Optimal. Leaf size=389 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}} \]
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Rubi [A]
time = 0.38, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 848, 820,
738, 212} \begin {gather*} \frac {\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rule 848
Rule 863
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx &=\int \frac {a e+c d x}{x^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\int \frac {-\frac {1}{2} a e \left (c d^2-7 a e^2\right )+3 a c d e^2 x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 a d e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\int \frac {-\frac {1}{4} a e \left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right )-a c d e^2 \left (c d^2-7 a e^2\right ) x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 a^2 d^2 e^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\int \frac {-\frac {1}{8} a e \left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right )-\frac {1}{4} a c d e^2 \left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{24 a^3 d^3 e^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}-\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 a^3 d^4 e^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 a^3 d^4 e^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 10.27, size = 273, normalized size = 0.70 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-15 c^3 d^6 x^3+a c^2 d^4 e x^2 (10 d-17 e x)+a^2 c d^2 e^2 x \left (-8 d^2+12 d e x-25 e^2 x^2\right )+a^3 e^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{x^4}+\frac {3 \left (5 c^4 d^8+4 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-35 a^4 e^8\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 a^{7/2} d^{9/2} 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. \(3470\) vs.
\(2(355)=710\).
time = 0.10, size = 3471, normalized size = 8.92
method | result | size |
default | \(\text {Expression too large to display}\) | \(3471\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 13.76, size = 719, normalized size = 1.85 \begin {gather*} \left [-\frac {{\left (3 \, {\left (5 \, c^{4} d^{8} x^{4} + 4 \, a c^{3} d^{6} x^{4} e^{2} + 6 \, a^{2} c^{2} d^{4} x^{4} e^{4} + 20 \, a^{3} c d^{2} x^{4} e^{6} - 35 \, a^{4} x^{4} e^{8}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (15 \, a c^{3} d^{7} x^{3} e - 10 \, a^{2} c^{2} d^{6} x^{2} e^{2} - 105 \, a^{4} d x^{3} e^{7} + 70 \, a^{4} d^{2} x^{2} e^{6} + {\left (25 \, a^{3} c d^{3} x^{3} - 56 \, a^{4} d^{3} x\right )} e^{5} - 12 \, {\left (a^{3} c d^{4} x^{2} - 4 \, a^{4} d^{4}\right )} e^{4} + {\left (17 \, a^{2} c^{2} d^{5} x^{3} + 8 \, a^{3} c d^{5} x\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 \, a^{4} d^{5} x^{4}}, -\frac {{\left (3 \, {\left (5 \, c^{4} d^{8} x^{4} + 4 \, a c^{3} d^{6} x^{4} e^{2} + 6 \, a^{2} c^{2} d^{4} x^{4} e^{4} + 20 \, a^{3} c d^{2} x^{4} e^{6} - 35 \, a^{4} x^{4} e^{8}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (15 \, a c^{3} d^{7} x^{3} e - 10 \, a^{2} c^{2} d^{6} x^{2} e^{2} - 105 \, a^{4} d x^{3} e^{7} + 70 \, a^{4} d^{2} x^{2} e^{6} + {\left (25 \, a^{3} c d^{3} x^{3} - 56 \, a^{4} d^{3} x\right )} e^{5} - 12 \, {\left (a^{3} c d^{4} x^{2} - 4 \, a^{4} d^{4}\right )} e^{4} + {\left (17 \, a^{2} c^{2} d^{5} x^{3} + 8 \, a^{3} c d^{5} x\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 \, a^{4} d^{5} x^{4}}\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. 1501 vs.
\(2 (347) = 694\).
time = 1.77, size = 1501, normalized size = 3.86 \begin {gather*} -\frac {{\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \arctan \left (-\frac {\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}}{\sqrt {-a d e}}\right ) e^{\left (-3\right )}}{64 \, \sqrt {-a d e} a^{3} d^{4}} + \frac {{\left (15 \, {\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 )} a^{3} c^{4} d^{11} e^{3} + 73 \, {\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 )}^{3} a^{2} c^{4} d^{10} e^{2} - 55 \, {\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 )}^{5} a c^{4} d^{9} e + 15 \, {\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 )}^{7} c^{4} d^{8} + 384 \, {\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 )}^{2} \sqrt {c d} a^{3} c^{3} d^{9} e^{\frac {7}{2}} + 396 \, {\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 )} a^{4} c^{3} d^{9} e^{5} + 980 \, {\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 )}^{3} a^{3} c^{3} d^{8} e^{4} - 44 \, {\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 )}^{5} a^{2} c^{3} d^{7} e^{3} + 12 \, {\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 )}^{7} a c^{3} d^{6} e^{2} + 128 \, \sqrt {c d} a^{5} c^{2} d^{8} e^{\frac {13}{2}} + 1792 \, {\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 )}^{2} \sqrt {c d} a^{4} c^{2} d^{7} e^{\frac {11}{2}} + 768 \, {\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 )}^{4} \sqrt {c d} a^{3} c^{2} d^{6} e^{\frac {9}{2}} + 1170 \, {\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 )} a^{5} c^{2} d^{7} e^{7} + 2238 \, {\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 )}^{3} a^{4} c^{2} d^{6} e^{6} - 66 \, {\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 )}^{5} a^{3} c^{2} d^{5} e^{5} + 18 \, {\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 )}^{7} a^{2} c^{2} d^{4} e^{4} + 256 \, \sqrt {c d} a^{6} c d^{6} e^{\frac {17}{2}} + 2432 \, {\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 )}^{2} \sqrt {c d} a^{5} c d^{5} e^{\frac {15}{2}} + 1212 \, {\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 )} a^{6} c d^{5} e^{9} + 292 \, {\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 )}^{3} a^{5} c d^{4} e^{8} - 220 \, {\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 )}^{5} a^{4} c d^{3} e^{7} + 60 \, {\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 )}^{7} a^{3} c d^{2} e^{6} + 384 \, \sqrt {c d} a^{7} d^{4} e^{\frac {21}{2}} + 279 \, {\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 )} a^{7} d^{3} e^{11} - 511 \, {\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 )}^{3} a^{6} d^{2} e^{10} + 385 \, {\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 )}^{5} a^{5} d e^{9} - 105 \, {\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 )}^{7} a^{4} e^{8}\right )} e^{\left (-3\right )}}{192 \, {\left (a d e - {\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 )}^{2}\right )}^{4} a^{3} 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 {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^5\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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