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

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]*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]))*ArcTanh[(Sqr
t[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*Sqrt[2]*Sqrt[d^2 + e^
2]*Sqrt[d + Sqrt[d^2 + e^2]])

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Rubi [A]  time = 0.655713, 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, Rules used = {827, 1169, 634, 618, 206, 628} \[ -\frac{\left (A e-B \left (\sqrt{d^2+e^2}+d\right )\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{d^2+e^2}+d} \sqrt{d+e x}+\sqrt{d^2+e^2}+d+e x\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{\sqrt{d^2+e^2}+d}}+\frac{\left (A e-B \left (\sqrt{d^2+e^2}+d\right )\right ) \log \left (\sqrt{2} \sqrt{\sqrt{d^2+e^2}+d} \sqrt{d+e x}+\sqrt{d^2+e^2}+d+e x\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{\sqrt{d^2+e^2}+d}}+\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{d^2+e^2}+d}-\sqrt{2} \sqrt{d+e x}}{\sqrt{d-\sqrt{d^2+e^2}}}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d-\sqrt{d^2+e^2}}}-\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{d^2+e^2}+d}+\sqrt{2} \sqrt{d+e x}}{\sqrt{d-\sqrt{d^2+e^2}}}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d-\sqrt{d^2+e^2}}} \]

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]*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]))*ArcTanh[(Sqr
t[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*Sqrt[2]*Sqrt[d^2 + e^
2]*Sqrt[d + Sqrt[d^2 + e^2]])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{A+B x}{\sqrt{d+e x} \left (1+x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{-B d+A e+B x^2}{d^2+e^2-2 d x^2+x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} (-B d+A e) \sqrt{d+\sqrt{d^2+e^2}}-\left (-B d+A e-B \sqrt{d^2+e^2}\right ) x}{\sqrt{d^2+e^2}-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} x+x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{2} \sqrt{d^2+e^2} \sqrt{d+\sqrt{d^2+e^2}}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} (-B d+A e) \sqrt{d+\sqrt{d^2+e^2}}+\left (-B d+A e-B \sqrt{d^2+e^2}\right ) x}{\sqrt{d^2+e^2}+\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} x+x^2} \, dx,x,\sqrt{d+e x}\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 ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d^2+e^2}-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} x+x^2} \, dx,x,\sqrt{d+e x}\right )}{2 \sqrt{d^2+e^2}}+\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d^2+e^2}+\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} x+x^2} \, dx,x,\sqrt{d+e x}\right )}{2 \sqrt{d^2+e^2}}-\frac{\left (A e-B \left (d+\sqrt{d^2+e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}}+2 x}{\sqrt{d^2+e^2}-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} x+x^2} \, dx,x,\sqrt{d+e x}\right )}{2 \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 ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}}+2 x}{\sqrt{d^2+e^2}+\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} x+x^2} \, dx,x,\sqrt{d+e x}\right )}{2 \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 ) \log \left (d+\sqrt{d^2+e^2}+e x-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} \sqrt{d+e x}\right )}{2 \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 ) \log \left (d+\sqrt{d^2+e^2}+e x+\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} \sqrt{d+e x}\right )}{2 \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 ) \operatorname{Subst}\left (\int \frac{1}{2 \left (d-\sqrt{d^2+e^2}\right )-x^2} \, dx,x,-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}}+2 \sqrt{d+e x}\right )}{\sqrt{d^2+e^2}}-\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (d-\sqrt{d^2+e^2}\right )-x^2} \, dx,x,\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}}+2 \sqrt{d+e x}\right )}{\sqrt{d^2+e^2}}\\ &=\frac{\left (A e-B \left (d-\sqrt{d^2+e^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+\sqrt{d^2+e^2}}-\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{d+\sqrt{d^2+e^2}}+\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 ) \log \left (d+\sqrt{d^2+e^2}+e x-\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} \sqrt{d+e x}\right )}{2 \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 ) \log \left (d+\sqrt{d^2+e^2}+e x+\sqrt{2} \sqrt{d+\sqrt{d^2+e^2}} \sqrt{d+e x}\right )}{2 \sqrt{2} \sqrt{d^2+e^2} \sqrt{d+\sqrt{d^2+e^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0968584, 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 = 0.091, size = 3518, normalized size = 8. \begin{align*} \text{output too large to display} \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 25.3628, size = 15524, normalized size = 35.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*(d^2 + e^2)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2
*d^2*e^2 + e^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 +
(A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*
B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))^(3/4)*arctan((sqrt(2)*sqrt(e*x
+ d)*((2*(A^3*B^2 + A*B^4)*d^6 - (3*A^4*B + 2*A^2*B^3 - B^5)*d^5*e + (A^5 + 4*A^3*B^2 + 3*A*B^4)*d^4*e^2 - 2*(
3*A^4*B + 2*A^2*B^3 - B^5)*d^3*e^3 + 2*(A^5 + A^3*B^2)*d^2*e^4 - (3*A^4*B + 2*A^2*B^3 - B^5)*d*e^5 + (A^5 - A*
B^4)*e^6)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*
sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)) + (2*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*d^5 - (A^6*B + A^4*B^3 - A^2*B^5
- B^7)*d^4*e + 4*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*d^3*e^2 - 2*(A^6*B + A^4*B^3 - A^2*B^5 - B^7)*d^2*e^3 + 2*(A^5*
B^2 + 2*A^3*B^4 + A*B^6)*d*e^4 - (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*e^5)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3
)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^
2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4
)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4
)/(d^2 + e^2))^(3/4) - sqrt(2)*sqrt(4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B
^7)*d^2*e + (A^8 - 2*A^4*B^4 + B^8)*d*e^2 + sqrt(2)*(4*(A^4*B^3 + A^2*B^5)*d^3 - 4*(2*A^5*B^2 + A^3*B^4 - A*B^
6)*d^2*e + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*d*e^2 - (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*e^3 + (4*A^2*B^3*d^
4 - 4*(A^3*B^2 - A*B^4)*d^3*e + (A^4*B + 2*A^2*B^3 + B^5)*d^2*e^2 - 4*(A^3*B^2 - A*B^4)*d*e^3 + (A^4*B - 2*A^2
*B^3 + B^5)*e^4)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))*sqrt(e*x + d)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 +
(A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^
2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^
2*B^2 + B^4)/(d^2 + e^2))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^2*e - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*
B^7)*d*e^2 + (A^8 - 2*A^4*B^4 + B^8)*e^3)*x + (4*(A^4*B^2 + A^2*B^4)*d^4 - 4*(A^5*B - A*B^5)*d^3*e + (A^6 + 3*
A^4*B^2 + 3*A^2*B^4 + B^6)*d^2*e^2 - 4*(A^5*B - A*B^5)*d*e^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*e^4)*sqrt((A^4
+ 2*A^2*B^2 + B^4)/(d^2 + e^2)))*((B*d^5 - A*d^4*e + 2*B*d^3*e^2 - 2*A*d^2*e^3 + B*d*e^4 - A*e^5)*sqrt((4*A^2*
B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*sqrt((A^4 + 2*A^2*B^2
+ B^4)/(d^2 + e^2)) + ((A^2*B + B^3)*d^4 + 2*(A^2*B + B^3)*d^2*e^2 + (A^2*B + B^3)*e^4)*sqrt((4*A^2*B^2*d^2 -
4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d
^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 +
2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 +
2*A^2*B^2 + B^4)/(d^2 + e^2))^(3/4) + (2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*d^5 - (A^8 + 2*A^6*B^2 - 2*A
^2*B^6 - B^8)*d^4*e + 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*d^3*e^2 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8
)*d^2*e^3 + 2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*d*e^4 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*e^5)*sqrt((4
*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*sqrt((A^4 + 2*A^2
*B^2 + B^4)/(d^2 + e^2)) + (2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*d^4 - (A^10 + 3*A^8*B^2 + 2*
A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B^10)*d^3*e + 2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*d^2*e^2
- (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B^10)*d*e^3)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)
*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4)))/(4*(A^10*B^2 + 4*A^8*B^4 + 6*A^6*B^6 + 4*A^4*B^8
+ A^2*B^10)*d^2*e - 4*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*d*e^2 + (A^12 + 2*A^1
0*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*e^3)) + 4*sqrt(2)*(d^2 + e^2)*sqrt((4*A^2*B^2*d^2 -
4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d
^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 +
2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 +
2*A^2*B^2 + B^4)/(d^2 + e^2))^(3/4)*arctan((sqrt(2)*sqrt(e*x + d)*((2*(A^3*B^2 + A*B^4)*d^6 - (3*A^4*B + 2*A^
2*B^3 - B^5)*d^5*e + (A^5 + 4*A^3*B^2 + 3*A*B^4)*d^4*e^2 - 2*(3*A^4*B + 2*A^2*B^3 - B^5)*d^3*e^3 + 2*(A^5 + A^
3*B^2)*d^2*e^4 - (3*A^4*B + 2*A^2*B^3 - B^5)*d*e^5 + (A^5 - A*B^4)*e^6)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3
)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)) + (2*(
A^5*B^2 + 2*A^3*B^4 + A*B^6)*d^5 - (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*d^4*e + 4*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*d
^3*e^2 - 2*(A^6*B + A^4*B^3 - A^2*B^5 - B^7)*d^2*e^3 + 2*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*d*e^4 - (A^6*B + A^4*B^
3 - A^2*B^5 - B^7)*e^5)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^
2*e^2 + e^4)))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A
^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^
3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))^(3/4) - sqrt(2)*sqrt(4*(A^6*B^2 +
2*A^4*B^4 + A^2*B^6)*d^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*d^2*e + (A^8 - 2*A^4*B^4 + B^8)*d*e^2 - sqrt
(2)*(4*(A^4*B^3 + A^2*B^5)*d^3 - 4*(2*A^5*B^2 + A^3*B^4 - A*B^6)*d^2*e + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)
*d*e^2 - (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*e^3 + (4*A^2*B^3*d^4 - 4*(A^3*B^2 - A*B^4)*d^3*e + (A^4*B + 2*A^2*B
^3 + B^5)*d^2*e^2 - 4*(A^3*B^2 - A*B^4)*d*e^3 + (A^4*B - 2*A^2*B^3 + B^5)*e^4)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d
^2 + e^2)))*sqrt(e*x + d)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A
*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A
^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))^(1/4) + (4*(A^6*B^2 +
2*A^4*B^4 + A^2*B^6)*d^2*e - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*d*e^2 + (A^8 - 2*A^4*B^4 + B^8)*e^3)*x + (4
*(A^4*B^2 + A^2*B^4)*d^4 - 4*(A^5*B - A*B^5)*d^3*e + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*d^2*e^2 - 4*(A^5*B -
A*B^5)*d*e^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*e^4)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))*((B*d^5 - A*d^4
*e + 2*B*d^3*e^2 - 2*A*d^2*e^3 + B*d*e^4 - A*e^5)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B
^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)) + ((A^2*B + B^3)*d^4 + 2*(A^
2*B + B^3)*d^2*e^2 + (A^2*B + B^3)*e^4)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*
e^2)/(d^4 + 2*d^2*e^2 + e^4)))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e
+ 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 -
4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))^(3/4) - (2*(A^7*B
+ 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*d^5 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*d^4*e + 4*(A^7*B + 3*A^5*B^3 + 3*A
^3*B^5 + A*B^7)*d^3*e^2 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*d^2*e^3 + 2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A
*B^7)*d*e^4 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*e^5)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*
A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2*e^2 + e^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)) - (2*(A^9*B + 4*A^7*B^3
+ 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*d^4 - (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B^10)*d^3*e +
2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*d^2*e^2 - (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*
A^2*B^8 - B^10)*d*e^3)*sqrt((4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2)/(d^4 + 2*d^2
*e^2 + e^4)))/(4*(A^10*B^2 + 4*A^8*B^4 + 6*A^6*B^6 + 4*A^4*B^8 + A^2*B^10)*d^2*e - 4*(A^11*B + 3*A^9*B^3 + 2*A
^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*d*e^2 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^
10 + B^12)*e^3)) - sqrt(2)*(A^4 + 2*A^2*B^2 + B^4 - (2*A*B*e + (A^2 - B^2)*d)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^
2 + e^2)))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 -
B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d
*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))^(1/4)*log(4*(A^6*B^2 + 2*A^4*B^4 + A^
2*B^6)*d^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*d^2*e + (A^8 - 2*A^4*B^4 + B^8)*d*e^2 + sqrt(2)*(4*(A^4*B^3
+ A^2*B^5)*d^3 - 4*(2*A^5*B^2 + A^3*B^4 - A*B^6)*d^2*e + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*d*e^2 - (A^7 -
A^5*B^2 - A^3*B^4 + A*B^6)*e^3 + (4*A^2*B^3*d^4 - 4*(A^3*B^2 - A*B^4)*d^3*e + (A^4*B + 2*A^2*B^3 + B^5)*d^2*e
^2 - 4*(A^3*B^2 - A*B^4)*d*e^3 + (A^4*B - 2*A^2*B^3 + B^5)*e^4)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))*sqr
t(e*x + d)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 + (A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 -
B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d
*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2
*B^6)*d^2*e - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*d*e^2 + (A^8 - 2*A^4*B^4 + B^8)*e^3)*x + (4*(A^4*B^2 + A^2
*B^4)*d^4 - 4*(A^5*B - A*B^5)*d^3*e + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*d^2*e^2 - 4*(A^5*B - A*B^5)*d*e^3 +
(A^6 - A^4*B^2 - A^2*B^4 + B^6)*e^4)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2))) + sqrt(2)*(A^4 + 2*A^2*B^2 + B
^4 - (2*A*B*e + (A^2 - B^2)*d)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 +
(A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^
2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^
2*B^2 + B^4)/(d^2 + e^2))^(1/4)*log(4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B
^7)*d^2*e + (A^8 - 2*A^4*B^4 + B^8)*d*e^2 - sqrt(2)*(4*(A^4*B^3 + A^2*B^5)*d^3 - 4*(2*A^5*B^2 + A^3*B^4 - A*B^
6)*d^2*e + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*d*e^2 - (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*e^3 + (4*A^2*B^3*d^
4 - 4*(A^3*B^2 - A*B^4)*d^3*e + (A^4*B + 2*A^2*B^3 + B^5)*d^2*e^2 - 4*(A^3*B^2 - A*B^4)*d*e^3 + (A^4*B - 2*A^2
*B^3 + B^5)*e^4)*sqrt((A^4 + 2*A^2*B^2 + B^4)/(d^2 + e^2)))*sqrt(e*x + d)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*d^2 +
(A^4 + 2*A^2*B^2 + B^4)*e^2 + (2*A*B*d^2*e + 2*A*B*e^3 + (A^2 - B^2)*d^3 + (A^2 - B^2)*d*e^2)*sqrt((A^4 + 2*A^
2*B^2 + B^4)/(d^2 + e^2)))/(4*A^2*B^2*d^2 - 4*(A^3*B - A*B^3)*d*e + (A^4 - 2*A^2*B^2 + B^4)*e^2))*((A^4 + 2*A^
2*B^2 + B^4)/(d^2 + e^2))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^2*e - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*
B^7)*d*e^2 + (A^8 - 2*A^4*B^4 + B^8)*e^3)*x + (4*(A^4*B^2 + A^2*B^4)*d^4 - 4*(A^5*B - A*B^5)*d^3*e + (A^6 + 3*
A^4*B^2 + 3*A^2*B^4 + B^6)*d^2*e^2 - 4*(A^5*B - A*B^5)*d*e^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*e^4)*sqrt((A^4
+ 2*A^2*B^2 + B^4)/(d^2 + e^2))))/(A^4 + 2*A^2*B^2 + B^4)

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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