Integrand size = 48, antiderivative size = 129 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{5/2}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 129, 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 = {886, 874} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)} \]
[In]
[Out]
Rule 874
Rule 886
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}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {(2 c d) \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}{7 (c d f-a e g)} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{5/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} (-5 a e g+c d (7 f+2 g x))}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +5 a e g -7 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}\) | \(99\) |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-2 g \,x^{2} c^{2} d^{2}+3 a c d e g x -7 c^{2} d^{2} f x +5 a^{2} e^{2} g -7 a c d e f \right ) \left (c d x +a e \right )}{35 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )^{2}}\) | \(100\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (113) = 226\).
Time = 0.50 (sec) , antiderivative size = 526, normalized size of antiderivative = 4.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {2 \, {\left (2 \, c^{3} d^{3} g x^{3} + 7 \, a^{2} c d e^{2} f - 5 \, a^{3} e^{3} g + {\left (7 \, c^{3} d^{3} f - a c^{2} d^{2} e g\right )} x^{2} + 2 \, {\left (7 \, a c^{2} d^{2} e f - 4 \, a^{2} c d e^{2} g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{35 \, {\left (c^{2} d^{3} f^{6} - 2 \, a c d^{2} e f^{5} g + a^{2} d e^{2} f^{4} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{5} + {\left (4 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} + {\left (c^{2} d^{3} - 8 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \, {\left (a c d^{2} e - 2 \, a^{2} e^{3}\right )} f g^{5}\right )} x^{4} + 2 \, {\left (3 \, c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \, {\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} f^{3} g^{3} - {\left (4 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} f^{2} g^{4}\right )} x^{3} + 2 \, {\left (2 \, c^{2} d^{2} e f^{5} g + 3 \, a^{2} d e^{2} f^{2} g^{4} + {\left (3 \, c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{4} g^{2} - 2 \, {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{6} + 4 \, a^{2} d e^{2} f^{3} g^{3} + 2 \, {\left (2 \, c^{2} d^{3} - a c d e^{2}\right )} f^{5} g - {\left (8 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{2}\right )} x\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (113) = 226\).
Time = 0.74 (sec) , antiderivative size = 1001, normalized size of antiderivative = 7.76 \[ \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)^{9/2}} \, dx=-\frac {2 \, {\left (7 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{5} e f {\left | c \right |} {\left | d \right |} - 14 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} e^{3} f {\left | c \right |} {\left | d \right |} + 7 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d e^{5} f {\left | c \right |} {\left | d \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} g {\left | c \right |} {\left | d \right |} - \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} g {\left | c \right |} {\left | d \right |} + 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} g {\left | c \right |} {\left | d \right |} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6} g {\left | c \right |} {\left | d \right |}\right )}}{35 \, {\left (\sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{2} e^{3} f^{5} - 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{3} e^{2} f^{4} g - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d e^{4} f^{4} g + 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{4} e f^{3} g^{2} + 6 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{2} e^{3} f^{3} g^{2} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} e^{5} f^{3} g^{2} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{5} f^{2} g^{3} - 6 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{3} e^{2} f^{2} g^{3} - 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d e^{4} f^{2} g^{3} + 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{4} e f g^{4} + 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d^{2} e^{3} f g^{4} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d^{3} e^{2} g^{5}\right )}} + \frac {2 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} {\left (\frac {2 \, {\left (c^{7} d^{7} e^{6} f g^{4} {\left | c \right |} {\left | d \right |} - a c^{6} d^{6} e^{7} g^{5} {\left | c \right |} {\left | d \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}}{c^{3} d^{3} e^{6} f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{7} f^{2} g^{4} + 3 \, a^{2} c d e^{8} f g^{5} - a^{3} e^{9} g^{6}} + \frac {7 \, {\left (c^{8} d^{8} e^{8} f^{2} g^{3} {\left | c \right |} {\left | d \right |} - 2 \, a c^{7} d^{7} e^{9} f g^{4} {\left | c \right |} {\left | d \right |} + a^{2} c^{6} d^{6} e^{10} g^{5} {\left | c \right |} {\left | d \right |}\right )}}{c^{3} d^{3} e^{6} f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{7} f^{2} g^{4} + 3 \, a^{2} c d e^{8} f g^{5} - a^{3} e^{9} g^{6}}\right )}}{35 \, {\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 {7}{2}}} \]
[In]
[Out]
Time = 12.66 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^2\,e^2\,\left (5\,a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c^3\,d^3\,x^3}{35\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {4\,a\,c\,d\,e\,x\,\left (4\,a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {3\,f^2\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]
[In]
[Out]