Integrand size = 39, antiderivative size = 315 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=-\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {3 c^4 d^4 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}} \]
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Time = 0.14 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 686, 674, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {3 c^4 d^4 \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]
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Rule 211
Rule 674
Rule 676
Rule 686
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}}{4 e (d+e x)^{11/2}}+\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)^{9/2}} \, dx}{8 e} \\ & = -\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 e^2} \\ & = -\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {\left (3 c^3 d^3\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{64 e^2 \left (c d^2-a e^2\right )} \\ & = -\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^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}}{4 e (d+e x)^{11/2}}+\frac {\left (3 c^4 d^4\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}{128 e^2 \left (c d^2-a e^2\right )^2} \\ & = -\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^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}}{4 e (d+e x)^{11/2}}+\frac {\left (3 c^4 d^4\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 )}{64 e \left (c d^2-a e^2\right )^2} \\ & = -\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^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}}{4 e (d+e x)^{11/2}}+\frac {3 c^4 d^4 \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 )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {c^4 d^4 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {e} \left (16 a^3 e^6-24 a^2 c d e^4 (d-e x)+2 a c^2 d^2 e^2 \left (d^2-22 d e x+e^2 x^2\right )+c^3 d^3 \left (3 d^3+11 d^2 e x-11 d e^2 x^2-3 e^3 x^3\right )\right )}{c^4 d^4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^4}+\frac {3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2} (a e+c d x)^{3/2}}\right )}{64 e^{5/2} (d+e x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(651\) vs. \(2(277)=554\).
Time = 2.81 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{4} x^{4}+12 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{3} x^{3}+18 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{2} x^{2}+12 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e x -3 c^{3} d^{3} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{8}+2 a \,c^{2} d^{2} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-11 c^{3} d^{4} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+24 a^{2} c d \,e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-44 a \,c^{2} d^{3} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+11 c^{3} d^{5} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+16 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} e^{6}-24 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}+2 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{64 \left (e x +d \right )^{\frac {9}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, e^{2} \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\) | \(652\) |
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Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (277) = 554\).
Time = 0.37 (sec) , antiderivative size = 1387, normalized size of antiderivative = 4.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {13}{2}}} \,d x } \]
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Time = 0.51 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {\frac {3 \, c^{5} d^{5} e {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{8} d^{11} e^{4} {\left | e \right |} - 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{7} d^{9} e^{6} {\left | e \right |} + 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{6} d^{7} e^{8} {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{5} d^{5} e^{10} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{7} d^{9} e^{3} {\left | e \right |} - 22 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{6} d^{7} e^{5} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{5} d^{5} e^{7} {\left | e \right |} - 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{6} d^{7} e^{2} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{4} {\left | e \right |} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{5} d^{5} e {\left | e \right |}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{4} c^{4} d^{4} e^{4}}}{64 \, c d e^{4}} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \]
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