Integrand size = 48, antiderivative size = 63 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {874} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]
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Rule 874
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \]
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Time = 0.58 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2}}{5 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (a e g -c d f \right )}\) | \(55\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a e g -c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}\) | \(63\) |
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (55) = 110\).
Time = 0.42 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{5 \, {\left (c d^{2} f^{4} - a d e f^{3} g + {\left (c d e f g^{3} - a e^{2} g^{4}\right )} x^{4} + {\left (3 \, c d e f^{2} g^{2} - a d e g^{4} + {\left (c d^{2} - 3 \, a e^{2}\right )} f g^{3}\right )} x^{3} + 3 \, {\left (c d e f^{3} g - a d e f g^{3} + {\left (c d^{2} - a e^{2}\right )} f^{2} g^{2}\right )} x^{2} + {\left (c d e f^{4} - 3 \, a d e f^{2} g^{2} + {\left (3 \, c d^{2} - a e^{2}\right )} f^{3} g\right )} x\right )}} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {7}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (55) = 110\).
Time = 0.95 (sec) , antiderivative size = 446, normalized size of antiderivative = 7.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=-\frac {2 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} {\left | c \right |} {\left | d \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} {\left | c \right |} {\left | d \right |} + \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4} {\left | c \right |} {\left | d \right |}\right )}}{5 \, {\left (\sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d e^{2} f^{3} - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d^{2} e f^{2} g - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a e^{3} f^{2} g + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d^{3} f g^{2} + 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a d e^{2} f g^{2} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a d^{2} e g^{3}\right )}} + \frac {2 \, {\left (c^{5} d^{5} e^{4} f g^{2} {\left | c \right |} {\left | d \right |} - a c^{4} d^{4} e^{5} g^{3} {\left | c \right |} {\left | d \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}}{5 \, {\left (c^{2} d^{2} e^{4} f^{2} g^{2} - 2 \, a c d e^{5} f g^{3} + a^{2} e^{6} g^{4}\right )} {\left (c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g\right )}^{\frac {5}{2}}} \]
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Time = 12.48 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=-\frac {\left (\frac {2\,a^2\,e^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {2\,c^2\,d^2\,x^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {4\,a\,c\,d\,e\,x}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (5\,c\,d\,f^3-5\,a\,e\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {x\,\sqrt {f+g\,x}\,\left (10\,a\,e\,f\,g^2-10\,c\,d\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}} \]
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