3.1466 \(\int \frac{A+B x}{\sqrt{d+e x} \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=440 \[ -\frac{\left (A e-B \left (\sqrt{d^2+e^2}+d\right )\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{d^2+e^2}+d} \sqrt{d+e x}+\sqrt{d^2+e^2}+d+e x\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{\sqrt{d^2+e^2}+d}}+\frac{\left (A e-B \left (\sqrt{d^2+e^2}+d\right )\right ) \log \left (\sqrt{2} \sqrt{\sqrt{d^2+e^2}+d} \sqrt{d+e x}+\sqrt{d^2+e^2}+d+e x\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{\sqrt{d^2+e^2}+d}}+\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{d^2+e^2}+d}-\sqrt{2} \sqrt{d+e x}}{\sqrt{d-\sqrt{d^2+e^2}}}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d-\sqrt{d^2+e^2}}}-\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{d^2+e^2}+d}+\sqrt{2} \sqrt{d+e x}}{\sqrt{d-\sqrt{d^2+e^2}}}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d-\sqrt{d^2+e^2}}} \]

[Out]

((A*e - B*(d - Sqrt[d^2 + e^2]))*ArcTanh[(Sqrt[d + Sqrt[d^2 + e^2]] - Sqrt[2]*Sq
rt[d + e*x])/Sqrt[d - Sqrt[d^2 + e^2]]])/(Sqrt[2]*Sqrt[d^2 + e^2]*Sqrt[d - Sqrt[
d^2 + e^2]]) - ((A*e - B*(d - Sqrt[d^2 + e^2]))*ArcTanh[(Sqrt[d + Sqrt[d^2 + e^2
]] + Sqrt[2]*Sqrt[d + e*x])/Sqrt[d - Sqrt[d^2 + e^2]]])/(Sqrt[2]*Sqrt[d^2 + e^2]
*Sqrt[d - Sqrt[d^2 + e^2]]) - ((A*e - B*(d + Sqrt[d^2 + e^2]))*Log[d + Sqrt[d^2
+ e^2] + e*x - Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2]]*Sqrt[d + e*x]])/(2*Sqrt[2]*Sqrt
[d^2 + e^2]*Sqrt[d + Sqrt[d^2 + e^2]]) + ((A*e - B*(d + Sqrt[d^2 + e^2]))*Log[d
+ Sqrt[d^2 + e^2] + e*x + Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2]]*Sqrt[d + e*x]])/(2*S
qrt[2]*Sqrt[d^2 + e^2]*Sqrt[d + Sqrt[d^2 + e^2]])

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Rubi [A]  time = 1.45985, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (A e-B \left (\sqrt{d^2+e^2}+d\right )\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{d^2+e^2}+d} \sqrt{d+e x}+\sqrt{d^2+e^2}+d+e x\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{\sqrt{d^2+e^2}+d}}+\frac{\left (A e-B \left (\sqrt{d^2+e^2}+d\right )\right ) \log \left (\sqrt{2} \sqrt{\sqrt{d^2+e^2}+d} \sqrt{d+e x}+\sqrt{d^2+e^2}+d+e x\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{\sqrt{d^2+e^2}+d}}+\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{d^2+e^2}+d}-\sqrt{2} \sqrt{d+e x}}{\sqrt{d-\sqrt{d^2+e^2}}}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d-\sqrt{d^2+e^2}}}-\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{d^2+e^2}+d}+\sqrt{2} \sqrt{d+e x}}{\sqrt{d-\sqrt{d^2+e^2}}}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d-\sqrt{d^2+e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

((A*e - B*(d - Sqrt[d^2 + e^2]))*ArcTanh[(Sqrt[d + Sqrt[d^2 + e^2]] - Sqrt[2]*Sq
rt[d + e*x])/Sqrt[d - Sqrt[d^2 + e^2]]])/(Sqrt[2]*Sqrt[d^2 + e^2]*Sqrt[d - Sqrt[
d^2 + e^2]]) - ((A*e - B*(d - Sqrt[d^2 + e^2]))*ArcTanh[(Sqrt[d + Sqrt[d^2 + e^2
]] + Sqrt[2]*Sqrt[d + e*x])/Sqrt[d - Sqrt[d^2 + e^2]]])/(Sqrt[2]*Sqrt[d^2 + e^2]
*Sqrt[d - Sqrt[d^2 + e^2]]) - ((A*e - B*(d + Sqrt[d^2 + e^2]))*Log[d + Sqrt[d^2
+ e^2] + e*x - Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2]]*Sqrt[d + e*x]])/(2*Sqrt[2]*Sqrt
[d^2 + e^2]*Sqrt[d + Sqrt[d^2 + e^2]]) + ((A*e - B*(d + Sqrt[d^2 + e^2]))*Log[d
+ Sqrt[d^2 + e^2] + e*x + Sqrt[2]*Sqrt[d + Sqrt[d^2 + e^2]]*Sqrt[d + e*x]])/(2*S
qrt[2]*Sqrt[d^2 + e^2]*Sqrt[d + Sqrt[d^2 + e^2]])

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Rubi in Sympy [A]  time = 141.943, size = 400, normalized size = 0.91 \[ - \frac{\sqrt{2} \left (A e - B d - B \sqrt{d^{2} + e^{2}}\right ) \log{\left (d + e x - \sqrt{2} \sqrt{d + e x} \sqrt{d + \sqrt{d^{2} + e^{2}}} + \sqrt{d^{2} + e^{2}} \right )}}{4 \sqrt{d + \sqrt{d^{2} + e^{2}}} \sqrt{d^{2} + e^{2}}} + \frac{\sqrt{2} \left (A e - B d - B \sqrt{d^{2} + e^{2}}\right ) \log{\left (d + e x + \sqrt{2} \sqrt{d + e x} \sqrt{d + \sqrt{d^{2} + e^{2}}} + \sqrt{d^{2} + e^{2}} \right )}}{4 \sqrt{d + \sqrt{d^{2} + e^{2}}} \sqrt{d^{2} + e^{2}}} - \frac{\sqrt{2} \left (A e - B d + B \sqrt{d^{2} + e^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{d + e x} - \frac{\sqrt{2} \sqrt{d + \sqrt{d^{2} + e^{2}}}}{2}\right )}{\sqrt{d - \sqrt{d^{2} + e^{2}}}} \right )}}{2 \sqrt{d - \sqrt{d^{2} + e^{2}}} \sqrt{d^{2} + e^{2}}} - \frac{\sqrt{2} \left (A e - B d + B \sqrt{d^{2} + e^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{d + e x} + \frac{\sqrt{2} \sqrt{d + \sqrt{d^{2} + e^{2}}}}{2}\right )}{\sqrt{d - \sqrt{d^{2} + e^{2}}}} \right )}}{2 \sqrt{d - \sqrt{d^{2} + e^{2}}} \sqrt{d^{2} + e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(A*e - B*d - B*sqrt(d**2 + e**2))*log(d + e*x - sqrt(2)*sqrt(d + e*x)*s
qrt(d + sqrt(d**2 + e**2)) + sqrt(d**2 + e**2))/(4*sqrt(d + sqrt(d**2 + e**2))*s
qrt(d**2 + e**2)) + sqrt(2)*(A*e - B*d - B*sqrt(d**2 + e**2))*log(d + e*x + sqrt
(2)*sqrt(d + e*x)*sqrt(d + sqrt(d**2 + e**2)) + sqrt(d**2 + e**2))/(4*sqrt(d + s
qrt(d**2 + e**2))*sqrt(d**2 + e**2)) - sqrt(2)*(A*e - B*d + B*sqrt(d**2 + e**2))
*atanh(sqrt(2)*(sqrt(d + e*x) - sqrt(2)*sqrt(d + sqrt(d**2 + e**2))/2)/sqrt(d -
sqrt(d**2 + e**2)))/(2*sqrt(d - sqrt(d**2 + e**2))*sqrt(d**2 + e**2)) - sqrt(2)*
(A*e - B*d + B*sqrt(d**2 + e**2))*atanh(sqrt(2)*(sqrt(d + e*x) + sqrt(2)*sqrt(d
+ sqrt(d**2 + e**2))/2)/sqrt(d - sqrt(d**2 + e**2)))/(2*sqrt(d - sqrt(d**2 + e**
2))*sqrt(d**2 + e**2))

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Mathematica [C]  time = 0.105374, size = 89, normalized size = 0.2 \[ \frac{i (A+i B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d+i e}}\right )}{\sqrt{d+i e}}-\frac{i (A-i B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d-i e}}\right )}{\sqrt{d-i e}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(A - I*B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - I*e]])/Sqrt[d - I*e] + (I*(A + I*
B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d + I*e]])/Sqrt[d + I*e]

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Maple [B]  time = 0.125, size = 3518, normalized size = 8. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/(d^2+e^2)^(3/2)*e^2*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^
2+e^2)^(1/2))*B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2/(d^2+e^2)^(3/2)*e^3/(2*(d^2+e^2)
^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2
+e^2)^(1/2)-2*d)^(1/2))*A-1/(d^2+e^2)^(1/2)/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan
(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*
B*d-2/(d^2+e^2)/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1
/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^2+1/4/(d^2+e^2)*ln((2*(d
^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^2+e^2)^(1/2))*B*(2*(d^2+e^2)^(1/
2)+2*d)^(1/2)*d+1/(d^2+e^2)^(1/2)*e/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^
2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*A+1/e^2/
(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1
/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^2-1/(d^2+e^2)*e^2/(2*(d^2+e^2)^(1/2)-2*d
)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2
)-2*d)^(1/2))*B+1/(d^2+e^2)^(3/2)/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+
e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^3-1/4/
(d^2+e^2)*e*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^2+e^2)^(1/2)
)*A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)-1/4/e^2*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+
d)^(1/2)-e*x-d-(d^2+e^2)^(1/2))*B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d+1/4/(d^2+e^2)^
(3/2)*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^2+e^2)^(1/2))*B*(2
*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2+1/4/(d^2+e^2)*e*ln(e*x+d+(2*(d^2+e^2)^(1/2)+2*d)
^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2))*A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)+1/4/e^2*ln
(e*x+d+(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2))*B*(2*(d^2+e^
2)^(1/2)+2*d)^(1/2)*d-1/(d^2+e^2)^(1/2)/e/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((
2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*A*
d^2+1/(d^2+e^2)^(3/2)/e/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2
*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*A*d^4-(d^2+e^2)^(1/2
)/e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2
*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d+1/(d^2+e^2)^(1/2)/e^2/(2*(d^2+e^2)
^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2
+e^2)^(1/2)-2*d)^(1/2))*B*d^3+1/(d^2+e^2)/e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arct
an((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)
)*B*d^4+1/4/(d^2+e^2)/e*ln(e*x+d+(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^
2+e^2)^(1/2))*A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2-1/(d^2+e^2)^(3/2)/e/(2*(d^2+e^
2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d
^2+e^2)^(1/2)-2*d)^(1/2))*A*d^4+(d^2+e^2)^(1/2)/e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2
)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)
^(1/2))*B*d-1/(d^2+e^2)^(1/2)/e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+
e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^3-1/(d
^2+e^2)/e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-
2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^4-3/(d^2+e^2)^(3/2)*e/(2*(d^
2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(
2*(d^2+e^2)^(1/2)-2*d)^(1/2))*A*d^2+1/(d^2+e^2)^(3/2)*e^2/(2*(d^2+e^2)^(1/2)-2*d
)^(1/2)*arctan(((2*(d^2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2
)-2*d)^(1/2))*B*d+1/4/(d^2+e^2)^(3/2)/e*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)
^(1/2)-e*x-d-(d^2+e^2)^(1/2))*A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3+1/4/(d^2+e^2)^
(3/2)*e*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^2+e^2)^(1/2))*A*
(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d-1/4/(d^2+e^2)^(3/2)/e*ln(e*x+d+(2*(d^2+e^2)^(1/2
)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2))*A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3-
1/4/(d^2+e^2)^(3/2)*e*ln(e*x+d+(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+
e^2)^(1/2))*A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d-1/4/(d^2+e^2)/e^2*ln(e*x+d+(2*(d^2
+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2))*B*(2*(d^2+e^2)^(1/2)+2*d)^
(1/2)*d^3+3/(d^2+e^2)^(3/2)*e/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1
/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*A*d^2-1/(d^2+e
^2)^(3/2)*e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2*(d^2+e^2)
^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d-1/4/(d^2+e^2)/e*ln((2*(d^2
+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^2+e^2)^(1/2))*A*(2*(d^2+e^2)^(1/2)
+2*d)^(1/2)*d^2+1/(d^2+e^2)^(1/2)/e/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan(((2*(d^
2+e^2)^(1/2)+2*d)^(1/2)-2*(e*x+d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*A*d^2+1/
4/(d^2+e^2)/e^2*ln((2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)-e*x-d-(d^2+e^2)^(
1/2))*B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3-1/4/(d^2+e^2)^(3/2)*ln(e*x+d+(2*(d^2+e
^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2))*B*(2*(d^2+e^2)^(1/2)+2*d)^(1
/2)*d^2-1/(d^2+e^2)^(3/2)/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+
(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^3-1/4/(d^2+e^2
)^(3/2)*e^2*ln(e*x+d+(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2)
)*B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)+1/(d^2+e^2)^(1/2)/(2*(d^2+e^2)^(1/2)-2*d)^(1/2
)*arctan((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)
^(1/2))*B*d+2/(d^2+e^2)/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2
*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^2-1/4/(d^2+e^2)*
ln(e*x+d+(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*(e*x+d)^(1/2)+(d^2+e^2)^(1/2))*B*(2*(d^2+
e^2)^(1/2)+2*d)^(1/2)*d-1/(d^2+e^2)^(1/2)*e/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan
((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*
A-1/e^2/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)
+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2))*B*d^2+1/(d^2+e^2)*e^2/(2*(d^2+e^2)^(
1/2)-2*d)^(1/2)*arctan((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e
^2)^(1/2)-2*d)^(1/2))*B+2/(d^2+e^2)^(3/2)*e^3/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)*arct
an((2*(e*x+d)^(1/2)+(2*(d^2+e^2)^(1/2)+2*d)^(1/2))/(2*(d^2+e^2)^(1/2)-2*d)^(1/2)
)*A

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/(sqrt(e*x + d)*(x^2 + 1)), 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((B*x + A)/(sqrt(e*x + d)*(x^2 + 1)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(sqrt(d + e*x)*(x**2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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