Optimal. Leaf size=261 \[ \frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {668, 684, 654,
626, 635, 212} \begin {gather*} -\frac {5 \left (c d^2-a e^2\right )^4 \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 )}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}+\frac {5 \left (c d^2-a e^2\right )^2 \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}}{64 c d e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 668
Rule 684
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int (a e+c d x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\\ &=\frac {(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}}{4 e}+\frac {\left (5 \left (2 a c d e^2-c d \left (c d^2+a e^2\right )\right )\right ) \int (a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{8 c d e}\\ &=\frac {5}{24} \left (a-\frac {c d^2}{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 {(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}}{4 e}+\frac {\left (5 \left (c d^2-a e^2\right )^2\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{16 e^2}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \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}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{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 {(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}}{4 e}-\frac {\left (5 \left (c d^2-a e^2\right )^4\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \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}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{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 {(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}}{4 e}-\frac {\left (5 \left (c d^2-a e^2\right )^4\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 )}{64 c d e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \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}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{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 {(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}}{4 e}-\frac {5 \left (c d^2-a e^2\right )^4 \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 )}{128 c^{3/2} d^{3/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 236, normalized size = 0.90 \begin {gather*} \frac {\left (c d^2-a e^2\right )^4 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6+a^2 c d e^4 (73 d+118 e x)+a c^2 d^2 e^2 \left (-55 d^2+36 d e x+136 e^2 x^2\right )+c^3 d^3 \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^4}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 c^{3/2} d^{3/2} e^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 408, normalized size = 1.56
method | result | size |
default | \(\frac {\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {10 c d e \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )}}{e^{2}}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 709 vs.
\(2 (230) = 460\).
time = 0.30, size = 709, normalized size = 2.72 \begin {gather*} -\frac {5 \, c^{4} d^{8} e^{\left (-\frac {7}{2}\right )} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{128 \, \left (c d\right )^{\frac {3}{2}}} + \frac {5 \, a c^{3} d^{6} e^{\left (-\frac {3}{2}\right )} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{32 \, \left (c d\right )^{\frac {3}{2}}} + \frac {5}{32} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c^{2} d^{4} x e^{\left (-2\right )} + \frac {5}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c^{2} d^{5} e^{\left (-3\right )} - \frac {15 \, a^{2} c^{2} d^{4} e^{\frac {1}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{64 \, \left (c d\right )^{\frac {3}{2}}} - \frac {5}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a c d^{3} e^{\left (-1\right )} + \frac {5 \, a^{3} c d^{2} e^{\frac {5}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{32 \, \left (c d\right )^{\frac {3}{2}}} - \frac {5}{16} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a c d^{2} x - \frac {5}{24} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} c d^{2} e^{\left (-2\right )} + \frac {5}{32} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{2} x e^{2} - \frac {5}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{2} d e - \frac {5 \, a^{4} e^{\frac {9}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{128 \, \left (c d\right )^{\frac {3}{2}}} + \frac {5}{24} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} a + \frac {5 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{3} e^{3}}{64 \, c d} + \frac {{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {5}{2}}}{4 \, {\left (x e^{2} + d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.03, size = 651, normalized size = 2.49 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\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 (10 \, c^{4} d^{6} x e^{2} - 15 \, c^{4} d^{7} e - 118 \, a^{2} c^{2} d^{2} x e^{6} - 15 \, a^{3} c d e^{7} - {\left (136 \, a c^{3} d^{3} x^{2} + 73 \, a^{2} c^{2} d^{3}\right )} e^{5} - 12 \, {\left (4 \, c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{4} x\right )} e^{4} - {\left (8 \, c^{4} d^{5} x^{2} - 55 \, a c^{3} d^{5}\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 )}}{768 \, c^{2} d^{2}}, \frac {{\left (15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\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 (10 \, c^{4} d^{6} x e^{2} - 15 \, c^{4} d^{7} e - 118 \, a^{2} c^{2} d^{2} x e^{6} - 15 \, a^{3} c d e^{7} - {\left (136 \, a c^{3} d^{3} x^{2} + 73 \, a^{2} c^{2} d^{3}\right )} e^{5} - 12 \, {\left (4 \, c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{4} x\right )} e^{4} - {\left (8 \, c^{4} d^{5} x^{2} - 55 \, a c^{3} d^{5}\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 )}}{384 \, c^{2} d^{2}}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1330 vs.
\(2 (230) = 460\).
time = 1.60, size = 1330, normalized size = 5.10 \begin {gather*} \frac {1}{192} \, {\left (\frac {15 \, {\left (c^{4} d^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4 \, a c^{3} d^{6} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 6 \, a^{2} c^{2} d^{4} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4 \, a^{3} c d^{2} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{4} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}}}{\sqrt {-c d e}}\right ) e^{\left (-4\right )}}{\sqrt {-c d e} c d} + \frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{7} d^{11} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 55 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c^{6} d^{10} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a c^{6} d^{9} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 73 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} c^{5} d^{9} e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 220 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a c^{5} d^{8} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} c^{4} d^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 90 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{2} c^{5} d^{7} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 292 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a c^{4} d^{7} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 330 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{6} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a c^{3} d^{6} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{3} c^{4} d^{5} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 438 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a^{2} c^{3} d^{5} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 220 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{3} c^{3} d^{4} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 90 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a^{2} c^{2} d^{4} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{4} c^{3} d^{3} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 292 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a^{3} c^{2} d^{3} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 55 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{4} c^{2} d^{2} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a^{3} c d^{2} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 73 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a^{4} c d e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a^{4} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-4\right )}}{{\left (\frac {c d^{2} e}{x e + d} - \frac {a e^{3}}{x e + d}\right )}^{4} c d}\right )} e \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 {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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