Integrand size = 46, antiderivative size = 170 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {\left (2 a e^2 g-c d (e f+d g)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{3/2} (c d f-a e g)^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {892, 888, 211} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} (c d f-a e g)^{3/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)} \]
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Rule 211
Rule 888
Rule 892
Rubi steps \begin{align*} \text {integral}& = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {3}{2} c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\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}{g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )} \\ & = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {\left (e^2 \left (2 a e^2 g-c d (e f+d g)\right )\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 )}{g (c d f-a e g)} \\ & = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {\left (2 a e^2 g-c d (e f+d g)\right ) \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 )}{g^{3/2} (c d f-a e g)^{3/2}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {\sqrt {g} (-e f+d g) (a e+c d x)}{(-c d f+a e g) (f+g x)}+\frac {\left (-2 a e^2 g+c d (e f+d g)\right ) \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{3/2}}\right )}{g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(336\) vs. \(2(154)=308\).
Time = 0.56 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {\left (-2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,e^{2} g^{2} x +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c \,d^{2} g^{2} x +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d e f g x -2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,e^{2} f g +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c \,d^{2} f g +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d e \,f^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, d g +\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, e f \right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, g \left (a e g -c d f \right ) \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}}\) | \(337\) |
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (154) = 308\).
Time = 0.32 (sec) , antiderivative size = 896, normalized size of antiderivative = 5.27 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {{\left (c d^{2} e f^{2} + {\left (c d^{3} - 2 \, a d e^{2}\right )} f g + {\left (c d e^{2} f g + {\left (c d^{2} e - 2 \, a e^{3}\right )} g^{2}\right )} x^{2} + {\left (c d e^{2} f^{2} + 2 \, {\left (c d^{2} e - a e^{3}\right )} f g + {\left (c d^{3} - 2 \, a d e^{2}\right )} g^{2}\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 d e f^{2} g + a d e g^{3} - {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{2 \, {\left (c^{2} d^{3} f^{3} g^{2} - 2 \, a c d^{2} e f^{2} g^{3} + a^{2} d e^{2} f g^{4} + {\left (c^{2} d^{2} e f^{2} g^{3} - 2 \, a c d e^{2} f g^{4} + a^{2} e^{3} g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{4}\right )} x\right )}}, -\frac {{\left (c d^{2} e f^{2} + {\left (c d^{3} - 2 \, a d e^{2}\right )} f g + {\left (c d e^{2} f g + {\left (c d^{2} e - 2 \, a e^{3}\right )} g^{2}\right )} x^{2} + {\left (c d e^{2} f^{2} + 2 \, {\left (c d^{2} e - a e^{3}\right )} f g + {\left (c d^{3} - 2 \, a d e^{2}\right )} g^{2}\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 d e f^{2} g + a d e g^{3} - {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c^{2} d^{3} f^{3} g^{2} - 2 \, a c d^{2} e f^{2} g^{3} + a^{2} d e^{2} f g^{4} + {\left (c^{2} d^{2} e f^{2} g^{3} - 2 \, a c d e^{2} f g^{4} + a^{2} e^{3} g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{4}\right )} x}\right ] \]
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\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{2}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (154) = 308\).
Time = 0.41 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.76 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {c d e^{2} f \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + c d^{2} e g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 2 \, a e^{3} g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} e}{\sqrt {c d f g - a e g^{2}} c d f g {\left | e \right |} - \sqrt {c d f g - a e g^{2}} a e g^{2} {\left | e \right |}} + \frac {e {\left (\frac {{\left (c^{2} d^{2} e^{2} f + c^{2} d^{3} e g - 2 \, a c d e^{3} g\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c d f g {\left | e \right |} - a e g^{2} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2} e^{2} f - \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{3} e g}{{\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )} {\left (c d f g {\left | e \right |} - a e g^{2} {\left | e \right |}\right )}}\right )}}{c d} \]
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Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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