Integrand size = 46, antiderivative size = 207 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {c^2 d^2 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\frac {c^2 d^2 \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2} \]
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Rule 211
Rule 876
Rule 886
Rule 888
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {(c d) \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 g} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g (c d f-a e g)} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (c^2 d^2 e^2\right ) \text {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{4 g (c d f-a e g)} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {c d f-a e g} \sqrt {a e+c d x} (2 a e g+c d (-f+g x))+c^2 d^2 (f+g x)^2 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 g^{3/2} (c d f-a e g)^{3/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \]
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Time = 0.54 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} g^{2} x^{2}+2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f g x +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d g x -2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a e g +\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right ) g \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\) | \(275\) |
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Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (181) = 362\).
Time = 0.40 (sec) , antiderivative size = 1056, normalized size of antiderivative = 5.10 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\left [\frac {{\left (c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + {\left (2 \, c^{2} d^{2} e f g + c^{2} d^{3} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, c^{2} d^{3} f g\right )} x\right )} \sqrt {-c d f g + a e g^{2}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d f g + a e g^{2}} \sqrt {e x + d}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, {\left (c^{2} d^{2} f^{2} g - 3 \, a c d e f g^{2} + 2 \, a^{2} e^{2} g^{3} - {\left (c^{2} d^{2} f g^{2} - a c d e g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{8 \, {\left (c^{2} d^{3} f^{4} g^{2} - 2 \, a c d^{2} e f^{3} g^{3} + a^{2} d e^{2} f^{2} g^{4} + {\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{3} + {\left (2 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g^{3} - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{4}\right )} x\right )}}, -\frac {{\left (c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + {\left (2 \, c^{2} d^{2} e f g + c^{2} d^{3} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, c^{2} d^{3} f g\right )} x\right )} \sqrt {c d f g - a e g^{2}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d f g - a e g^{2}} \sqrt {e x + d}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + {\left (c^{2} d^{2} f^{2} g - 3 \, a c d e f g^{2} + 2 \, a^{2} e^{2} g^{3} - {\left (c^{2} d^{2} f g^{2} - a c d e g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{4 \, {\left (c^{2} d^{3} f^{4} g^{2} - 2 \, a c d^{2} e f^{3} g^{3} + a^{2} d e^{2} f^{2} g^{4} + {\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{3} + {\left (2 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g^{3} - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{4}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (181) = 362\).
Time = 0.49 (sec) , antiderivative size = 662, normalized size of antiderivative = 3.20 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\frac {{\left (\frac {c^{2} d^{2} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c d f g - a e g^{2}\right )}^{\frac {3}{2}}} - \frac {c^{2} d^{2} e^{3} f^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 2 \, c^{2} d^{3} e^{2} f g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + c^{2} d^{4} e g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d e^{2} f - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d^{2} e g + 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a e^{3} g}{\sqrt {c d f g - a e g^{2}} c d e^{2} f^{3} g - 2 \, \sqrt {c d f g - a e g^{2}} c d^{2} e f^{2} g^{2} - \sqrt {c d f g - a e g^{2}} a e^{3} f^{2} g^{2} + \sqrt {c d f g - a e g^{2}} c d^{3} f g^{3} + 2 \, \sqrt {c d f g - a e g^{2}} a d e^{2} f g^{3} - \sqrt {c d f g - a e g^{2}} a d^{2} e g^{4}} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3} e^{4} f - \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{5} g - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} g}{{\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )}^{2} {\left (c d f g - a e g^{2}\right )}}\right )} {\left | e \right |}}{4 \, e^{2}} \]
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Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (f+g\,x\right )}^3\,\sqrt {d+e\,x}} \,d x \]
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