Optimal. Leaf size=88 \[ \frac {e f \tan ^{-1}\left (\frac {a b+\left (4 a^2+b^2-2 a c\right ) x+a b x^2}{2 a \sqrt {2 a-c} \sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{a \sqrt {2 a-c} d} \]
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Rubi [A]
time = 0.16, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2109}
\begin {gather*} \frac {e f \text {ArcTan}\left (\frac {x \left (4 a^2-2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt {2 a-c} \sqrt {a x^4+a+b x^3+b x+c x^2}}\right )}{a d \sqrt {2 a-c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2109
Rubi steps
\begin {align*} \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx &=\frac {e f \tan ^{-1}\left (\frac {a b+\left (4 a^2+b^2-2 a c\right ) x+a b x^2}{2 a \sqrt {2 a-c} \sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{a \sqrt {2 a-c} d}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 83, normalized size = 0.94 \begin {gather*} -\frac {2 e f \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {2 a-c} x}{a+b x+a x^2-\sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{a \sqrt {2 a-c} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.37, size = 242984, normalized size = 2761.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(242984\) |
elliptic | \(\text {Expression too large to display}\) | \(254498\) |
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 = 2.23, size = 326, normalized size = 3.70 \begin {gather*} \left [-\frac {\sqrt {-2 \, a + c} f e \log \left (\frac {2 \, a b^{3} x^{3} + 2 \, a b^{3} x - {\left (8 \, a^{4} - a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4} - 8 \, a^{4} + a^{2} b^{2} + 4 \, a^{3} c + {\left (16 \, a^{4} + 10 \, a^{2} b^{2} + b^{4} + 8 \, a^{2} c^{2} - 4 \, {\left (6 \, a^{3} + a b^{2}\right )} c\right )} x^{2} - 4 \, {\left (a^{2} b x^{2} + a^{2} b + {\left (4 \, a^{3} + a b^{2} - 2 \, a^{2} c\right )} x\right )} \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-2 \, a + c}}{a^{2} x^{4} + 2 \, a b x^{3} + 2 \, a b x + {\left (2 \, a^{2} + b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (2 \, a^{2} - a c\right )} d}, -\frac {\sqrt {2 \, a - c} f \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {2 \, a - c} a}{a b x^{2} + a b + {\left (4 \, a^{2} + b^{2} - 2 \, a c\right )} x}\right ) e}{{\left (2 \, a^{2} - a c\right )} d}\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*} - \frac {e f \left (\int \frac {x^{2}}{a x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + a \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + b x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx + \int \left (- \frac {1}{a x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + a \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + b x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {e\,f-e\,f\,x^2}{\left (a\,d\,x^2+b\,d\,x+a\,d\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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