3.1.23 \(\int \frac {A+B x}{(a+b x+c x^2) \sqrt {d+f x^2}} \, dx\) [23]

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

[Out]

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

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Rubi [A]
time = 0.55, 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 = {1048, 739, 212} \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 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 739

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 1048

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 ) \text {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 ) \text {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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.39, size = 195, normalized size = 0.65 \begin {gather*} -\text {RootSum}\left [c d^2+2 b d \sqrt {f} \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 b \sqrt {f} \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {B d \log \left (-\sqrt {f} x+\sqrt {d+f x^2}-\text {$\#$1}\right )+2 A \sqrt {f} \log \left (-\sqrt {f} x+\sqrt {d+f x^2}-\text {$\#$1}\right ) \text {$\#$1}-B \log \left (-\sqrt {f} x+\sqrt {d+f x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b d \sqrt {f}-2 c d \text {$\#$1}+4 a f \text {$\#$1}-3 b \sqrt {f} \text {$\#$1}^2+2 c \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(638\) vs. \(2(266)=532\).
time = 0.16, size = 639, normalized size = 2.12

method	result	size
default	\(-\frac {\left (2 A c +B \sqrt {-4 a c +b^{2}}-b B \right ) \ln \left (\frac {-\frac {b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d}{c^{2}}-\frac {f \left (b -\sqrt {-4 a c +b^{2}}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}+\frac {\sqrt {-\frac {2 \left (b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d \right )}{c^{2}}}\, \sqrt {4 f \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\frac {4 f \left (b -\sqrt {-4 a c +b^{2}}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}-\frac {2 \left (b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d \right )}{c^{2}}}}{2}}{x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}\right )}{\sqrt {-4 a c +b^{2}}\, c \sqrt {-\frac {2 \left (b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d \right )}{c^{2}}}}-\frac {\left (-2 A c +B \sqrt {-4 a c +b^{2}}+b B \right ) \ln \left (\frac {-\frac {-b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d}{c^{2}}-\frac {f \left (b +\sqrt {-4 a c +b^{2}}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}+\frac {\sqrt {-\frac {2 \left (-b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d \right )}{c^{2}}}\, \sqrt {4 f \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\frac {4 f \left (b +\sqrt {-4 a c +b^{2}}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}-\frac {2 \left (-b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d \right )}{c^{2}}}}{2}}{x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}\right )}{\sqrt {-4 a c +b^{2}}\, c \sqrt {-\frac {2 \left (-b f \sqrt {-4 a c +b^{2}}+2 a c f -b^{2} f -2 c^{2} d \right )}{c^{2}}}}\)	\(639\)

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,method=_RETURNVERBOSE)

[Out]

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

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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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8977 vs. \(2 (263) = 526\).
time = 33.48, size = 8977, normalized size = 29.73 \begin {gather*} \text {Too large to display} \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="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 +...

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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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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,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%{poly1[%%%{-4,[3,2,0]%%%}+%%%{16,[1,3,1]%%%},%%%{4,[4,
2,0]%%%}+%%

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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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