Integrand size = 28, antiderivative size = 549 \[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {c \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (\sqrt {c} f-\sqrt {-a} g\right )^{3/2}}-\frac {c \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right )^{3/2} \left (\sqrt {c} f+\sqrt {-a} g\right )^{3/2}} \]
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Time = 0.96 (sec) , antiderivative size = 543, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {926, 106, 157, 12, 95, 214} \[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {c \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (\sqrt {c} f-\sqrt {-a} g\right )^{3/2}}-\frac {c \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} g+\sqrt {c} f\right )^{3/2}}-\frac {e}{\sqrt {-a} \sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {e}{\sqrt {-a} \sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) (e f-d g)}+\frac {g \sqrt {d+e x} \left (2 a e g-\sqrt {-a} \sqrt {c} (d g+e f)\right )}{a \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (\sqrt {-a} g+\sqrt {c} f\right ) (e f-d g)^2}+\frac {g \sqrt {d+e x} \left (\sqrt {-a} \sqrt {c} (d g+e f)+2 a e g\right )}{a \sqrt {f+g x} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2} \]
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Rule 12
Rule 95
Rule 106
Rule 157
Rule 214
Rule 926
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}}+\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}}\right ) \, dx \\ & = -\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}} \, dx}{2 \sqrt {-a}} \\ & = -\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}-\frac {\int \frac {\frac {1}{2} \left (2 \sqrt {-a} e g+\sqrt {c} (e f-d g)\right )+\sqrt {c} e g x}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}} \, dx}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {\int \frac {\frac {1}{2} \left (2 \sqrt {-a} e g-\sqrt {c} (e f-d g)\right )-\sqrt {c} e g x}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}} \, dx}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g)} \\ & = -\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {2 \int -\frac {c (e f-d g)^2}{4 \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2}+\frac {2 \int -\frac {c (e f-d g)^2}{4 \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2} \\ & = -\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}-\frac {c \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right )}-\frac {c \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \left (\sqrt {-a} c d f+(-a)^{3/2} e g+a \sqrt {c} (e f+d g)\right )} \\ & = -\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}-\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {c} d+\sqrt {-a} e-\left (\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right )}-\frac {c \text {Subst}\left (\int \frac {1}{-\sqrt {c} d+\sqrt {-a} e-\left (-\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} c d f+(-a)^{3/2} e g+a \sqrt {c} (e f+d g)} \\ & = -\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (\sqrt {-a} c d f+(-a)^{3/2} e g+a \sqrt {c} (e f+d g)\right )}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right )^{3/2} \left (\sqrt {c} f+\sqrt {-a} g\right )^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.81 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 \left (c \left (d^3 g^3+d^2 e g^3 x+e^3 f^2 (f+g x)\right )+a e^2 g^2 (d g+e (f+2 g x))\right )}{\left (c d^2+a e^2\right ) (e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {d+e x} \sqrt {f+g x}}-\frac {i c \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {c d^2+a e^2} \left (\sqrt {c} f-i \sqrt {a} g\right )^2}+\frac {i c \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {c d^2+a e^2} \left (\sqrt {c} f+i \sqrt {a} g\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(30647\) vs. \(2(433)=866\).
Time = 0.46 (sec) , antiderivative size = 30648, normalized size of antiderivative = 55.83
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Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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