3.1.23 \(\int \frac {A+B x}{(a+b x+c x^2) \sqrt {d+f x^2}} \, dx\)

Optimal. Leaf size=302 \[ \frac {\left (-B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 c d-f x \left (b-\sqrt {b^2-4 a c}\right )}{\sqrt {2} \sqrt {d+f x^2} \sqrt {b f \left (b-\sqrt {b^2-4 a c}\right )-2 a c f+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b f \left (b-\sqrt {b^2-4 a c}\right )-2 a c f+2 c^2 d}}+\frac {\left (2 A c-B \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {2 c d-f x \left (\sqrt {b^2-4 a c}+b\right )}{\sqrt {2} \sqrt {d+f x^2} \sqrt {b f \left (\sqrt {b^2-4 a c}+b\right )-2 a c f+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b f \left (\sqrt {b^2-4 a c}+b\right )-2 a c f+2 c^2 d}} \]

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Rubi [A]  time = 0.84, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1034, 725, 206} \begin {gather*} \frac {\left (-B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 c d-f x \left (b-\sqrt {b^2-4 a c}\right )}{\sqrt {2} \sqrt {d+f x^2} \sqrt {b f \left (b-\sqrt {b^2-4 a c}\right )-2 a c f+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b f \left (b-\sqrt {b^2-4 a c}\right )-2 a c f+2 c^2 d}}+\frac {\left (2 A c-B \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {2 c d-f x \left (\sqrt {b^2-4 a c}+b\right )}{\sqrt {2} \sqrt {d+f x^2} \sqrt {b f \left (\sqrt {b^2-4 a c}+b\right )-2 a c f+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b f \left (\sqrt {b^2-4 a c}+b\right )-2 a c f+2 c^2 d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d - (b - Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a
*c*f + b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - 2*a*c*f + b*(
b - Sqrt[b^2 - 4*a*c])*f]) + ((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(2*c*d - (b + Sqrt[b^2 - 4*a*c])*f*x
)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]
*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f])

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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1034

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c
*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]

Rubi steps

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

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Mathematica [A]  time = 0.43, size = 283, normalized size = 0.94 \begin {gather*} \frac {\sqrt {2} \left (-\frac {\left (B \sqrt {b^2-4 a c}+2 A c-b B\right ) \tanh ^{-1}\left (\frac {f x \left (\sqrt {b^2-4 a c}-b\right )+2 c d}{\sqrt {d+f x^2} \sqrt {2 b f \left (b-\sqrt {b^2-4 a c}\right )-4 a c f+4 c^2 d}}\right )}{2 \sqrt {b f \left (b-\sqrt {b^2-4 a c}\right )-2 a c f+2 c^2 d}}-\frac {\left (B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 c d-f x \left (\sqrt {b^2-4 a c}+b\right )}{\sqrt {d+f x^2} \sqrt {2 b f \left (\sqrt {b^2-4 a c}+b\right )-4 a c f+4 c^2 d}}\right )}{2 \sqrt {b f \left (\sqrt {b^2-4 a c}+b\right )-2 a c f+2 c^2 d}}\right )}{\sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

(Sqrt[2]*(-1/2*((-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[4*
c^2*d - 4*a*c*f + 2*b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/Sqrt[2*c^2*d - 2*a*c*f + b*(b - Sqrt[b^2 -
4*a*c])*f] - ((b*B - 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d - (b + Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[4*c^2*d
- 4*a*c*f + 2*b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(2*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4
*a*c])*f])))/Sqrt[b^2 - 4*a*c]

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IntegrateAlgebraic [C]  time = 0.44, size = 195, normalized size = 0.65 \begin {gather*} -\text {RootSum}\left [\text {$\#$1}^4 c-2 \text {$\#$1}^3 b \sqrt {f}+4 \text {$\#$1}^2 a f-2 \text {$\#$1}^2 c d+2 \text {$\#$1} b d \sqrt {f}+c d^2\&,\frac {\text {$\#$1}^2 (-B) \log \left (-\text {$\#$1}+\sqrt {d+f x^2}-\sqrt {f} x\right )+2 \text {$\#$1} A \sqrt {f} \log \left (-\text {$\#$1}+\sqrt {d+f x^2}-\sqrt {f} x\right )+B d \log \left (-\text {$\#$1}+\sqrt {d+f x^2}-\sqrt {f} x\right )}{2 \text {$\#$1}^3 c-3 \text {$\#$1}^2 b \sqrt {f}+4 \text {$\#$1} a f-2 \text {$\#$1} c d+b d \sqrt {f}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

-RootSum[c*d^2 + 2*b*d*Sqrt[f]*#1 - 2*c*d*#1^2 + 4*a*f*#1^2 - 2*b*Sqrt[f]*#1^3 + c*#1^4 & , (B*d*Log[-(Sqrt[f]
*x) + Sqrt[d + f*x^2] - #1] + 2*A*Sqrt[f]*Log[-(Sqrt[f]*x) + Sqrt[d + f*x^2] - #1]*#1 - B*Log[-(Sqrt[f]*x) + S
qrt[d + f*x^2] - #1]*#1^2)/(b*d*Sqrt[f] - 2*c*d*#1 + 4*a*f*#1 - 3*b*Sqrt[f]*#1^2 + 2*c*#1^3) & ]

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fricas [B]  time = 60.52, size = 8977, normalized size = 29.73

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*sqrt(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a + A*B*b)*c)*d + (2*B^2*a^2 - 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c
)*f + ((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 -
4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B
^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f +
(b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b
^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2))*
log((2*(B^4*a*b^2 - A*B^3*b^3 - 2*A^3*B*b*c^2 - (2*A*B^3*a*b - 3*A^2*B^2*b^2)*c)*d^2 + 2*(2*A*B^3*a^2*b - 3*A^
2*B^2*a*b^2 + A^3*B*b^3 + (2*A^3*B*a*b - A^4*b^2)*c)*d*f + sqrt(2)*((B^3*b^4 - 8*A^2*B*a*c^3 + 2*(6*A*B^2*a*b
+ A^2*B*b^2)*c^2 - (4*B^3*a*b^2 + 3*A*B^2*b^3)*c)*d^2 + (3*A*B^2*a*b^3 - A^2*B*b^4 + 4*(4*A^2*B*a^2 - A^3*a*b)
*c^2 - (12*A*B^2*a^2*b - A^3*b^3)*c)*d*f + (2*A^2*B*a^2*b^2 - A^3*a*b^3 - 4*(2*A^2*B*a^3 - A^3*a^2*b)*c)*f^2 -
((B*b^4*c^2 + 4*(2*B*a^2 + A*a*b)*c^4 - (6*B*a*b^2 + A*b^3)*c^3)*d^3 + (B*b^6 - 4*(6*B*a^3 + A*a^2*b)*c^3 + (
22*B*a^2*b^2 + 5*A*a*b^3)*c^2 - (8*B*a*b^4 + A*b^5)*c)*d^2*f + (3*B*a^2*b^4 - A*a*b^5 + 4*(6*B*a^4 - A*a^3*b)*
c^2 - (18*B*a^3*b^2 - 5*A*a^2*b^3)*c)*d*f^2 + (2*B*a^4*b^2 - A*a^3*b^3 - 4*(2*B*a^5 - A*a^4*b)*c)*f^3)*sqrt(((
B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f +
(4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)
*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3
+ (a^4*b^2 - 4*a^5*c)*f^4)))*sqrt(f*x^2 + d)*sqrt(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a + A*B*b)*c)*d + (2*B^2*a^2
- 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c)*f + ((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^
2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*
B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 -
6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3
*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2
)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)) - 4*((B^4*a^2*b - A*B^3*a*b^2 - 2*A^3*B*a*c^2 - (2*A*B^3*a^2 - 3*A^2*B^2*a*b
)*c)*d*f + (2*A*B^3*a^3 - 3*A^2*B^2*a^2*b + A^3*B*a*b^2 + (2*A^3*B*a^2 - A^4*a*b)*c)*f^2)*x + 2*((4*A^2*a*c^4
+ (4*B^2*a^2 - 4*A*B*a*b - A^2*b^2)*c^3 - (B^2*a*b^2 - A*B*b^3)*c^2)*d^3 - (B^2*a*b^4 - A*B*b^5 + 8*A^2*a^2*c^
3 + 2*(4*B^2*a^3 - 4*A*B*a^2*b - 3*A^2*a*b^2)*c^2 - (6*B^2*a^2*b^2 - 6*A*B*a*b^3 - A^2*b^4)*c)*d^2*f - (B^2*a^
3*b^2 - A*B*a^2*b^3 - 4*A^2*a^3*c^2 - (4*B^2*a^4 - 4*A*B*a^3*b - A^2*a^2*b^2)*c)*d*f^2)*sqrt(((B^4*b^2 - 4*A*B
^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2
- 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 -
8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4
*a^5*c)*f^4)))/x) - 1/4*sqrt(2)*sqrt(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a + A*B*b)*c)*d + (2*B^2*a^2 - 2*A*B*a*b +
A^2*b^2 - 2*A^2*a*c)*f + ((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f
^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*
b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 +
8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4
*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b
^2 - 4*a^3*c)*f^2))*log((2*(B^4*a*b^2 - A*B^3*b^3 - 2*A^3*B*b*c^2 - (2*A*B^3*a*b - 3*A^2*B^2*b^2)*c)*d^2 + 2*(
2*A*B^3*a^2*b - 3*A^2*B^2*a*b^2 + A^3*B*b^3 + (2*A^3*B*a*b - A^4*b^2)*c)*d*f - sqrt(2)*((B^3*b^4 - 8*A^2*B*a*c
^3 + 2*(6*A*B^2*a*b + A^2*B*b^2)*c^2 - (4*B^3*a*b^2 + 3*A*B^2*b^3)*c)*d^2 + (3*A*B^2*a*b^3 - A^2*B*b^4 + 4*(4*
A^2*B*a^2 - A^3*a*b)*c^2 - (12*A*B^2*a^2*b - A^3*b^3)*c)*d*f + (2*A^2*B*a^2*b^2 - A^3*a*b^3 - 4*(2*A^2*B*a^3 -
A^3*a^2*b)*c)*f^2 - ((B*b^4*c^2 + 4*(2*B*a^2 + A*a*b)*c^4 - (6*B*a*b^2 + A*b^3)*c^3)*d^3 + (B*b^6 - 4*(6*B*a^
3 + A*a^2*b)*c^3 + (22*B*a^2*b^2 + 5*A*a*b^3)*c^2 - (8*B*a*b^4 + A*b^5)*c)*d^2*f + (3*B*a^2*b^4 - A*a*b^5 + 4*
(6*B*a^4 - A*a^3*b)*c^2 - (18*B*a^3*b^2 - 5*A*a^2*b^3)*c)*d*f^2 + (2*B*a^4*b^2 - A*a^3*b^3 - 4*(2*B*a^5 - A*a^
4*b)*c)*f^3)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a
- A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*
b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*
c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))*sqrt(f*x^2 + d)*sqrt(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a + A*B*
b)*c)*d + (2*B^2*a^2 - 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c)*f + ((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^
2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2
*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5
)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2
+ 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6
*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)) - 4*((B^4*a^2*b - A*B^3*a*b^2 - 2*A^3*B*a*c^2 - (2*A*B^3
*a^2 - 3*A^2*B^2*a*b)*c)*d*f + (2*A*B^3*a^3 - 3*A^2*B^2*a^2*b + A^3*B*a*b^2 + (2*A^3*B*a^2 - A^4*a*b)*c)*f^2)*
x + 2*((4*A^2*a*c^4 + (4*B^2*a^2 - 4*A*B*a*b - A^2*b^2)*c^3 - (B^2*a*b^2 - A*B*b^3)*c^2)*d^3 - (B^2*a*b^4 - A*
B*b^5 + 8*A^2*a^2*c^3 + 2*(4*B^2*a^3 - 4*A*B*a^2*b - 3*A^2*a*b^2)*c^2 - (6*B^2*a^2*b^2 - 6*A*B*a*b^3 - A^2*b^4
)*c)*d^2*f - (B^2*a^3*b^2 - A*B*a^2*b^3 - 4*A^2*a^3*c^2 - (4*B^2*a^4 - 4*A*B*a^3*b - A^2*a^2*b^2)*c)*d*f^2)*sq
rt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*
d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2
*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*
d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/x) + 1/4*sqrt(2)*sqrt(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a + A*B*b)*c)*d + (2*B
^2*a^2 - 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c)*f - ((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (
a^2*b^2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(
2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4
*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 -
6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a
^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2))*log((2*(B^4*a*b^2 - A*B^3*b^3 - 2*A^3*B*b*c^2 - (2*A*B^3*a*b - 3*A^2*B
^2*b^2)*c)*d^2 + 2*(2*A*B^3*a^2*b - 3*A^2*B^2*a*b^2 + A^3*B*b^3 + (2*A^3*B*a*b - A^4*b^2)*c)*d*f + sqrt(2)*((B
^3*b^4 - 8*A^2*B*a*c^3 + 2*(6*A*B^2*a*b + A^2*B*b^2)*c^2 - (4*B^3*a*b^2 + 3*A*B^2*b^3)*c)*d^2 + (3*A*B^2*a*b^3
- A^2*B*b^4 + 4*(4*A^2*B*a^2 - A^3*a*b)*c^2 - (12*A*B^2*a^2*b - A^3*b^3)*c)*d*f + (2*A^2*B*a^2*b^2 - A^3*a*b^
3 - 4*(2*A^2*B*a^3 - A^3*a^2*b)*c)*f^2 + ((B*b^4*c^2 + 4*(2*B*a^2 + A*a*b)*c^4 - (6*B*a*b^2 + A*b^3)*c^3)*d^3
+ (B*b^6 - 4*(6*B*a^3 + A*a^2*b)*c^3 + (22*B*a^2*b^2 + 5*A*a*b^3)*c^2 - (8*B*a*b^4 + A*b^5)*c)*d^2*f + (3*B*a^
2*b^4 - A*a*b^5 + 4*(6*B*a^4 - A*a^3*b)*c^2 - (18*B*a^3*b^2 - 5*A*a^2*b^3)*c)*d*f^2 + (2*B*a^4*b^2 - A*a^3*b^3
- 4*(2*B*a^5 - A*a^4*b)*c)*f^3)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*
b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4
+ 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(
a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))*sqrt(f*x^2 + d)*sqrt(((B^2*b^2 + 2*A^2*c
^2 - 2*(B^2*a + A*B*b)*c)*d + (2*B^2*a^2 - 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c)*f - ((b^2*c^2 - 4*a*c^3)*d^2 + (b^
4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 +
2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)
/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 -
24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a
*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)) - 4*((B^4*a^2*b - A*B^3*a*b^2 - 2*A^
3*B*a*c^2 - (2*A*B^3*a^2 - 3*A^2*B^2*a*b)*c)*d*f + (2*A*B^3*a^3 - 3*A^2*B^2*a^2*b + A^3*B*a*b^2 + (2*A^3*B*a^2
- A^4*a*b)*c)*f^2)*x - 2*((4*A^2*a*c^4 + (4*B^2*a^2 - 4*A*B*a*b - A^2*b^2)*c^3 - (B^2*a*b^2 - A*B*b^3)*c^2)*d
^3 - (B^2*a*b^4 - A*B*b^5 + 8*A^2*a^2*c^3 + 2*(4*B^2*a^3 - 4*A*B*a^2*b - 3*A^2*a*b^2)*c^2 - (6*B^2*a^2*b^2 - 6
*A*B*a*b^3 - A^2*b^4)*c)*d^2*f - (B^2*a^3*b^2 - A*B*a^2*b^3 - 4*A^2*a^3*c^2 - (4*B^2*a^4 - 4*A*B*a^3*b - A^2*a
^2*b^2)*c)*d*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*
B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 -
6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3
*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/x) - 1/4*sqrt(2)*sqrt(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a
+ A*B*b)*c)*d + (2*B^2*a^2 - 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c)*f - ((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c
+ 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b
- A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4
*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^
2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b
^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2))*log((2*(B^4*a*b^2 - A*B^3*b^3 - 2*A^3*B*b*c^2 - (2
*A*B^3*a*b - 3*A^2*B^2*b^2)*c)*d^2 + 2*(2*A*B^3*a^2*b - 3*A^2*B^2*a*b^2 + A^3*B*b^3 + (2*A^3*B*a*b - A^4*b^2)*
c)*d*f - sqrt(2)*((B^3*b^4 - 8*A^2*B*a*c^3 + 2*(6*A*B^2*a*b + A^2*B*b^2)*c^2 - (4*B^3*a*b^2 + 3*A*B^2*b^3)*c)*
d^2 + (3*A*B^2*a*b^3 - A^2*B*b^4 + 4*(4*A^2*B*a^2 - A^3*a*b)*c^2 - (12*A*B^2*a^2*b - A^3*b^3)*c)*d*f + (2*A^2*
B*a^2*b^2 - A^3*a*b^3 - 4*(2*A^2*B*a^3 - A^3*a^2*b)*c)*f^2 + ((B*b^4*c^2 + 4*(2*B*a^2 + A*a*b)*c^4 - (6*B*a*b^
2 + A*b^3)*c^3)*d^3 + (B*b^6 - 4*(6*B*a^3 + A*a^2*b)*c^3 + (22*B*a^2*b^2 + 5*A*a*b^3)*c^2 - (8*B*a*b^4 + A*b^5
)*c)*d^2*f + (3*B*a^2*b^4 - A*a*b^5 + 4*(6*B*a^4 - A*a^3*b)*c^2 - (18*B*a^3*b^2 - 5*A*a^2*b^3)*c)*d*f^2 + (2*B
*a^4*b^2 - A*a^3*b^3 - 4*(2*B*a^5 - A*a^4*b)*c)*f^3)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*
A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^
2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^
3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))*sqrt(f*x^2 + d)*sqrt
(((B^2*b^2 + 2*A^2*c^2 - 2*(B^2*a + A*B*b)*c)*d + (2*B^2*a^2 - 2*A*B*a*b + A^2*b^2 - 2*A^2*a*c)*f - ((b^2*c^2
- 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4
*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B
*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c
+ 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2 + 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^
4)))/((b^2*c^2 - 4*a*c^3)*d^2 + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*f + (a^2*b^2 - 4*a^3*c)*f^2)) - 4*((B^4*a^2*b
- A*B^3*a*b^2 - 2*A^3*B*a*c^2 - (2*A*B^3*a^2 - 3*A^2*B^2*a*b)*c)*d*f + (2*A*B^3*a^3 - 3*A^2*B^2*a^2*b + A^3*B*
a*b^2 + (2*A^3*B*a^2 - A^4*a*b)*c)*f^2)*x - 2*((4*A^2*a*c^4 + (4*B^2*a^2 - 4*A*B*a*b - A^2*b^2)*c^3 - (B^2*a*b
^2 - A*B*b^3)*c^2)*d^3 - (B^2*a*b^4 - A*B*b^5 + 8*A^2*a^2*c^3 + 2*(4*B^2*a^3 - 4*A*B*a^2*b - 3*A^2*a*b^2)*c^2
- (6*B^2*a^2*b^2 - 6*A*B*a*b^3 - A^2*b^4)*c)*d^2*f - (B^2*a^3*b^2 - A*B*a^2*b^3 - 4*A^2*a^3*c^2 - (4*B^2*a^4 -
4*A*B*a^3*b - A^2*a^2*b^2)*c)*d*f^2)*sqrt(((B^4*b^2 - 4*A*B^3*b*c + 4*A^2*B^2*c^2)*d^2 + 2*(2*A*B^3*a*b - A^2
*B^2*b^2 - 2*(2*A^2*B^2*a - A^3*B*b)*c)*d*f + (4*A^2*B^2*a^2 - 4*A^3*B*a*b + A^4*b^2)*f^2)/((b^2*c^4 - 4*a*c^5
)*d^4 + 2*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*f + (b^6 - 8*a*b^4*c + 22*a^2*b^2*c^2 - 24*a^3*c^3)*d^2*f^2
+ 2*(a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*d*f^3 + (a^4*b^2 - 4*a^5*c)*f^4)))/x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%{poly1[%%%{-4,[3,2,0]%%%}+%%%{16,[1,3,1]%%%},%%%{4,[4,2,0]%%%
}+%%%{-24,[2,3,1]%%%}+%%%{32,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[2,1,0,0]%%%}+%%%{%%{[
%%%{-4,[5,0,0]%%%}+%%%{24,[3,1,1]%%%}+%%%{-32,[1,2,2]%%%},%%%{4,[6,0,0]%%%}+%%%{-32,[4,1,1]%%%}+%%%{72,[2,2,2]
%%%}+%%%{-32,[0,3,3]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[2,0,0,1]%%%}+%%%{%%{poly1[%%%{8,[2,3,
0]%%%}+%%%{-32,[0,4,1]%%%},%%%{-8,[3,3,0]%%%}+%%%{32,[1,4,1]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%
},[1,1,1,0]%%%}+%%%{%%{[%%%{8,[4,1,0]%%%}+%%%{-40,[2,2,1]%%%}+%%%{32,[0,3,2]%%%},%%%{-8,[5,1,0]%%%}+%%%{56,[3,
2,1]%%%}+%%%{-96,[1,3,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[1,0,1,1]%%%}+%%%{%%%{8,[2,4,0]%%%
}+%%%{-32,[0,5,1]%%%},[0,1,2,0]%%%}+%%%{%%{poly1[%%%{-4,[3,2,0]%%%}+%%%{16,[1,3,1]%%%},%%%{4,[4,2,0]%%%}+%%%{-
24,[2,3,1]%%%}+%%%{32,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[0,0,2,1]%%%} / %%%{%%{[%%%{-
4,[3,2,0]%%%}+%%%{16,[1,3,1]%%%},%%%{4,[4,2,0]%%%}+%%%{-24,[2,3,1]%%%}+%%%{32,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]
%%%}+%%%{4,[0,1,1]%%%}]%%},[0,1,0,0]%%%}+%%%{%%{[%%%{-4,[5,0,0]%%%}+%%%{24,[3,1,1]%%%}+%%%{-32,[1,2,2]%%%},%%%
{4,[6,0,0]%%%}+%%%{-32,[4,1,1]%%%}+%%%{72,[2,2,2]%%%}+%%%{-32,[0,3,3]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,
1]%%%}]%%},[0,0,0,1]%%%} Error: Bad Argument ValueUnable to divide, perhaps due to rounding error%%%{%%{[%%%{1
,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%},%%%{1,[4,0,0]%%%}+%%%{-6,[2,1,1]%%%}+%%%{8,[0,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%
}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,2]%%%},[2,1,0,0]%%%}+%%%{%%%{1,[2,0,0]%%%}+%%%{-4,[0,1,1]%%%}/2,[2,0,0,1]%%
%}+%%%{%%{[%%%{-1,[4,0,0]%%%}+%%%{5,[2,1,1]%%%}+%%%{-4,[0,2,2]%%%},%%%{-1,[5,0,0]%%%}+%%%{7,[3,1,1]%%%}+%%%{-1
2,[1,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{2,[0,0,3]%%%},[1,1,1,0]%%%}+%%%{%%{[%%%{-1,[2
,0,0]%%%}+%%%{4,[0,1,1]%%%},%%%{-1,[3,0,0]%%%}+%%%{4,[1,1,1]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%
}/%%%{2,[0,0,1]%%%},[1,0,1,1]%%%}+%%%{%%{[%%%{1,[5,0,0]%%%}+%%%{-6,[3,1,1]%%%}+%%%{8,[1,2,2]%%%},%%%{1,[6,0,0]
%%%}+%%%{-8,[4,1,1]%%%}+%%%{18,[2,2,2]%%%}+%%%{-8,[0,3,3]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%
%%{4,[0,0,4]%%%},[0,1,2,0]%%%}+%%%{%%{[%%%{1,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%},%%%{1,[4,0,0]%%%}+%%%{-6,[2,1,1]%%
%}+%%%{8,[0,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,2]%%%},[0,0,2,1]%%%} / %%%{%%{[
%%%{1,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%},%%%{1,[4,0,0]%%%}+%%%{-6,[2,1,1]%%%}+%%%{8,[0,2,2]%%%}]:[1,0,%%%{-1,[2,0,
0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,4]%%%},[0,1,0,0]%%%}+%%%{%%%{1,[2,0,0]%%%}+%%%{-4,[0,1,1]%%%}/%%%{2,[0
,0,2]%%%},[0,0,0,1]%%%} Error: Bad Argument Value

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maple [B]  time = 0.03, size = 1771, normalized size = 5.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x)

[Out]

2/(-4*a*c+b^2)^(1/2)/(-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(-(-4*a*c+b^2)^(1/2)*
b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-(-4*a*c+b
^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b+(-4*a*c+b^2)^(1
/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x+1/2*(
b+(-4*a*c+b^2)^(1/2))/c))*A-1/c/(-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(-(-4*a*c+
b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*
(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b+(-4*
a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2
))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*B-1/(-4*a*c+b^2)^(1/2)/c/(-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2
*d)/c^2)^(1/2)*ln((-(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(
-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+
b^2)^(1/2))/c)^2*f-4*f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-(-4*a*c+b^2)^(1/2)*b*f+2*
a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*b*B-2/(-4*a*c+b^2)^(1/2)/(-2*((-4*a*c+b^2)^
(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b-(-4*a
*c+b^2)^(1/2))/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)
^(1/2)*(4*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)
-2*((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))*A-1/c/(-2*((-
4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2
-f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*
c^2*d)/c^2)^(1/2)*(4*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2*(-b+(-4*a*c+b^2
)^(1/2))/c)-2*((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))*B+
1/(-4*a*c+b^2)^(1/2)/c/(-2*((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-((-4*a*c+b^2)^(1/2)*
b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*((-4*a*c+b
^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b-(-4*a*c+b^2)^(
1/2))/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)-2*((-4*a*c+b^2)^(1/2)*b*f+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x-1/2*
(-b+(-4*a*c+b^2)^(1/2))/c))*b*B

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{\sqrt {f\,x^2+d}\,\left (c\,x^2+b\,x+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((d + f*x^2)^(1/2)*(a + b*x + c*x^2)),x)

[Out]

int((A + B*x)/((d + f*x^2)^(1/2)*(a + b*x + c*x^2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+b*x+a)/(f*x**2+d)**(1/2),x)

[Out]

Integral((A + B*x)/(sqrt(d + f*x**2)*(a + b*x + c*x**2)), x)

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