3.1010 \(\int \frac {e f-e f x^2}{(-a d+b d x-a d x^2) \sqrt {-a+b x+c x^2+b x^3-a x^4}} \, dx\)

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

[Out]

e*f*arctanh(1/2*(a*b-(4*a^2+2*a*c+b^2)*x+a*b*x^2)/a/(2*a+c)^(1/2)/(-a*x^4+b*x^3+c*x^2+b*x-a)^(1/2))/a/d/(2*a+c
)^(1/2)

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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.

[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*ArcTanh[(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)

Rule 2085

Int[((f_) + (g_.)*(x_)^2)/(((d_) + (e_.)*(x_) + (d_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (b_.)*(x
_)^3 + (a_.)*(x_)^4]), x_Symbol] :> -Simp[(a*f*ArcTanh[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2*Rt[-(a^2*(
2*a - c)), 2]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(d*Rt[-(a^2*(2*a - c)), 2]), x] /; FreeQ[{a, b, c, d, e
, f, g}, x] && EqQ[b*d - a*e, 0] && EqQ[f + g, 0] && NegQ[a^2*(2*a - c)]

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.

[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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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.

[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, algorithm="fricas")

[Out]

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

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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.

[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, 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)

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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.

[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 = 0.00, 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.

[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, 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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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.

[In]

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

[Out]

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

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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.

[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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