^{6} \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(267\) |
default | \(\frac {\sqrt {d x -c}\, \left (-15 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,d^{6} x^{6}-12 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) b \,c^{2} d^{4} x^{6}+15 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,c^{2} d^{4} x^{4}+12 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) b \,c^{4} d^{2} x^{4}+15 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, a \,d^{4} x^{4}+12 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, b \,c^{2} d^{2} x^{4}-5 a \,c^{2} d^{2} x^{2} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}-4 b \,c^{4} x^{2} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}-2 a \,c^{4} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{8 c^{6} \left (-d x +c \right ) \sqrt {-c^{2}}\, x^{4} \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {d x +c}}\) | \(395\) |
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Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {{\left (2 \, a c^{5} - 3 \, {\left (4 \, b c^{3} d^{2} + 5 \, a c d^{4}\right )} x^{4} + {\left (4 \, b c^{5} + 5 \, a c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c} - 6 \, {\left ({\left (4 \, b c^{2} d^{4} + 5 \, a d^{6}\right )} x^{6} - {\left (4 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{4}\right )} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right )}{8 \, {\left (c^{7} d^{2} x^{6} - c^{9} x^{4}\right )}} \]
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Timed out. \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 \, b d^{2} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{2 \, c^{5}} + \frac {15 \, a d^{4} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{8 \, c^{7}} - \frac {3 \, b d^{2}}{2 \, \sqrt {d^{2} x^{2} - c^{2}} c^{4}} - \frac {15 \, a d^{4}}{8 \, \sqrt {d^{2} x^{2} - c^{2}} c^{6}} + \frac {b}{2 \, \sqrt {d^{2} x^{2} - c^{2}} c^{2} x^{2}} + \frac {5 \, a d^{2}}{8 \, \sqrt {d^{2} x^{2} - c^{2}} c^{4} x^{2}} + \frac {a}{4 \, \sqrt {d^{2} x^{2} - c^{2}} c^{2} x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (142) = 284\).
Time = 0.41 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.42 \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 \, {\left (4 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right )}{4 \, c^{7}} - \frac {{\left (b c^{2} d^{2} + a d^{4}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{7}} + \frac {2 \, {\left (b c^{2} d^{2} + a d^{4}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} c^{6}} + \frac {4 \, b c^{2} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 7 \, a d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 16 \, b c^{4} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} + 60 \, a c^{2} d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} - 64 \, b c^{6} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 240 \, a c^{4} d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 256 \, b c^{8} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} - 448 \, a c^{6} d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, {\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{6}} \]
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Timed out. \[ \int \frac {a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {b\,x^2+a}{x^5\,{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \]
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