Optimal. Leaf size=452 \[ \frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {865, 846, 793,
626, 635, 212} \begin {gather*} \frac {\left (-35 a^2 e^4-10 c d e x \left (9 c d^2-5 a e^2\right )-20 a c d^2 e^2+63 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 793
Rule 846
Rule 865
Rubi steps
\begin {align*} \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx &=\int x^2 (a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\\ &=\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\int x \left (-2 a c d^2 e-\frac {1}{2} c d \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{7 c d e}\\ &=\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{48 c^2 d^2 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^4}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^4 d^4 e^5}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^4 d^4 e^5}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 1.24, size = 479, normalized size = 1.06 \begin {gather*} \frac {\left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-525 a^6 e^{12}+350 a^5 c d e^{10} (4 d+e x)-35 a^4 c^2 d^2 e^8 \left (15 d^2+26 d e x+8 e^2 x^2\right )-60 a^3 c^3 d^3 e^6 \left (10 d^3-5 d^2 e x-12 d e^2 x^2-4 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (3689 d^4-2332 d^3 e x+1824 d^2 e^2 x^2+33520 d e^3 x^3+23680 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (-1680 d^5+1099 d^4 e x-872 d^3 e^2 x^2+744 d^2 e^3 x^3+24320 d e^4 x^4+18560 e^5 x^5\right )+3 c^6 d^6 \left (315 d^6-210 d^5 e x+168 d^4 e^2 x^2-144 d^3 e^3 x^3+128 d^2 e^4 x^4+6400 d e^5 x^5+5120 e^6 x^6\right )\right )}{\left (c d^2-a e^2\right )^5 (a e+c d x) (d+e x)}-\frac {105 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{107520 c^{9/2} d^{9/2} e^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1079\) vs.
\(2(418)=836\).
time = 0.08, size = 1080, normalized size = 2.39
method | result | size |
default | \(\frac {\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{7 c d e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 c d e}}{e}-\frac {d \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}\right )}{e^{3}}\) | \(1080\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.36, size = 1235, normalized size = 2.73 \begin {gather*} \left [-\frac {{\left (105 \, {\left (9 \, c^{7} d^{14} - 35 \, a c^{6} d^{12} e^{2} + 45 \, a^{2} c^{5} d^{10} e^{4} - 15 \, a^{3} c^{4} d^{8} e^{6} - 5 \, a^{4} c^{3} d^{6} e^{8} - 9 \, a^{5} c^{2} d^{4} e^{10} + 15 \, a^{6} c d^{2} e^{12} - 5 \, a^{7} e^{14}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (630 \, c^{7} d^{12} x e^{2} - 945 \, c^{7} d^{13} e - 350 \, a^{5} c^{2} d^{2} x e^{12} + 525 \, a^{6} c d e^{13} + 280 \, {\left (a^{4} c^{3} d^{3} x^{2} - 5 \, a^{5} c^{2} d^{3}\right )} e^{11} - 10 \, {\left (24 \, a^{3} c^{4} d^{4} x^{3} - 91 \, a^{4} c^{3} d^{4} x\right )} e^{10} - 5 \, {\left (4736 \, a^{2} c^{5} d^{5} x^{4} + 144 \, a^{3} c^{4} d^{5} x^{2} - 105 \, a^{4} c^{3} d^{5}\right )} e^{9} - 20 \, {\left (1856 \, a c^{6} d^{6} x^{5} + 1676 \, a^{2} c^{5} d^{6} x^{3} + 15 \, a^{3} c^{4} d^{6} x\right )} e^{8} - 8 \, {\left (1920 \, c^{7} d^{7} x^{6} + 6080 \, a c^{6} d^{7} x^{4} + 228 \, a^{2} c^{5} d^{7} x^{2} - 75 \, a^{3} c^{4} d^{7}\right )} e^{7} - 4 \, {\left (4800 \, c^{7} d^{8} x^{5} + 372 \, a c^{6} d^{8} x^{3} - 583 \, a^{2} c^{5} d^{8} x\right )} e^{6} - {\left (384 \, c^{7} d^{9} x^{4} - 1744 \, a c^{6} d^{9} x^{2} + 3689 \, a^{2} c^{5} d^{9}\right )} e^{5} + 2 \, {\left (216 \, c^{7} d^{10} x^{3} - 1099 \, a c^{6} d^{10} x\right )} e^{4} - 168 \, {\left (3 \, c^{7} d^{11} x^{2} - 20 \, a c^{6} d^{11}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-6\right )}}{430080 \, c^{5} d^{5}}, \frac {{\left (105 \, {\left (9 \, c^{7} d^{14} - 35 \, a c^{6} d^{12} e^{2} + 45 \, a^{2} c^{5} d^{10} e^{4} - 15 \, a^{3} c^{4} d^{8} e^{6} - 5 \, a^{4} c^{3} d^{6} e^{8} - 9 \, a^{5} c^{2} d^{4} e^{10} + 15 \, a^{6} c d^{2} e^{12} - 5 \, a^{7} e^{14}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (630 \, c^{7} d^{12} x e^{2} - 945 \, c^{7} d^{13} e - 350 \, a^{5} c^{2} d^{2} x e^{12} + 525 \, a^{6} c d e^{13} + 280 \, {\left (a^{4} c^{3} d^{3} x^{2} - 5 \, a^{5} c^{2} d^{3}\right )} e^{11} - 10 \, {\left (24 \, a^{3} c^{4} d^{4} x^{3} - 91 \, a^{4} c^{3} d^{4} x\right )} e^{10} - 5 \, {\left (4736 \, a^{2} c^{5} d^{5} x^{4} + 144 \, a^{3} c^{4} d^{5} x^{2} - 105 \, a^{4} c^{3} d^{5}\right )} e^{9} - 20 \, {\left (1856 \, a c^{6} d^{6} x^{5} + 1676 \, a^{2} c^{5} d^{6} x^{3} + 15 \, a^{3} c^{4} d^{6} x\right )} e^{8} - 8 \, {\left (1920 \, c^{7} d^{7} x^{6} + 6080 \, a c^{6} d^{7} x^{4} + 228 \, a^{2} c^{5} d^{7} x^{2} - 75 \, a^{3} c^{4} d^{7}\right )} e^{7} - 4 \, {\left (4800 \, c^{7} d^{8} x^{5} + 372 \, a c^{6} d^{8} x^{3} - 583 \, a^{2} c^{5} d^{8} x\right )} e^{6} - {\left (384 \, c^{7} d^{9} x^{4} - 1744 \, a c^{6} d^{9} x^{2} + 3689 \, a^{2} c^{5} d^{9}\right )} e^{5} + 2 \, {\left (216 \, c^{7} d^{10} x^{3} - 1099 \, a c^{6} d^{10} x\right )} e^{4} - 168 \, {\left (3 \, c^{7} d^{11} x^{2} - 20 \, a c^{6} d^{11}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-6\right )}}{215040 \, c^{5} d^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 612, normalized size = 1.35 \begin {gather*} \frac {1}{107520} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, c^{2} d^{2} x e + \frac {{\left (15 \, c^{8} d^{9} e^{6} + 29 \, a c^{7} d^{7} e^{8}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (3 \, c^{8} d^{10} e^{5} + 380 \, a c^{7} d^{8} e^{7} + 185 \, a^{2} c^{6} d^{6} e^{9}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac {{\left (27 \, c^{8} d^{11} e^{4} - 93 \, a c^{7} d^{9} e^{6} - 2095 \, a^{2} c^{6} d^{7} e^{8} - 15 \, a^{3} c^{5} d^{5} e^{10}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (63 \, c^{8} d^{12} e^{3} - 218 \, a c^{7} d^{10} e^{5} + 228 \, a^{2} c^{6} d^{8} e^{7} + 90 \, a^{3} c^{5} d^{6} e^{9} - 35 \, a^{4} c^{4} d^{4} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac {{\left (315 \, c^{8} d^{13} e^{2} - 1099 \, a c^{7} d^{11} e^{4} + 1166 \, a^{2} c^{6} d^{9} e^{6} - 150 \, a^{3} c^{5} d^{7} e^{8} + 455 \, a^{4} c^{4} d^{5} e^{10} - 175 \, a^{5} c^{3} d^{3} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (945 \, c^{8} d^{14} e - 3360 \, a c^{7} d^{12} e^{3} + 3689 \, a^{2} c^{6} d^{10} e^{5} - 600 \, a^{3} c^{5} d^{8} e^{7} - 525 \, a^{4} c^{4} d^{6} e^{9} + 1400 \, a^{5} c^{3} d^{4} e^{11} - 525 \, a^{6} c^{2} d^{2} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} + \frac {{\left (9 \, c^{7} d^{14} - 35 \, a c^{6} d^{12} e^{2} + 45 \, a^{2} c^{5} d^{10} e^{4} - 15 \, a^{3} c^{4} d^{8} e^{6} - 5 \, a^{4} c^{3} d^{6} e^{8} - 9 \, a^{5} c^{2} d^{4} e^{10} + 15 \, a^{6} c d^{2} e^{12} - 5 \, a^{7} e^{14}\right )} e^{\left (-\frac {11}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{2048 \, \sqrt {c d} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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