3.1009 \(\int \frac{e f-e f x^2}{(a d+b d x+a d x^2) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx\)

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

[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 = 0.249151, 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, Rules used = {2084} \[ \frac{e f \tan ^{-1}\left (\frac{x \left (4 a^2-2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt{2 a-c} \sqrt{a x^4+a+b x^3+b x+c x^2}}\right )}{a d \sqrt{2 a-c}} \]

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)

Rule 2084

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*ArcTan[(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] && PosQ[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 \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*}

Mathematica [C]  time = 6.50512, 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 = 0.125, size = 242984, normalized size = 2761.2 \begin{align*} \text{output too large to display} \end{align*}

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. \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 31.0476, size = 698, normalized size = 7.93 \begin{align*} \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} \sqrt{2 \, a - c} a}{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 ] \end{align*}

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)*sqrt
(2*a - c)*a/(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 - \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}}}\, dx\right )}{d} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError