Optimal. Leaf size=414 \[ \frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626,
635, 212} \begin {gather*} -\frac {45 \left (c d^2-a e^2\right )^8 \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 )}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac {45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \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}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \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 )^{5/2}}{128 c^3 d^3 e}+\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 684
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac {\left (9 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (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}{16 d}\\ &=\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac {\left (9 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{32 d^2}\\ &=\frac {3 \left (c d^2-a e^2\right )^2 \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 )^{5/2}}{128 c^3 d^3 e}+\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {\left (15 \left (c d^2-a e^2\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{256 c^3 d^3 e}\\ &=-\frac {15 \left (c d^2-a e^2\right )^4 \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}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \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 )^{5/2}}{128 c^3 d^3 e}+\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac {\left (45 \left (c d^2-a e^2\right )^6\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^4 d^4 e^2}\\ &=\frac {45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \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}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \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 )^{5/2}}{128 c^3 d^3 e}+\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {\left (45 \left (c d^2-a e^2\right )^8\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32768 c^5 d^5 e^3}\\ &=\frac {45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \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}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \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 )^{5/2}}{128 c^3 d^3 e}+\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {\left (45 \left (c d^2-a e^2\right )^8\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 )}{16384 c^5 d^5 e^3}\\ &=\frac {45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \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}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \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 )^{5/2}}{128 c^3 d^3 e}+\frac {9 \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 )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \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 )}{32768 c^{11/2} d^{11/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.63, size = 351, normalized size = 0.85 \begin {gather*} \frac {\left (c d^2-a e^2\right )^8 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^5 \left (315 e^7-\frac {2415 c d e^6 (d+e x)}{a e+c d x}+\frac {8043 c^2 d^2 e^5 (d+e x)^2}{(a e+c d x)^2}-\frac {15159 c^3 d^3 e^4 (d+e x)^3}{(a e+c d x)^3}+\frac {17609 c^4 d^4 e^3 (d+e x)^4}{(a e+c d x)^4}+\frac {8043 c^5 d^5 e^2 (d+e x)^5}{(a e+c d x)^5}-\frac {2415 c^6 d^6 e (d+e x)^6}{(a e+c d x)^6}+\frac {315 c^7 d^7 (d+e x)^7}{(a e+c d x)^7}\right )}{\left (c d^2-a e^2\right )^8 (d+e x)^2}-\frac {315 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{114688 c^{11/2} d^{11/2} e^{7/2}} \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. \(1559\) vs.
\(2(376)=752\).
time = 0.70, size = 1560, normalized size = 3.77
method | result | size |
default | \(\text {Expression too large to display}\) | \(1560\) |
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 = 2.88, size = 1483, normalized size = 3.58 \begin {gather*} \left [\frac {{\left (315 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\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 (210 \, c^{8} d^{14} x e^{2} - 315 \, c^{8} d^{15} e + 210 \, a^{6} c^{2} d^{2} x e^{14} - 315 \, a^{7} c d e^{15} - 21 \, {\left (8 \, a^{5} c^{3} d^{3} x^{2} - 115 \, a^{6} c^{2} d^{3}\right )} e^{13} + 12 \, {\left (12 \, a^{4} c^{4} d^{4} x^{3} - 133 \, a^{5} c^{3} d^{4} x\right )} e^{12} - {\left (128 \, a^{3} c^{5} d^{5} x^{4} - 1272 \, a^{4} c^{4} d^{5} x^{2} + 8043 \, a^{5} c^{3} d^{5}\right )} e^{11} - 2 \, {\left (10368 \, a^{2} c^{6} d^{6} x^{5} + 544 \, a^{3} c^{5} d^{6} x^{3} - 2631 \, a^{4} c^{4} d^{6} x\right )} e^{10} - 3 \, {\left (11264 \, a c^{7} d^{7} x^{6} + 34432 \, a^{2} c^{6} d^{7} x^{4} + 1392 \, a^{3} c^{5} d^{7} x^{2} - 5053 \, a^{4} c^{4} d^{7}\right )} e^{9} - 56 \, {\left (256 \, c^{8} d^{8} x^{7} + 2880 \, a c^{7} d^{8} x^{5} + 3660 \, a^{2} c^{6} d^{8} x^{3} + 175 \, a^{3} c^{5} d^{8} x\right )} e^{8} - {\left (66560 \, c^{8} d^{9} x^{6} + 299904 \, a c^{7} d^{9} x^{4} + 200784 \, a^{2} c^{6} d^{9} x^{2} + 17609 \, a^{3} c^{5} d^{9}\right )} e^{7} - 6 \, {\left (19840 \, c^{8} d^{10} x^{5} + 43872 \, a c^{7} d^{10} x^{3} + 15507 \, a^{2} c^{6} d^{10} x\right )} e^{6} - {\left (98432 \, c^{8} d^{11} x^{4} + 97032 \, a c^{7} d^{11} x^{2} + 8043 \, a^{2} c^{6} d^{11}\right )} e^{5} - 4 \, {\left (8156 \, c^{8} d^{12} x^{3} + 399 \, a c^{7} d^{12} x\right )} e^{4} - 21 \, {\left (8 \, c^{8} d^{13} x^{2} - 115 \, a c^{7} d^{13}\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 (-4\right )}}{458752 \, c^{6} d^{6}}, \frac {{\left (315 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\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 (210 \, c^{8} d^{14} x e^{2} - 315 \, c^{8} d^{15} e + 210 \, a^{6} c^{2} d^{2} x e^{14} - 315 \, a^{7} c d e^{15} - 21 \, {\left (8 \, a^{5} c^{3} d^{3} x^{2} - 115 \, a^{6} c^{2} d^{3}\right )} e^{13} + 12 \, {\left (12 \, a^{4} c^{4} d^{4} x^{3} - 133 \, a^{5} c^{3} d^{4} x\right )} e^{12} - {\left (128 \, a^{3} c^{5} d^{5} x^{4} - 1272 \, a^{4} c^{4} d^{5} x^{2} + 8043 \, a^{5} c^{3} d^{5}\right )} e^{11} - 2 \, {\left (10368 \, a^{2} c^{6} d^{6} x^{5} + 544 \, a^{3} c^{5} d^{6} x^{3} - 2631 \, a^{4} c^{4} d^{6} x\right )} e^{10} - 3 \, {\left (11264 \, a c^{7} d^{7} x^{6} + 34432 \, a^{2} c^{6} d^{7} x^{4} + 1392 \, a^{3} c^{5} d^{7} x^{2} - 5053 \, a^{4} c^{4} d^{7}\right )} e^{9} - 56 \, {\left (256 \, c^{8} d^{8} x^{7} + 2880 \, a c^{7} d^{8} x^{5} + 3660 \, a^{2} c^{6} d^{8} x^{3} + 175 \, a^{3} c^{5} d^{8} x\right )} e^{8} - {\left (66560 \, c^{8} d^{9} x^{6} + 299904 \, a c^{7} d^{9} x^{4} + 200784 \, a^{2} c^{6} d^{9} x^{2} + 17609 \, a^{3} c^{5} d^{9}\right )} e^{7} - 6 \, {\left (19840 \, c^{8} d^{10} x^{5} + 43872 \, a c^{7} d^{10} x^{3} + 15507 \, a^{2} c^{6} d^{10} x\right )} e^{6} - {\left (98432 \, c^{8} d^{11} x^{4} + 97032 \, a c^{7} d^{11} x^{2} + 8043 \, a^{2} c^{6} d^{11}\right )} e^{5} - 4 \, {\left (8156 \, c^{8} d^{12} x^{3} + 399 \, a c^{7} d^{12} x\right )} e^{4} - 21 \, {\left (8 \, c^{8} d^{13} x^{2} - 115 \, a c^{7} d^{13}\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 (-4\right )}}{229376 \, c^{6} d^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 736, normalized size = 1.78 \begin {gather*} \frac {1}{114688} \, \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 (2 \, {\left (4 \, {\left (14 \, c^{2} d^{2} x e^{4} + \frac {{\left (65 \, c^{9} d^{10} e^{10} + 33 \, a c^{8} d^{8} e^{12}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {3 \, {\left (155 \, c^{9} d^{11} e^{9} + 210 \, a c^{8} d^{9} e^{11} + 27 \, a^{2} c^{7} d^{7} e^{13}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (769 \, c^{9} d^{12} e^{8} + 2343 \, a c^{8} d^{10} e^{10} + 807 \, a^{2} c^{7} d^{8} e^{12} + a^{3} c^{6} d^{6} e^{14}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (2039 \, c^{9} d^{13} e^{7} + 16452 \, a c^{8} d^{11} e^{9} + 12810 \, a^{2} c^{7} d^{9} e^{11} + 68 \, a^{3} c^{6} d^{7} e^{13} - 9 \, a^{4} c^{5} d^{5} e^{15}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {3 \, {\left (7 \, c^{9} d^{14} e^{6} + 4043 \, a c^{8} d^{12} e^{8} + 8366 \, a^{2} c^{7} d^{10} e^{10} + 174 \, a^{3} c^{6} d^{8} e^{12} - 53 \, a^{4} c^{5} d^{6} e^{14} + 7 \, a^{5} c^{4} d^{4} e^{16}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (105 \, c^{9} d^{15} e^{5} - 798 \, a c^{8} d^{13} e^{7} - 46521 \, a^{2} c^{7} d^{11} e^{9} - 4900 \, a^{3} c^{6} d^{9} e^{11} + 2631 \, a^{4} c^{5} d^{7} e^{13} - 798 \, a^{5} c^{4} d^{5} e^{15} + 105 \, a^{6} c^{3} d^{3} e^{17}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (315 \, c^{9} d^{16} e^{4} - 2415 \, a c^{8} d^{14} e^{6} + 8043 \, a^{2} c^{7} d^{12} e^{8} + 17609 \, a^{3} c^{6} d^{10} e^{10} - 15159 \, a^{4} c^{5} d^{8} e^{12} + 8043 \, a^{5} c^{4} d^{6} e^{14} - 2415 \, a^{6} c^{3} d^{4} e^{16} + 315 \, a^{7} c^{2} d^{2} e^{18}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} + \frac {45 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\right )} e^{\left (-\frac {7}{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 )}{32768 \, \sqrt {c d} c^{5} d^{5}} \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 {\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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