3.15.66 \(\int \frac {A+B x}{\sqrt {d+e x} (1+x^2)} \, dx\) [1466]

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

[Out]

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

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Rubi [A]
time = 0.43, antiderivative size = 440, normalized size of antiderivative = 1.00, 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 = {841, 1183, 648, 632, 212, 642} \begin {gather*} -\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}}} \end {gather*}

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 212

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

Rule 632

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 642

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]

Rule 648

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 841

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 1183

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]

Rubi steps

\begin {align*} \int \frac {A+B x}{\sqrt {d+e x} \left (1+x^2\right )} \, dx &=2 \text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 91, normalized size = 0.21 \begin {gather*} \frac {(-i A+B) \tan ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {-d-i e}}\right )}{\sqrt {-d-i e}}+\frac {(i A+B) \tan ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {-d+i e}}\right )}{\sqrt {-d+i e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1364\) vs. \(2(353)=706\).
time = 0.74, size = 1365, normalized size = 3.10 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,method=_RETURNVERBOSE)

[Out]

1/2/(d^2+e^2)^(3/2)/e^2*(1/2*(A*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e+A*(d^2+e^2)^(1/2)*(2*(d^2+
e^2)^(1/2)+2*d)^(1/2)*e^3-A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3*e-A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d*e^3+B*(d^2+e
^2)^(3/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d-B*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3-B*(d^2+e^2)^(1/2
)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d*e^2-B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e^2-B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*
e^4)*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))+2*(2*A*d^2*e^3+2*A*e^5-2*B*d^3*e^2-
2*B*d*e^4-1/2*(A*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e+A*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)
^(1/2)*e^3-A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3*e-A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d*e^3+B*(d^2+e^2)^(3/2)*(2*(d
^2+e^2)^(1/2)+2*d)^(1/2)*d-B*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3-B*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^
(1/2)+2*d)^(1/2)*d*e^2-B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e^2-B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*e^4)*(2*(d^2+e^
2)^(1/2)+2*d)^(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)))+1/2/(d^2+e^2)^(3/2)/e^2*(1/2*(-A*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2
*e-A*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*e^3+A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3*e+A*(2*(d^2+e^2)^(1
/2)+2*d)^(1/2)*d*e^3-B*(d^2+e^2)^(3/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d+B*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*
d)^(1/2)*d^3+B*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d*e^2+B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e^2+B*(
2*(d^2+e^2)^(1/2)+2*d)^(1/2)*e^4)*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))+2*(2*A
*d^2*e^3+2*A*e^5-2*B*d^3*e^2-2*B*d*e^4+1/2*(-A*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e-A*(d^2+e^2)
^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*e^3+A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3*e+A*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)
*d*e^3-B*(d^2+e^2)^(3/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d+B*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^3+B
*(d^2+e^2)^(1/2)*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d*e^2+B*(2*(d^2+e^2)^(1/2)+2*d)^(1/2)*d^2*e^2+B*(2*(d^2+e^2)^(1
/2)+2*d)^(1/2)*e^4)*(2*(d^2+e^2)^(1/2)+2*d)^(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)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)/((x^2 + 1)*sqrt(x*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7581 vs. \(2 (345) = 690\).
time = 8.49, size = 7581, normalized size = 17.23 \begin {gather*} \text {Too large to display} \end {gather*}

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 + (A^2 - B^2)*
d^3 + 2*A*B*e^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(4*(
A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^3 + (A^8 - 2*A^4*B^4 + B^8)*x*e^3 + 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(x*e + 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 + (A^2 - B^2)*d^3 + 2*A*B*e^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^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*d*x - (A^8 -
2*A^4*B^4 + B^8)*d)*e^2 + 4*((A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^2*x - (A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*d^2)
*e + (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 + (A^2 - B^2)*d^3 + 2*A*B*e^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) - sq
rt(2)*((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(x*e + 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 + (A^2 - B^2)*d^3 + 2*A*B*e^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)) + 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 + (A^2 - B^2)*d^3 + 2*A*B*e^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(4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*d^3 + (A^8 - 2*A^4
*B^4 + B^8)*x*e^3 - 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*(...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\sqrt {d + e x} \left (x^{2} + 1\right )}\, dx \end {gather*}

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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Mupad [B]
time = 2.17, size = 1244, normalized size = 2.83 \begin {gather*} -\mathrm {atan}\left (\frac {\left (\left (32\,B\,d\,e^2-32\,A\,e^3+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}+\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\,1{}\mathrm {i}+\left (\left (32\,A\,e^3-32\,B\,d\,e^2+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}+\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\,1{}\mathrm {i}}{16\,B^3\,e^2+\left (\left (32\,A\,e^3-32\,B\,d\,e^2+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}+\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}-\left (\left (32\,B\,d\,e^2-32\,A\,e^3+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}+\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}+16\,A^2\,B\,e^2}\right )\,\sqrt {\frac {-A^2\,1{}\mathrm {i}+2\,A\,B+B^2\,1{}\mathrm {i}}{4\,\left (-e+d\,1{}\mathrm {i}\right )}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\left (\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}+\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,\left (32\,B\,d\,e^2-32\,A\,e^3+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\right )\right )\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,1{}\mathrm {i}+\left (\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}+\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,\left (32\,A\,e^3-32\,B\,d\,e^2+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\right )\right )\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,1{}\mathrm {i}}{16\,B^3\,e^2-\left (\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}+\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,\left (32\,B\,d\,e^2-32\,A\,e^3+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\right )\right )\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}+\left (\left (16\,A^2\,e^2-16\,B^2\,e^2\right )\,\sqrt {d+e\,x}+\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,\left (32\,A\,e^3-32\,B\,d\,e^2+64\,d\,e^2\,\sqrt {d+e\,x}\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\right )\right )\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}+16\,A^2\,B\,e^2}\right )\,\sqrt {\frac {-A^2+A\,B\,2{}\mathrm {i}+B^2}{4\,\left (d-e\,1{}\mathrm {i}\right )}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan((((32*B*d*e^2 - 32*A*e^3 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*(
(B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + (16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2))*((B^2*1i - A^2*1i
+ 2*A*B)/(4*(d*1i - e)))^(1/2)*1i + ((32*A*e^3 - 32*B*d*e^2 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2*1i + 2*
A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + (16*A^2*e^2 - 16*B^2*e^2)*(d +
e*x)^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2)*1i)/(((32*A*e^3 - 32*B*d*e^2 + 64*d*e^2*(d + e*x)
^(1/2)*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) + (1
6*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2) - ((32*B*d*e^2 - 32*
A*e^3 + 64*d*e^2*(d + e*x)^(1/2)*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/
(4*(d*1i - e)))^(1/2) + (16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^
(1/2) + 16*B^3*e^2 + 16*A^2*B*e^2))*((B^2*1i - A^2*1i + 2*A*B)/(4*(d*1i - e)))^(1/2)*2i - atan((((16*A^2*e^2 -
16*B^2*e^2)*(d + e*x)^(1/2) + ((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)*(32*B*d*e^2 - 32*A*e^3 + 64*d*e^2*(
d + e*x)^(1/2)*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)))*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)*1i +
((16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2) + ((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)*(32*A*e^3 - 32*B*d*e^
2 + 64*d*e^2*(d + e*x)^(1/2)*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)))*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)
))^(1/2)*1i)/(16*B^3*e^2 - ((16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2) + ((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^
(1/2)*(32*B*d*e^2 - 32*A*e^3 + 64*d*e^2*(d + e*x)^(1/2)*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)))*((B^2 -
A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2) + ((16*A^2*e^2 - 16*B^2*e^2)*(d + e*x)^(1/2) + ((B^2 - A^2 + A*B*2i)/(4*(d
- e*1i)))^(1/2)*(32*A*e^3 - 32*B*d*e^2 + 64*d*e^2*(d + e*x)^(1/2)*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)
))*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2) + 16*A^2*B*e^2))*((B^2 - A^2 + A*B*2i)/(4*(d - e*1i)))^(1/2)*2i

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