3.772 \(\int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x)^2 \sqrt{3+4 x^4}} \, dx\)

Optimal. Leaf size=268 \[ \frac{\left (\frac{1}{2}-\frac{i}{2}\right ) d \sqrt{\sqrt{3}-2 i x^2}}{\left (-\sqrt{3} d^2+2 i c^2\right ) (c+d x)}-\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) d \sqrt{\sqrt{3}+2 i x^2}}{\left (\sqrt{3} d^2+2 i c^2\right ) (c+d x)}+\frac{(1+i) c \tan ^{-1}\left (\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2} \sqrt{-\sqrt{3} d^2+2 i c^2}}\right )}{\left (-\sqrt{3} d^2+2 i c^2\right )^{3/2}}+\frac{(1-i) c \tanh ^{-1}\left (\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2} \sqrt{\sqrt{3} d^2+2 i c^2}}\right )}{\left (\sqrt{3} d^2+2 i c^2\right )^{3/2}} \]

[Out]

((1/2 - I/2)*d*Sqrt[Sqrt[3] - (2*I)*x^2])/(((2*I)*c^2 - Sqrt[3]*d^2)*(c + d*x))
- ((1/2 + I/2)*d*Sqrt[Sqrt[3] + (2*I)*x^2])/(((2*I)*c^2 + Sqrt[3]*d^2)*(c + d*x)
) + ((1 + I)*c*ArcTan[(Sqrt[3]*d + (2*I)*c*x)/(Sqrt[(2*I)*c^2 - Sqrt[3]*d^2]*Sqr
t[Sqrt[3] - (2*I)*x^2])])/((2*I)*c^2 - Sqrt[3]*d^2)^(3/2) + ((1 - I)*c*ArcTanh[(
Sqrt[3]*d - (2*I)*c*x)/(Sqrt[(2*I)*c^2 + Sqrt[3]*d^2]*Sqrt[Sqrt[3] + (2*I)*x^2])
])/((2*I)*c^2 + Sqrt[3]*d^2)^(3/2)

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Rubi [A]  time = 0.625938, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (\frac{1}{2}-\frac{i}{2}\right ) d \sqrt{\sqrt{3}-2 i x^2}}{\left (-\sqrt{3} d^2+2 i c^2\right ) (c+d x)}-\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) d \sqrt{\sqrt{3}+2 i x^2}}{\left (\sqrt{3} d^2+2 i c^2\right ) (c+d x)}+\frac{(1+i) c \tan ^{-1}\left (\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2} \sqrt{-\sqrt{3} d^2+2 i c^2}}\right )}{\left (-\sqrt{3} d^2+2 i c^2\right )^{3/2}}+\frac{(1-i) c \tanh ^{-1}\left (\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2} \sqrt{\sqrt{3} d^2+2 i c^2}}\right )}{\left (\sqrt{3} d^2+2 i c^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]),x]

[Out]

((1/2 - I/2)*d*Sqrt[Sqrt[3] - (2*I)*x^2])/(((2*I)*c^2 - Sqrt[3]*d^2)*(c + d*x))
- ((1/2 + I/2)*d*Sqrt[Sqrt[3] + (2*I)*x^2])/(((2*I)*c^2 + Sqrt[3]*d^2)*(c + d*x)
) + ((1 + I)*c*ArcTan[(Sqrt[3]*d + (2*I)*c*x)/(Sqrt[(2*I)*c^2 - Sqrt[3]*d^2]*Sqr
t[Sqrt[3] - (2*I)*x^2])])/((2*I)*c^2 - Sqrt[3]*d^2)^(3/2) + ((1 - I)*c*ArcTanh[(
Sqrt[3]*d - (2*I)*c*x)/(Sqrt[(2*I)*c^2 + Sqrt[3]*d^2]*Sqrt[Sqrt[3] + (2*I)*x^2])
])/((2*I)*c^2 + Sqrt[3]*d^2)^(3/2)

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Rubi in Sympy [A]  time = 26.9435, size = 231, normalized size = 0.86 \[ \frac{c \left (1 - i\right ) \operatorname{atanh}{\left (\frac{- 2 i c x + \sqrt{3} d}{\sqrt{2 i c^{2} + \sqrt{3} d^{2}} \sqrt{2 i x^{2} + \sqrt{3}}} \right )}}{\left (2 i c^{2} + \sqrt{3} d^{2}\right )^{\frac{3}{2}}} + \frac{i c \left (1 - i\right ) \operatorname{atanh}{\left (\frac{2 i c x + \sqrt{3} d}{\sqrt{- 2 i c^{2} + \sqrt{3} d^{2}} \sqrt{- 2 i x^{2} + \sqrt{3}}} \right )}}{\left (- 2 i c^{2} + \sqrt{3} d^{2}\right )^{\frac{3}{2}}} - \frac{d \left (1 + i\right ) \sqrt{2 i x^{2} + \sqrt{3}}}{2 \left (c + d x\right ) \left (2 i c^{2} + \sqrt{3} d^{2}\right )} - \frac{d \left (1 - i\right ) \sqrt{- 2 i x^{2} + \sqrt{3}}}{2 \left (c + d x\right ) \left (- 2 i c^{2} + \sqrt{3} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2+(4*x**4+3)**(1/2))**(1/2)/(d*x+c)**2/(4*x**4+3)**(1/2),x)

[Out]

c*(1 - I)*atanh((-2*I*c*x + sqrt(3)*d)/(sqrt(2*I*c**2 + sqrt(3)*d**2)*sqrt(2*I*x
**2 + sqrt(3))))/(2*I*c**2 + sqrt(3)*d**2)**(3/2) + I*c*(1 - I)*atanh((2*I*c*x +
sqrt(3)*d)/(sqrt(-2*I*c**2 + sqrt(3)*d**2)*sqrt(-2*I*x**2 + sqrt(3))))/(-2*I*c*
*2 + sqrt(3)*d**2)**(3/2) - d*(1 + I)*sqrt(2*I*x**2 + sqrt(3))/(2*(c + d*x)*(2*I
*c**2 + sqrt(3)*d**2)) - d*(1 - I)*sqrt(-2*I*x**2 + sqrt(3))/(2*(c + d*x)*(-2*I*
c**2 + sqrt(3)*d**2))

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Mathematica [A]  time = 0.0979583, size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x)^2 \sqrt{3+4 x^4}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]),x]

[Out]

Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]), x]

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( dx+c \right ) ^{2}}\sqrt{2\,{x}^{2}+\sqrt{4\,{x}^{4}+3}}{\frac{1}{\sqrt{4\,{x}^{4}+3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+(4*x^4+3)^(1/2))^(1/2)/(d*x+c)^2/(4*x^4+3)^(1/2),x)

[Out]

int((2*x^2+(4*x^4+3)^(1/2))^(1/2)/(d*x+c)^2/(4*x^4+3)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(2*x^2 + sqrt(4*x^4 + 3))/(sqrt(4*x^4 + 3)*(d*x + c)^2), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2+(4*x**4+3)**(1/2))**(1/2)/(d*x+c)**2/(4*x**4+3)**(1/2),x)

[Out]

Integral(sqrt(2*x**2 + sqrt(4*x**4 + 3))/((c + d*x)**2*sqrt(4*x**4 + 3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(2*x^2 + sqrt(4*x^4 + 3))/(sqrt(4*x^4 + 3)*(d*x + c)^2), x)