Integrand size = 39, antiderivative size = 185 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {3 c^2 d^2 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 e^{5/2} \sqrt {c d^2-a e^2}} \]
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Time = 0.07 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {676, 674, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {3 c^2 d^2 \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{4 e^{5/2} \sqrt {c d^2-a e^2}}-\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]
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Rule 211
Rule 674
Rule 676
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {(3 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx}{4 e} \\ & = -\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {\left (3 c^2 d^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}{8 e^2} \\ & = -\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {\left (3 c^2 d^2\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 )}{4 e} \\ & = -\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {3 c^2 d^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 )}{4 e^{5/2} \sqrt {c d^2-a e^2}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {e} \left (2 a e^2+c d (3 d+5 e x)\right )+\frac {3 c^2 d^2 (d+e x)^2 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}\right )}{4 e^{5/2} (d+e x)^{5/2}} \]
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Time = 3.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.46
method | result | size |
default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{2} x^{2}+6 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{4}+5 c d e x \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}+2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,e^{2}+3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c \,d^{2}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, e^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(271\) |
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Time = 0.32 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.51 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\left [-\frac {3 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} - 2 \, a^{2} e^{5} + 5 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{8 \, {\left (c d^{5} e^{3} - a d^{3} e^{5} + {\left (c d^{2} e^{6} - a e^{8}\right )} x^{3} + 3 \, {\left (c d^{3} e^{5} - a d e^{7}\right )} x^{2} + 3 \, {\left (c d^{4} e^{4} - a d^{2} e^{6}\right )} x\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} - 2 \, a^{2} e^{5} + 5 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{4 \, {\left (c d^{5} e^{3} - a d^{3} e^{5} + {\left (c d^{2} e^{6} - a e^{8}\right )} x^{3} + 3 \, {\left (c d^{3} e^{5} - a d e^{7}\right )} x^{2} + 3 \, {\left (c d^{4} e^{4} - a d^{2} e^{6}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {\frac {3 \, c^{3} d^{3} e {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{5} e^{2} {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{4} {\left | e \right |} + 5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} e {\left | e \right |}}{{\left (e x + d\right )}^{2} c^{2} d^{2} e^{2}}}{4 \, c d e^{4}} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
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