Optimal. Leaf size=286 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\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 )}{16 a^{5/2} d^{7/2} e^{5/2}}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d 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^4 (d+e x)} \, dx &=\int \frac {a e+c d x}{x^4 \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}}{3 d x^3}-\frac {\int \frac {-\frac {1}{2} a e \left (c d^2-5 a e^2\right )+2 a c d e^2 x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a d e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\int \frac {-\frac {1}{4} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right )-\frac {1}{2} a c d e^2 \left (c d^2-5 a e^2\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 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}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}+\frac {\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\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}{16 a^2 d^3 e^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\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 )}{8 a^2 d^3 e^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\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 )}{16 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 10.20, size = 210, normalized size = 0.73 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (3 c^2 d^4 x^2-2 a c d^2 e x (d-2 e x)+a^2 e^2 \left (-8 d^2+10 d e x-15 e^2 x^2\right )\right )}{x^3}-\frac {3 \left (c^3 d^6+a c^2 d^4 e^2+3 a^2 c d^2 e^4-5 a^3 e^6\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 )}{24 a^{5/2} d^{7/2} e^{5/2}} \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. \(2058\) vs.
\(2(256)=512\).
time = 0.07, size = 2059, normalized size = 7.20
method | result | size |
default | \(\text {Expression too large to display}\) | \(2059\) |
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 = 6.44, size = 565, normalized size = 1.98 \begin {gather*} \left [-\frac {{\left (3 \, {\left (c^{3} d^{6} x^{3} + a c^{2} d^{4} x^{3} e^{2} + 3 \, a^{2} c d^{2} x^{3} e^{4} - 5 \, a^{3} x^{3} e^{6}\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 (3 \, a c^{2} d^{5} x^{2} e - 2 \, a^{2} c d^{4} x e^{2} - 15 \, a^{3} d x^{2} e^{5} + 10 \, a^{3} d^{2} x e^{4} + 4 \, {\left (a^{2} c d^{3} x^{2} - 2 \, a^{3} d^{3}\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 (-3\right )}}{96 \, a^{3} d^{4} x^{3}}, \frac {{\left (3 \, {\left (c^{3} d^{6} x^{3} + a c^{2} d^{4} x^{3} e^{2} + 3 \, a^{2} c d^{2} x^{3} e^{4} - 5 \, a^{3} x^{3} e^{6}\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 (3 \, a c^{2} d^{5} x^{2} e - 2 \, a^{2} c d^{4} x e^{2} - 15 \, a^{3} d x^{2} e^{5} + 10 \, a^{3} d^{2} x e^{4} + 4 \, {\left (a^{2} c d^{3} x^{2} - 2 \, a^{3} d^{3}\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 (-3\right )}}{48 \, a^{3} d^{4} x^{3}}\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. 925 vs.
\(2 (252) = 504\).
time = 1.49, size = 925, normalized size = 3.23 \begin {gather*} \frac {{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\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 (-2\right )}}{8 \, \sqrt {-a d e} a^{2} d^{3}} - \frac {{\left (3 \, {\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^{2} c^{3} d^{8} e^{2} + 8 \, {\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 c^{3} d^{7} e - 3 \, {\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} c^{3} d^{6} + 48 \, {\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^{2} c^{2} d^{6} e^{\frac {5}{2}} + 51 \, {\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^{2} d^{6} e^{4} + 72 \, {\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^{2} d^{5} e^{3} - 3 \, {\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^{2} d^{4} e^{2} + 16 \, \sqrt {c d} a^{4} c d^{5} e^{\frac {11}{2}} + 144 \, {\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 d^{4} e^{\frac {9}{2}} + 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 )} a^{4} c d^{4} e^{6} + 24 \, {\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 d^{3} e^{5} - 9 \, {\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 d^{2} e^{4} + 48 \, \sqrt {c d} a^{5} d^{3} e^{\frac {15}{2}} + 33 \, {\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} d^{2} e^{8} - 40 \, {\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} d e^{7} + 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 )}^{5} a^{3} e^{6}\right )} e^{\left (-2\right )}}{24 \, {\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 )}^{3} a^{2} d^{3}} \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^4\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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