Integrand size = 19, antiderivative size = 223 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {3 c e^2 \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {755, 849, 821, 739, 212} \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {3 c e^2 \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}+\frac {c d e \sqrt {a+c x^2} \left (2 c d^2-13 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (2 c d^2-3 a e^2\right )}{2 a (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )} \]
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Rule 212
Rule 739
Rule 755
Rule 821
Rule 849
Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}-\frac {\int \frac {-3 a e^2-2 c d e x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {\int \frac {10 a c d e^2+c e \left (2 c d^2-3 a e^2\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{2 a \left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (3 c e^2 \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (3 c e^2 \left (4 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {3 c e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \\ \end{align*}
Time = 10.46 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {-a^3 e^5+2 c^3 d^3 x (d+e x)^2-a^2 c e^3 \left (10 d^2+11 d e x+3 e^2 x^2\right )+a c^2 d e \left (6 d^3+6 d^2 e x-14 d e^2 x^2-13 e^3 x^3\right )}{a \left (c d^2+a e^2\right )^3 (d+e x)^2 \sqrt {a+c x^2}}+\frac {3 c e^2 \left (4 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{7/2}}+\frac {3 c e^2 \left (-4 c d^2+a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{7/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(205)=410\).
Time = 2.04 (sec) , antiderivative size = 933, normalized size of antiderivative = 4.18
method | result | size |
default | \(\frac {-\frac {e^{2}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {5 c d e \left (-\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {3 c d e \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}-\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}-\frac {3 c \,e^{2} \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}}{e^{3}}\) | \(933\) |
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Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (206) = 412\).
Time = 0.61 (sec) , antiderivative size = 1558, normalized size of antiderivative = 6.99 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (206) = 412\).
Time = 0.23 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.90 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\frac {15 \, c^{3} d^{3} x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{3} d^{6} + 3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e^{2} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{4} e^{6}\right )}} + \frac {15 \, c^{2} d^{2}}{2 \, {\left (\frac {\sqrt {c x^{2} + a} c^{3} d^{6}}{e} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} e + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{3} e^{5}\right )}} - \frac {13 \, c^{2} d x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{3} e^{4}\right )}} - \frac {5 \, c d}{2 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{4} x + 2 \, \sqrt {c x^{2} + a} a c d^{2} e^{2} x + \sqrt {c x^{2} + a} a^{2} e^{4} x + \frac {\sqrt {c x^{2} + a} c^{2} d^{5}}{e} + 2 \, \sqrt {c x^{2} + a} a c d^{3} e + \sqrt {c x^{2} + a} a^{2} d e^{3}\right )}} - \frac {3 \, c}{2 \, {\left (\frac {\sqrt {c x^{2} + a} c^{2} d^{4}}{e} + 2 \, \sqrt {c x^{2} + a} a c d^{2} e + \sqrt {c x^{2} + a} a^{2} e^{3}\right )}} - \frac {1}{2 \, {\left (\sqrt {c x^{2} + a} c d^{2} e x^{2} + \sqrt {c x^{2} + a} a e^{3} x^{2} + 2 \, \sqrt {c x^{2} + a} c d^{3} x + 2 \, \sqrt {c x^{2} + a} a d e^{2} x + \frac {\sqrt {c x^{2} + a} c d^{4}}{e} + \sqrt {c x^{2} + a} a d^{2} e\right )}} + \frac {15 \, c^{2} d^{2} \operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {7}{2}} e^{5}} - \frac {3 \, c \operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{2}} e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (206) = 412\).
Time = 0.30 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.05 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (c^{6} d^{9} - 6 \, a^{2} c^{4} d^{5} e^{4} - 8 \, a^{3} c^{3} d^{3} e^{6} - 3 \, a^{4} c^{2} d e^{8}\right )} x}{a c^{6} d^{12} + 6 \, a^{2} c^{5} d^{10} e^{2} + 15 \, a^{3} c^{4} d^{8} e^{4} + 20 \, a^{4} c^{3} d^{6} e^{6} + 15 \, a^{5} c^{2} d^{4} e^{8} + 6 \, a^{6} c d^{2} e^{10} + a^{7} e^{12}} + \frac {3 \, a c^{5} d^{8} e + 8 \, a^{2} c^{4} d^{6} e^{3} + 6 \, a^{3} c^{3} d^{4} e^{5} - a^{5} c e^{9}}{a c^{6} d^{12} + 6 \, a^{2} c^{5} d^{10} e^{2} + 15 \, a^{3} c^{4} d^{8} e^{4} + 20 \, a^{4} c^{3} d^{6} e^{6} + 15 \, a^{5} c^{2} d^{4} e^{8} + 6 \, a^{6} c d^{2} e^{10} + a^{7} e^{12}}}{\sqrt {c x^{2} + a}} + \frac {3 \, {\left (4 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d^{2} e^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c e^{5} + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d^{3} e^{2} - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d e^{4} - 22 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d^{2} e^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c e^{5} + 7 \, a^{2} c^{\frac {3}{2}} d e^{4}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{2}} \]
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Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^3} \,d x \]
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