Integrand size = 22, antiderivative size = 140 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {5 d}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {5 b d}{2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {5 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {455, 44, 53, 65, 214} \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {5 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}}-\frac {5 b d}{2 \sqrt {c+d x^2} (b c-a d)^3}-\frac {1}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {5 d}{6 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right ) \\ & = -\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {(5 d) \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )}{4 (b c-a d)} \\ & = -\frac {5 d}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {(5 b d) \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 (b c-a d)^2} \\ & = -\frac {5 d}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {5 b d}{2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\left (5 b^2 d\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 (b c-a d)^3} \\ & = -\frac {5 d}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {5 b d}{2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 (b c-a d)^3} \\ & = -\frac {5 d}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {5 b d}{2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {5 b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.98 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {2 a^2 d^2-2 a b d \left (7 c+5 d x^2\right )-b^2 \left (3 c^2+20 c d x^2+15 d^2 x^4\right )}{6 (b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {5 b^{3/2} d \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{2 (-b c+a d)^{7/2}} \]
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Time = 3.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(d \left (\frac {\sqrt {d \,x^{2}+c}\, b^{2}}{2 \left (b \,x^{2}+a \right ) d \left (a d -b c \right )^{3}}+\frac {5 \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) b^{2}}{2 \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right )^{3}}-\frac {1}{3 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a d -b c \right )^{3} \sqrt {d \,x^{2}+c}}\right )\) | \(134\) |
default | \(\text {Expression too large to display}\) | \(2101\) |
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Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (116) = 232\).
Time = 0.45 (sec) , antiderivative size = 895, normalized size of antiderivative = 6.39 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\left (b^{2} d^{3} x^{6} + a b c^{2} d + {\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} x^{4} + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (15 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} + 10 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, -\frac {15 \, {\left (b^{2} d^{3} x^{6} + a b c^{2} d + {\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} x^{4} + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, {\left (15 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} + 10 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.61 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {5 \, b^{2} d \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {\sqrt {d x^{2} + c} b^{2} d}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac {6 \, {\left (d x^{2} + c\right )} b d + b c d - a d^{2}}{3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
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Time = 5.73 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.22 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {5\,b^2\,d\,{\left (d\,x^2+c\right )}^2}{2\,{\left (a\,d-b\,c\right )}^3}-\frac {d}{3\,\left (a\,d-b\,c\right )}+\frac {5\,b\,d\,\left (d\,x^2+c\right )}{3\,{\left (a\,d-b\,c\right )}^2}}{b\,{\left (d\,x^2+c\right )}^{5/2}+{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-b\,c\right )}+\frac {5\,b^{3/2}\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^{7/2}}\right )}{2\,{\left (a\,d-b\,c\right )}^{7/2}} \]
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