3.1464 \(\int \frac{A+B x}{\sqrt{\frac{A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=133 \[ \frac{\sqrt{2} \sqrt{A} \sqrt{B} \tan ^{-1}\left (\frac{\sqrt{A} \sqrt{e \left (\frac{A}{B}-\frac{B}{A}+2 x\right )}}{\sqrt{B} \sqrt{e}}+\frac{A}{B}\right )}{\sqrt{e}}-\frac{\sqrt{2} \sqrt{A} \sqrt{B} \tan ^{-1}\left (\frac{A}{B}-\frac{\sqrt{A} \sqrt{e \left (\frac{A}{B}-\frac{B}{A}+2 x\right )}}{\sqrt{B} \sqrt{e}}\right )}{\sqrt{e}} \]

[Out]

-((Sqrt[2]*Sqrt[A]*Sqrt[B]*ArcTan[A/B - (Sqrt[A]*Sqrt[e*(A/B - B/A + 2*x)])/(Sqr
t[B]*Sqrt[e])])/Sqrt[e]) + (Sqrt[2]*Sqrt[A]*Sqrt[B]*ArcTan[A/B + (Sqrt[A]*Sqrt[e
*(A/B - B/A + 2*x)])/(Sqrt[B]*Sqrt[e])])/Sqrt[e]

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Rubi [A]  time = 0.81627, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093 \[ \frac{\sqrt{2} \sqrt{A} \sqrt{B} \tan ^{-1}\left (\frac{\sqrt{A} \sqrt{e \left (\frac{A}{B}-\frac{B}{A}+2 x\right )}}{\sqrt{B} \sqrt{e}}+\frac{A}{B}\right )}{\sqrt{e}}-\frac{\sqrt{2} \sqrt{A} \sqrt{B} \tan ^{-1}\left (\frac{A}{B}-\frac{\sqrt{A} \sqrt{e \left (\frac{A}{B}-\frac{B}{A}+2 x\right )}}{\sqrt{B} \sqrt{e}}\right )}{\sqrt{e}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[(A^2*e - B^2*e)/(2*A*B) + e*x]*(1 + x^2)),x]

[Out]

-((Sqrt[2]*Sqrt[A]*Sqrt[B]*ArcTan[A/B - (Sqrt[A]*Sqrt[e*(A/B - B/A + 2*x)])/(Sqr
t[B]*Sqrt[e])])/Sqrt[e]) + (Sqrt[2]*Sqrt[A]*Sqrt[B]*ArcTan[A/B + (Sqrt[A]*Sqrt[e
*(A/B - B/A + 2*x)])/(Sqrt[B]*Sqrt[e])])/Sqrt[e]

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Rubi in Sympy [A]  time = 121.97, size = 168, normalized size = 1.26 \[ \frac{\sqrt{2} \sqrt{A} \sqrt{B} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{A} \left (- \frac{\sqrt{2} \sqrt{A} \sqrt{e}}{2} + \frac{\sqrt{B} \sqrt{4 e x + \frac{2 e \left (A^{2} - B^{2}\right )}{A B}}}{2}\right )}{B \sqrt{e}} \right )}}{\sqrt{e}} + \frac{\sqrt{2} \sqrt{A} \sqrt{B} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{A} \left (\frac{\sqrt{2} \sqrt{A} \sqrt{e}}{2} + \frac{\sqrt{B} \sqrt{4 e x + \frac{2 e \left (A^{2} - B^{2}\right )}{A B}}}{2}\right )}{B \sqrt{e}} \right )}}{\sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(2*(B*x+A)/(x**2+1)/(2*(A**2*e-B**2*e)/A/B+4*e*x)**(1/2),x)

[Out]

sqrt(2)*sqrt(A)*sqrt(B)*atan(sqrt(2)*sqrt(A)*(-sqrt(2)*sqrt(A)*sqrt(e)/2 + sqrt(
B)*sqrt(4*e*x + 2*e*(A**2 - B**2)/(A*B))/2)/(B*sqrt(e)))/sqrt(e) + sqrt(2)*sqrt(
A)*sqrt(B)*atan(sqrt(2)*sqrt(A)*(sqrt(2)*sqrt(A)*sqrt(e)/2 + sqrt(B)*sqrt(4*e*x
+ 2*e*(A**2 - B**2)/(A*B))/2)/(B*sqrt(e)))/sqrt(e)

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Mathematica [C]  time = 0.18894, size = 142, normalized size = 1.07 \[ -\frac{i \sqrt{2} \sqrt{A} \sqrt{B} \sqrt{\frac{A}{B}-\frac{B}{A}+2 x} \left (\tanh ^{-1}\left (\frac{\sqrt{A} \sqrt{B} \sqrt{\frac{A}{B}-\frac{B}{A}+2 x}}{A-i B}\right )-\tanh ^{-1}\left (\frac{\sqrt{A} \sqrt{B} \sqrt{\frac{A}{B}-\frac{B}{A}+2 x}}{A+i B}\right )\right )}{\sqrt{e \left (\frac{A}{B}-\frac{B}{A}+2 x\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[(A^2*e - B^2*e)/(2*A*B) + e*x]*(1 + x^2)),x]

[Out]

((-I)*Sqrt[2]*Sqrt[A]*Sqrt[B]*Sqrt[A/B - B/A + 2*x]*(ArcTanh[(Sqrt[A]*Sqrt[B]*Sq
rt[A/B - B/A + 2*x])/(A - I*B)] - ArcTanh[(Sqrt[A]*Sqrt[B]*Sqrt[A/B - B/A + 2*x]
)/(A + I*B)]))/Sqrt[e*(A/B - B/A + 2*x)]

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Maple [A]  time = 0.086, size = 128, normalized size = 1. \[{\sqrt{2}AB\arctan \left ({\frac{1}{2\,B} \left ( 2\,\sqrt{2\,ex+{\frac{e \left ({A}^{2}-{B}^{2} \right ) }{AB}}}AB-2\,\sqrt{{A}^{3}Be} \right ){\frac{1}{\sqrt{AeB}}}} \right ){\frac{1}{\sqrt{AeB}}}}+{\sqrt{2}AB\arctan \left ({\frac{1}{2\,B} \left ( 2\,\sqrt{2\,ex+{\frac{e \left ({A}^{2}-{B}^{2} \right ) }{AB}}}AB+2\,\sqrt{{A}^{3}Be} \right ){\frac{1}{\sqrt{AeB}}}} \right ){\frac{1}{\sqrt{AeB}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(2*(B*x+A)/(x^2+1)/(2*(A^2*e-B^2*e)/A/B+4*e*x)^(1/2),x)

[Out]

2^(1/2)*A*B/(A*e*B)^(1/2)*arctan(1/2*(2*(2*e*x+e*(A^2-B^2)/A/B)^(1/2)*A*B-2*(A^3
*B*e)^(1/2))/B/(A*e*B)^(1/2))+2^(1/2)*A*B/(A*e*B)^(1/2)*arctan(1/2*(2*(2*e*x+e*(
A^2-B^2)/A/B)^(1/2)*A*B+2*(A^3*B*e)^(1/2))/B/(A*e*B)^(1/2))

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Maxima [A]  time = 1.09537, size = 560, normalized size = 4.21 \[ \frac{{\left (\frac{2 \, \sqrt{2} A^{2} B \arctan \left (\frac{A + \sqrt{2 \, A B x + A^{2} - B^{2}}}{B}\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}} + \frac{2 \, \sqrt{2} A^{2} B \arctan \left (-\frac{A - \sqrt{2 \, A B x + A^{2} - B^{2}}}{B}\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}} + \frac{\sqrt{2} A B^{2} \log \left (2 \, A B x + 2 \, A^{2} + 2 \, \sqrt{2 \, A B x + A^{2} - B^{2}} A\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}} - \frac{\sqrt{2} A B^{2} \log \left (2 \, A B x + 2 \, A^{2} - 2 \, \sqrt{2 \, A B x + A^{2} - B^{2}} A\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}}\right )} \sqrt{A}}{2 \, \sqrt{B}} + \frac{\frac{2 \, \sqrt{2} A^{2} B^{3} \arctan \left (\frac{A + \sqrt{2 \, A B x + A^{2} - B^{2}}}{B}\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}} + \frac{2 \, \sqrt{2} A^{2} B^{3} \arctan \left (-\frac{A - \sqrt{2 \, A B x + A^{2} - B^{2}}}{B}\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}} - \frac{\sqrt{2} A^{3} B^{2} \log \left (2 \, A B x + 2 \, A^{2} + 2 \, \sqrt{2 \, A B x + A^{2} - B^{2}} A\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}} + \frac{\sqrt{2} A^{3} B^{2} \log \left (2 \, A B x + 2 \, A^{2} - 2 \, \sqrt{2 \, A B x + A^{2} - B^{2}} A\right )}{{\left (A^{2} + B^{2}\right )} \sqrt{e}}}{2 \, A^{\frac{3}{2}} \sqrt{B}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(B*x + A)/(sqrt(4*e*x + 2*(A^2*e - B^2*e)/(A*B))*(x^2 + 1)),x, algorithm="maxima")

[Out]

1/2*(2*sqrt(2)*A^2*B*arctan((A + sqrt(2*A*B*x + A^2 - B^2))/B)/((A^2 + B^2)*sqrt
(e)) + 2*sqrt(2)*A^2*B*arctan(-(A - sqrt(2*A*B*x + A^2 - B^2))/B)/((A^2 + B^2)*s
qrt(e)) + sqrt(2)*A*B^2*log(2*A*B*x + 2*A^2 + 2*sqrt(2*A*B*x + A^2 - B^2)*A)/((A
^2 + B^2)*sqrt(e)) - sqrt(2)*A*B^2*log(2*A*B*x + 2*A^2 - 2*sqrt(2*A*B*x + A^2 -
B^2)*A)/((A^2 + B^2)*sqrt(e)))*sqrt(A)/sqrt(B) + 1/2*(2*sqrt(2)*A^2*B^3*arctan((
A + sqrt(2*A*B*x + A^2 - B^2))/B)/((A^2 + B^2)*sqrt(e)) + 2*sqrt(2)*A^2*B^3*arct
an(-(A - sqrt(2*A*B*x + A^2 - B^2))/B)/((A^2 + B^2)*sqrt(e)) - sqrt(2)*A^3*B^2*l
og(2*A*B*x + 2*A^2 + 2*sqrt(2*A*B*x + A^2 - B^2)*A)/((A^2 + B^2)*sqrt(e)) + sqrt
(2)*A^3*B^2*log(2*A*B*x + 2*A^2 - 2*sqrt(2*A*B*x + A^2 - B^2)*A)/((A^2 + B^2)*sq
rt(e)))/(A^(3/2)*sqrt(B))

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Fricas [A]  time = 0.298443, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{A B}{e}} \log \left (\frac{A^{2} x^{2} - 4 \, A B x - A^{2} + 2 \, B^{2} + 2 \,{\left (A x - B\right )} \sqrt{-\frac{A B}{e}} \sqrt{\frac{2 \, A B e x +{\left (A^{2} - B^{2}\right )} e}{A B}}}{x^{2} + 1}\right ), \sqrt{2} \sqrt{\frac{A B}{e}} \arctan \left (\frac{A x - B}{\sqrt{\frac{A B}{e}} \sqrt{\frac{2 \, A B e x +{\left (A^{2} - B^{2}\right )} e}{A B}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(B*x + A)/(sqrt(4*e*x + 2*(A^2*e - B^2*e)/(A*B))*(x^2 + 1)),x, algorithm="fricas")

[Out]

[1/2*sqrt(2)*sqrt(-A*B/e)*log((A^2*x^2 - 4*A*B*x - A^2 + 2*B^2 + 2*(A*x - B)*sqr
t(-A*B/e)*sqrt((2*A*B*e*x + (A^2 - B^2)*e)/(A*B)))/(x^2 + 1)), sqrt(2)*sqrt(A*B/
e)*arctan((A*x - B)/(sqrt(A*B/e)*sqrt((2*A*B*e*x + (A^2 - B^2)*e)/(A*B))))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{2} \left (\int \frac{A}{x^{2} \sqrt{\frac{A e}{B} + 2 e x - \frac{B e}{A}} + \sqrt{\frac{A e}{B} + 2 e x - \frac{B e}{A}}}\, dx + \int \frac{B x}{x^{2} \sqrt{\frac{A e}{B} + 2 e x - \frac{B e}{A}} + \sqrt{\frac{A e}{B} + 2 e x - \frac{B e}{A}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(B*x+A)/(x**2+1)/(2*(A**2*e-B**2*e)/A/B+4*e*x)**(1/2),x)

[Out]

sqrt(2)*(Integral(A/(x**2*sqrt(A*e/B + 2*e*x - B*e/A) + sqrt(A*e/B + 2*e*x - B*e
/A)), x) + Integral(B*x/(x**2*sqrt(A*e/B + 2*e*x - B*e/A) + sqrt(A*e/B + 2*e*x -
B*e/A)), x))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \,{\left (B x + A\right )}}{\sqrt{4 \, e x + \frac{2 \,{\left (A^{2} e - B^{2} e\right )}}{A B}}{\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(B*x + A)/(sqrt(4*e*x + 2*(A^2*e - B^2*e)/(A*B))*(x^2 + 1)),x, algorithm="giac")

[Out]

integrate(2*(B*x + A)/(sqrt(4*e*x + 2*(A^2*e - B^2*e)/(A*B))*(x^2 + 1)), x)