Integrand size = 13, antiderivative size = 91 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d} \]
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Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 205, 211} \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d}+\frac {3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac {c+d x}{4 a d \left (a+b (c+d x)^2\right )^2} \]
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Rule 205
Rule 211
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,c+d x\right )}{4 a d} \\ & = \frac {c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,c+d x\right )}{8 a^2 d} \\ & = \frac {c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {\frac {\sqrt {a} (c+d x) \left (5 a+3 b (c+d x)^2\right )}{\left (a+b (c+d x)^2\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {b}}}{8 a^{5/2} d} \]
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Time = 0.96 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.53
method | result | size |
default | \(\frac {2 b \,d^{2} x +2 b c d}{8 a b \,d^{2} \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{2}}+\frac {\frac {3 \left (2 b \,d^{2} x +2 b c d \right )}{16 a b \,d^{2} \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )}+\frac {3 \arctan \left (\frac {2 b \,d^{2} x +2 b c d}{2 d \sqrt {a b}}\right )}{8 d a \sqrt {a b}}}{a}\) | \(139\) |
risch | \(\frac {\frac {3 b \,d^{2} x^{3}}{8 a^{2}}+\frac {9 c \,x^{2} b d}{8 a^{2}}+\frac {\left (9 b \,c^{2}+5 a \right ) x}{8 a^{2}}+\frac {c \left (3 b \,c^{2}+5 a \right )}{8 d \,a^{2}}}{\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{2}}-\frac {3 \ln \left (b d x +b c +\sqrt {-a b}\right )}{16 \sqrt {-a b}\, d \,a^{2}}+\frac {3 \ln \left (-b d x -b c +\sqrt {-a b}\right )}{16 \sqrt {-a b}\, d \,a^{2}}\) | \(145\) |
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Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (77) = 154\).
Time = 0.28 (sec) , antiderivative size = 595, normalized size of antiderivative = 6.54 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\left [\frac {6 \, a b^{2} d^{3} x^{3} + 18 \, a b^{2} c d^{2} x^{2} + 6 \, a b^{2} c^{3} + 10 \, a^{2} b c + 2 \, {\left (9 \, a b^{2} c^{2} + 5 \, a^{2} b\right )} d x - 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + b^{2} c^{4} + 2 \, {\left (3 \, b^{2} c^{2} + a b\right )} d^{2} x^{2} + 2 \, a b c^{2} + 4 \, {\left (b^{2} c^{3} + a b c\right )} d x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b d^{2} x^{2} + 2 \, b c d x + b c^{2} - 2 \, \sqrt {-a b} {\left (d x + c\right )} - a}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}\right )}{16 \, {\left (a^{3} b^{3} d^{5} x^{4} + 4 \, a^{3} b^{3} c d^{4} x^{3} + 2 \, {\left (3 \, a^{3} b^{3} c^{2} + a^{4} b^{2}\right )} d^{3} x^{2} + 4 \, {\left (a^{3} b^{3} c^{3} + a^{4} b^{2} c\right )} d^{2} x + {\left (a^{3} b^{3} c^{4} + 2 \, a^{4} b^{2} c^{2} + a^{5} b\right )} d\right )}}, \frac {3 \, a b^{2} d^{3} x^{3} + 9 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{3} + 5 \, a^{2} b c + {\left (9 \, a b^{2} c^{2} + 5 \, a^{2} b\right )} d x + 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + b^{2} c^{4} + 2 \, {\left (3 \, b^{2} c^{2} + a b\right )} d^{2} x^{2} + 2 \, a b c^{2} + 4 \, {\left (b^{2} c^{3} + a b c\right )} d x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (d x + c\right )}}{a}\right )}{8 \, {\left (a^{3} b^{3} d^{5} x^{4} + 4 \, a^{3} b^{3} c d^{4} x^{3} + 2 \, {\left (3 \, a^{3} b^{3} c^{2} + a^{4} b^{2}\right )} d^{3} x^{2} + 4 \, {\left (a^{3} b^{3} c^{3} + a^{4} b^{2} c\right )} d^{2} x + {\left (a^{3} b^{3} c^{4} + 2 \, a^{4} b^{2} c^{2} + a^{5} b\right )} d\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (82) = 164\).
Time = 0.62 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.82 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {5 a c + 3 b c^{3} + 9 b c d^{2} x^{2} + 3 b d^{3} x^{3} + x \left (5 a d + 9 b c^{2} d\right )}{8 a^{4} d + 16 a^{3} b c^{2} d + 8 a^{2} b^{2} c^{4} d + 32 a^{2} b^{2} c d^{4} x^{3} + 8 a^{2} b^{2} d^{5} x^{4} + x^{2} \cdot \left (16 a^{3} b d^{3} + 48 a^{2} b^{2} c^{2} d^{3}\right ) + x \left (32 a^{3} b c d^{2} + 32 a^{2} b^{2} c^{3} d^{2}\right )} + \frac {- \frac {3 \sqrt {- \frac {1}{a^{5} b}} \log {\left (x + \frac {- 3 a^{3} \sqrt {- \frac {1}{a^{5} b}} + 3 c}{3 d} \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} b}} \log {\left (x + \frac {3 a^{3} \sqrt {- \frac {1}{a^{5} b}} + 3 c}{3 d} \right )}}{16}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (77) = 154\).
Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {3 \, b d^{3} x^{3} + 9 \, b c d^{2} x^{2} + 3 \, b c^{3} + {\left (9 \, b c^{2} + 5 \, a\right )} d x + 5 \, a c}{8 \, {\left (a^{2} b^{2} d^{5} x^{4} + 4 \, a^{2} b^{2} c d^{4} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{2} + a^{3} b\right )} d^{3} x^{2} + 4 \, {\left (a^{2} b^{2} c^{3} + a^{3} b c\right )} d^{2} x + {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} + a^{4}\right )} d\right )}} + \frac {3 \, \arctan \left (\frac {b d^{2} x + b c d}{\sqrt {a b} d}\right )}{8 \, \sqrt {a b} a^{2} d} \]
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Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {3 \, \arctan \left (\frac {b d x + b c}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} d} + \frac {3 \, b d^{3} x^{3} + 9 \, b c d^{2} x^{2} + 9 \, b c^{2} d x + 3 \, b c^{3} + 5 \, a d x + 5 \, a c}{8 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{2} a^{2} d} \]
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Time = 10.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.99 \[ \int \frac {1}{\left (a+b (c+d x)^2\right )^3} \, dx=\frac {\frac {x\,\left (9\,b\,c^2+5\,a\right )}{8\,a^2}+\frac {3\,b\,c^3+5\,a\,c}{8\,a^2\,d}+\frac {3\,b\,d^2\,x^3}{8\,a^2}+\frac {9\,b\,c\,d\,x^2}{8\,a^2}}{x^2\,\left (6\,b^2\,c^2\,d^2+2\,a\,b\,d^2\right )+x\,\left (4\,d\,b^2\,c^3+4\,a\,d\,b\,c\right )+a^2+b^2\,c^4+b^2\,d^4\,x^4+2\,a\,b\,c^2+4\,b^2\,c\,d^3\,x^3}+\frac {3\,\mathrm {atan}\left (\frac {8\,a^2\,\left (\frac {3\,\sqrt {b}\,c}{8\,a^{5/2}}+\frac {3\,\sqrt {b}\,d\,x}{8\,a^{5/2}}\right )}{3}\right )}{8\,a^{5/2}\,\sqrt {b}\,d} \]
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