Integrand size = 39, antiderivative size = 257 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {680, 686, 674, 211} \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {5 c d e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 e}{3 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rule 211
Rule 674
Rule 680
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(5 e) \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 \left (c d^2-a e^2\right )} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(5 c d e) \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )^2} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 c d e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 \left (c d^2-a e^2\right )^3} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 c d e^3\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 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 )}{\left (c d^2-a e^2\right )^3} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (\sqrt {c d^2-a e^2} \left (3 a^2 e^4+2 a c d e^2 (7 d+10 e x)+c^2 d^2 \left (-2 d^2+10 d e x+15 e^2 x^2\right )\right )+15 c d e^{3/2} (a e+c d x)^{3/2} (d+e x) \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 \left (c d^2-a e^2\right )^{7/2} ((a e+c d x) (d+e x))^{3/2}} \]
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Time = 2.78 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.65
method | result | size |
default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} e^{3} x^{2}+15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c d \,e^{4} x \sqrt {c d x +a e}+15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x +15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c \,d^{2} e^{3} \sqrt {c d x +a e}-15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}-20 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c d \,e^{3} x -10 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{3} e x -3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}-14 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}+2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(423\) |
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Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (229) = 458\).
Time = 0.40 (sec) , antiderivative size = 1236, normalized size of antiderivative = 4.81 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\left (c^{3} d^{3} e^{3} x^{4} + a^{2} c d^{3} e^{3} + 2 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{3} + {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} + 2 \, {\left (a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4}\right )} x\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 14 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{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}}{6 \, {\left (a^{2} c^{3} d^{8} e^{2} - 3 \, a^{3} c^{2} d^{6} e^{4} + 3 \, a^{4} c d^{4} e^{6} - a^{5} d^{2} e^{8} + {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{4} + 2 \, {\left (c^{5} d^{9} e - 2 \, a c^{4} d^{7} e^{3} + 2 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x^{3} + {\left (c^{5} d^{10} + a c^{4} d^{8} e^{2} - 8 \, a^{2} c^{3} d^{6} e^{4} + 8 \, a^{3} c^{2} d^{4} e^{6} - a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} x^{2} + 2 \, {\left (a c^{4} d^{9} e - 2 \, a^{2} c^{3} d^{7} e^{3} + 2 \, a^{4} c d^{3} e^{7} - a^{5} d e^{9}\right )} x\right )}}, \frac {15 \, {\left (c^{3} d^{3} e^{3} x^{4} + a^{2} c d^{3} e^{3} + 2 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{3} + {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} + 2 \, {\left (a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4}\right )} x\right )} \sqrt {\frac {e}{c d^{2} - a e^{2}}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {e}{c d^{2} - a e^{2}}}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + {\left (15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 14 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{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}}{3 \, {\left (a^{2} c^{3} d^{8} e^{2} - 3 \, a^{3} c^{2} d^{6} e^{4} + 3 \, a^{4} c d^{4} e^{6} - a^{5} d^{2} e^{8} + {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{4} + 2 \, {\left (c^{5} d^{9} e - 2 \, a c^{4} d^{7} e^{3} + 2 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x^{3} + {\left (c^{5} d^{10} + a c^{4} d^{8} e^{2} - 8 \, a^{2} c^{3} d^{6} e^{4} + 8 \, a^{3} c^{2} d^{4} e^{6} - a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} x^{2} + 2 \, {\left (a c^{4} d^{9} e - 2 \, a^{2} c^{3} d^{7} e^{3} + 2 \, a^{4} c d^{3} e^{7} - a^{5} d e^{9}\right )} x\right )}}\right ] \]
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\[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d + e x}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.43 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {15 \, c d e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4} - 6 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d e\right )}}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} {\left (e x + d\right )}}\right )} e^{2} \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]
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