Optimal. Leaf size=88 \[ \frac {e f \tanh ^{-1}\left (\frac {-x \left (4 a^2+2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt {2 a+c} \sqrt {-a x^4-a+b x^3+b x+c x^2}}\right )}{a d \sqrt {2 a+c}} \]
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Rubi [A] time = 0.33, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2085} \[ \frac {e f \tanh ^{-1}\left (\frac {-x \left (4 a^2+2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt {2 a+c} \sqrt {-a x^4-a+b x^3+b x+c x^2}}\right )}{a d \sqrt {2 a+c}} \]
Antiderivative was successfully verified.
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Rule 2085
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 \tanh ^{-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 [C] time = 6.55, size = 15147, normalized size = 172.12 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 6.03, size = 331, normalized size = 3.76 \[ \left [\frac {\sqrt {2 \, a + c} e f \log \left (\frac {2 \, a b^{3} x^{3} + 2 \, a b^{3} x + {\left (8 \, a^{4} - a^{2} b^{2} + 4 \, a^{3} c\right )} x^{4} + 8 \, a^{4} - a^{2} b^{2} + 4 \, a^{3} c - {\left (16 \, a^{4} + 10 \, a^{2} b^{2} + b^{4} + 8 \, a^{2} c^{2} + 4 \, {\left (6 \, a^{3} + a b^{2}\right )} c\right )} x^{2} - 4 \, {\left (a^{2} b x^{2} + a^{2} b - {\left (4 \, a^{3} + a b^{2} + 2 \, a^{2} c\right )} x\right )} \sqrt {-a x^{4} + b x^{3} + c x^{2} + b x - a} \sqrt {2 \, a + c}}{a^{2} x^{4} - 2 \, a b x^{3} - 2 \, a b x + {\left (2 \, a^{2} + b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (2 \, a^{2} + a c\right )} d}, -\frac {\sqrt {-2 \, a - c} e f \arctan \left (\frac {2 \, \sqrt {-a x^{4} + b x^{3} + c x^{2} + b x - a} a \sqrt {-2 \, a - c}}{a b x^{2} + a b - {\left (4 \, a^{2} + b^{2} + 2 \, a c\right )} x}\right )}{{\left (2 \, a^{2} + a c\right )} d}\right ] \]
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 \[ \int \frac {e f x^{2} - e f}{\sqrt {-a x^{4} + b x^{3} + c x^{2} + b x - a} {\left (a d x^{2} - b d x + a d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 269221, normalized size = 3059.33 \[ \text {output too large to display} \]
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 \[ \int \frac {e f x^{2} - e f}{\sqrt {-a x^{4} + b x^{3} + c x^{2} + b x - a} {\left (a d x^{2} - b d x + a d\right )}}\,{d x} \]
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 \[ \int -\frac {e\,f-e\,f\,x^2}{\left (a\,d\,x^2-b\,d\,x+a\,d\right )\,\sqrt {-a\,x^4+b\,x^3+c\,x^2+b\,x-a}} \,d x \]
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 \[ \frac {e f \left (\int \frac {x^{2}}{a x^{2} \sqrt {- a x^{4} - a + b x^{3} + b x + c x^{2}} + a \sqrt {- a x^{4} - a + b x^{3} + b x + c x^{2}} - b x \sqrt {- a x^{4} - a + b x^{3} + b x + c x^{2}}}\, dx + \int \left (- \frac {1}{a x^{2} \sqrt {- a x^{4} - a + b x^{3} + b x + c x^{2}} + a \sqrt {- a x^{4} - a + b x^{3} + b x + c x^{2}} - b x \sqrt {- a x^{4} - a + b x^{3} + b x + c x^{2}}}\right )\, dx\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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