3.871 \(\int \frac{e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx\)

Optimal. Leaf size=88 \[ \frac{e f \tan ^{-1}\left (\frac{x \left (4 a^2-2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt{2 a-c} \sqrt{a x^4+a+b x^3+b x+c x^2}}\right )}{a d \sqrt{2 a-c}} \]

[Out]

(e*f*ArcTan[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2*a*Sqrt[2*a - c]*Sqrt[a
+ b*x + c*x^2 + b*x^3 + a*x^4])])/(a*Sqrt[2*a - c]*d)

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Rubi [A]  time = 0.417129, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019 \[ \frac{e f \tan ^{-1}\left (\frac{x \left (4 a^2-2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt{2 a-c} \sqrt{a x^4+a+b x^3+b x+c x^2}}\right )}{a d \sqrt{2 a-c}} \]

Antiderivative was successfully verified.

[In]  Int[(e*f - e*f*x^2)/((a*d + b*d*x + a*d*x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

(e*f*ArcTan[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2*a*Sqrt[2*a - c]*Sqrt[a
+ b*x + c*x^2 + b*x^3 + a*x^4])])/(a*Sqrt[2*a - c]*d)

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Rubi in Sympy [A]  time = 33.8519, size = 76, normalized size = 0.86 \[ \frac{e f \operatorname{atan}{\left (\frac{a b x^{2} + a b + x \left (4 a^{2} - 2 a c + b^{2}\right )}{2 a \sqrt{2 a - c} \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}}} \right )}}{a d \sqrt{2 a - c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e*f*x**2+e*f)/(a*d*x**2+b*d*x+a*d)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)

[Out]

e*f*atan((a*b*x**2 + a*b + x*(4*a**2 - 2*a*c + b**2))/(2*a*sqrt(2*a - c)*sqrt(a*
x**4 + a + b*x**3 + b*x + c*x**2)))/(a*d*sqrt(2*a - c))

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Mathematica [C]  time = 6.32351, size = 13884, normalized size = 157.77 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*f - e*f*x^2)/((a*d + b*d*x + a*d*x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

Result too large to show

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Maple [C]  time = 0.181, size = 242984, normalized size = 2761.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e*f*x^2+e*f)/(a*d*x^2+b*d*x+a*d)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{e f x^{2} - e f}{\sqrt{a x^{4} + b x^{3} + c x^{2} + b x + a}{\left (a d x^{2} + b d x + a d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*f*x^2 - e*f)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(a*d*x^2 + b*d*x + a*d)),x, algorithm="maxima")

[Out]

-integrate((e*f*x^2 - e*f)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(a*d*x^2 + b*d
*x + a*d)), x)

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Fricas [A]  time = 3.41962, size = 1, normalized size = 0.01 \[ \left [\frac{e f \log \left (\frac{4 \, \sqrt{a x^{4} + b x^{3} + c x^{2} + b x + a}{\left (2 \, a^{3} b - a^{2} b c +{\left (2 \, a^{3} b - a^{2} b c\right )} x^{2} +{\left (8 \, a^{4} + 2 \, a^{2} b^{2} + 2 \, a^{2} c^{2} -{\left (8 \, a^{3} + a b^{2}\right )} c\right )} x\right )} +{\left (2 \, a b^{3} x^{3} + 2 \, a b^{3} x -{\left (8 \, a^{4} - a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4} - 8 \, a^{4} + a^{2} b^{2} + 4 \, a^{3} c +{\left (16 \, a^{4} + 10 \, a^{2} b^{2} + b^{4} + 8 \, a^{2} c^{2} - 4 \,{\left (6 \, a^{3} + a b^{2}\right )} c\right )} x^{2}\right )} \sqrt{-2 \, a + c}}{a^{2} x^{4} + 2 \, a b x^{3} + 2 \, a b x +{\left (2 \, a^{2} + b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, a \sqrt{-2 \, a + c} d}, -\frac{e f \arctan \left (\frac{2 \, \sqrt{a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt{2 \, a - c} a}{a b x^{2} + a b +{\left (4 \, a^{2} + b^{2} - 2 \, a c\right )} x}\right )}{\sqrt{2 \, a - c} a d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*f*x^2 - e*f)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(a*d*x^2 + b*d*x + a*d)),x, algorithm="fricas")

[Out]

[1/2*e*f*log((4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(2*a^3*b - a^2*b*c + (2*a^
3*b - a^2*b*c)*x^2 + (8*a^4 + 2*a^2*b^2 + 2*a^2*c^2 - (8*a^3 + a*b^2)*c)*x) + (2
*a*b^3*x^3 + 2*a*b^3*x - (8*a^4 - a^2*b^2 - 4*a^3*c)*x^4 - 8*a^4 + a^2*b^2 + 4*a
^3*c + (16*a^4 + 10*a^2*b^2 + b^4 + 8*a^2*c^2 - 4*(6*a^3 + a*b^2)*c)*x^2)*sqrt(-
2*a + c))/(a^2*x^4 + 2*a*b*x^3 + 2*a*b*x + (2*a^2 + b^2)*x^2 + a^2))/(a*sqrt(-2*
a + c)*d), -e*f*arctan(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 + (4*a^2 + b^2 - 2*a*c)*x))/(sqrt(2*a - c)*a*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{e f \left (\int \frac{x^{2}}{a x^{2} \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}} + a \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}} + b x \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx + \int \left (- \frac{1}{a x^{2} \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}} + a \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}} + b x \sqrt{a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e*f*x**2+e*f)/(a*d*x**2+b*d*x+a*d)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)

[Out]

-e*f*(Integral(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)
), x) + Integral(-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)
), x))/d

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{e f x^{2} - e f}{\sqrt{a x^{4} + b x^{3} + c x^{2} + b x + a}{\left (a d x^{2} + b d x + a d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*f*x^2 - e*f)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(a*d*x^2 + b*d*x + a*d)),x, algorithm="giac")

[Out]

integrate(-(e*f*x^2 - e*f)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(a*d*x^2 + b*d
*x + a*d)), x)