Integrand size = 48, antiderivative size = 124 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}} \]
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Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {882, 874} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {4 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
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Rule 874
Rule 882
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(2 g) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d f-a e g} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (a e g+c d (f+2 g x))}{(c d f-a e g)^2 \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \]
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Time = 0.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (2 c d g x +a e g +c d f \right )}{\sqrt {e x +d}\, \sqrt {g x +f}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{2}}\) | \(70\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (2 c d g x +a e g +c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{\sqrt {g x +f}\, \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(97\) |
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (112) = 224\).
Time = 0.35 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.62 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{a c^{2} d^{3} e f^{3} - 2 \, a^{2} c d^{2} e^{2} f^{2} g + a^{3} d e^{3} f g^{2} + {\left (c^{3} d^{3} e f^{2} g - 2 \, a c^{2} d^{2} e^{2} f g^{2} + a^{2} c d e^{3} g^{3}\right )} x^{3} + {\left (c^{3} d^{3} e f^{3} + {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{2} g - {\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g^{2} + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{3}\right )} x^{2} + {\left (a^{3} d e^{3} g^{3} + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{3} - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f^{2} g - {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{2}\right )} x} \]
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\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (112) = 224\).
Time = 0.38 (sec) , antiderivative size = 793, normalized size of antiderivative = 6.40 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-2 \, {\left (\frac {\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} c d g^{2}}{{\left (c^{2} d^{2} e^{2} f^{2} {\left | g \right |} - 2 \, a c d e^{3} f g {\left | g \right |} + a^{2} e^{4} g^{2} {\left | g \right |}\right )} \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} + \frac {2 \, \sqrt {c d g} g^{2}}{{\left (c d e^{2} f g - a e^{3} g^{2} + {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )} {\left (c d f {\left | g \right |} - a e g {\left | g \right |}\right )}}\right )} e^{2} + \frac {2 \, {\left (\sqrt {e^{2} f - d e g} c^{2} d^{2} e f g^{2} - \sqrt {e^{2} f - d e g} c^{2} d^{3} g^{3} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} c d^{2} g^{2} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} a e^{2} g^{2}\right )}}{\sqrt {e^{2} f - d e g} \sqrt {c d g} c^{3} d^{4} f^{2} g {\left | g \right |} - \sqrt {e^{2} f - d e g} \sqrt {c d g} a c^{2} d^{2} e^{2} f^{2} g {\left | g \right |} - 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a c^{2} d^{3} e f g^{2} {\left | g \right |} + 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} c d e^{3} f g^{2} {\left | g \right |} + \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} c d^{2} e^{2} g^{3} {\left | g \right |} - \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{3} e^{4} g^{3} {\left | g \right |} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{3} d^{3} e f^{3} {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{3} d^{4} f^{2} g {\left | g \right |} - 2 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c^{2} d^{2} e^{2} f^{2} g {\left | g \right |} + 2 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c^{2} d^{3} e f g^{2} {\left | g \right |} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} c d e^{3} f g^{2} {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} c d^{2} e^{2} g^{3} {\left | g \right |}} \]
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Time = 13.59 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\left (\frac {4\,g\,x\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {\left (2\,a\,e\,g+2\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}+\frac {a\,\sqrt {f+g\,x}}{c}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )}{c\,d\,e}} \]
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