3.1.21 \(\int \frac {A+B x}{(a+b x+c x^2) \sqrt {d+e x+f x^2}} \, dx\) [21]

Optimal. Leaf size=416 \[ \frac {\left (b B-2 A c-B \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {4 c d-\left (b-\sqrt {b^2-4 a c}\right ) e+2 \left (c e-\left (b-\sqrt {b^2-4 a c}\right ) f\right ) x}{2 \sqrt {2} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f+\sqrt {b^2-4 a c} (c e-b f)} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f+\sqrt {b^2-4 a c} (c e-b f)}}+\frac {\left (2 A c-B \left (b+\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {4 c d-\left (b+\sqrt {b^2-4 a c}\right ) e+2 \left (c e-\left (b+\sqrt {b^2-4 a c}\right ) f\right ) x}{2 \sqrt {2} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)}} \]

[Out]

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

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Rubi [A]
time = 1.79, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1046, 738, 212} \begin {gather*} \frac {\left (-B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 x \left (c e-f \left (b-\sqrt {b^2-4 a c}\right )\right )-e \left (b-\sqrt {b^2-4 a c}\right )+4 c d}{2 \sqrt {2} \sqrt {d+e x+f x^2} \sqrt {\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+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 x \left (c e-f \left (\sqrt {b^2-4 a c}+b\right )\right )-e \left (\sqrt {b^2-4 a c}+b\right )+4 c d}{2 \sqrt {2} \sqrt {d+e x+f x^2} \sqrt {-\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(4*c*d - (b - Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b - Sqrt[b^2 - 4*a
*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x + f
*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]) +
((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(4*c*d - (b + Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b + Sqrt[b^2 - 4*
a*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x +
f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*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 738

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

Rule 1046

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> 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 + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 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+e x+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+e x+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+e x+f x^2}} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2 \left (b B-2 A c-B \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{16 c^2 d-8 c \left (b-\sqrt {b^2-4 a c}\right ) e+4 \left (b-\sqrt {b^2-4 a c}\right )^2 f-x^2} \, dx,x,\frac {4 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-\left (-2 c e+2 \left (b-\sqrt {b^2-4 a c}\right ) f\right ) x}{\sqrt {d+e x+f x^2}}\right )}{\sqrt {b^2-4 a c}}+\frac {\left (2 \left (2 A c-B \left (b+\sqrt {b^2-4 a c}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{16 c^2 d-8 c \left (b+\sqrt {b^2-4 a c}\right ) e+4 \left (b+\sqrt {b^2-4 a c}\right )^2 f-x^2} \, dx,x,\frac {4 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-\left (-2 c e+2 \left (b+\sqrt {b^2-4 a c}\right ) f\right ) x}{\sqrt {d+e x+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 {4 c d-\left (b-\sqrt {b^2-4 a c}\right ) e+2 \left (c e-\left (b-\sqrt {b^2-4 a c}\right ) f\right ) x}{2 \sqrt {2} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f+\sqrt {b^2-4 a c} (c e-b f)} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f+\sqrt {b^2-4 a c} (c e-b f)}}+\frac {\left (2 A c-B \left (b+\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {4 c d-\left (b+\sqrt {b^2-4 a c}\right ) e+2 \left (c e-\left (b+\sqrt {b^2-4 a c}\right ) f\right ) x}{2 \sqrt {2} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b 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 = 0.45, size = 278, normalized size = 0.67 \begin {gather*} -\text {RootSum}\left [c d^2-b d e+a e^2+2 b d \sqrt {f} \text {$\#$1}-4 a e \sqrt {f} \text {$\#$1}-2 c d \text {$\#$1}^2+b e \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 b \sqrt {f} \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {B d \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )-A e \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )+2 A \sqrt {f} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}-B \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b d \sqrt {f}-2 a e \sqrt {f}-2 c d \text {$\#$1}+b e \text {$\#$1}+4 a f \text {$\#$1}-3 b \sqrt {f} \text {$\#$1}^2+2 c \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-RootSum[c*d^2 - b*d*e + a*e^2 + 2*b*d*Sqrt[f]*#1 - 4*a*e*Sqrt[f]*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4*a*f*#1^2 - 2*
b*Sqrt[f]*#1^3 + c*#1^4 & , (B*d*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] - A*e*Log[-(Sqrt[f]*x) + Sqrt[
d + e*x + f*x^2] - #1] + 2*A*Sqrt[f]*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1 - B*Log[-(Sqrt[f]*x) +
Sqrt[d + e*x + f*x^2] - #1]*#1^2)/(b*d*Sqrt[f] - 2*a*e*Sqrt[f] - 2*c*d*#1 + b*e*#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. \(804\) vs. \(2(370)=740\).
time = 0.20, size = 805, normalized size = 1.94

method	result	size
default	\(-\frac {\left (2 A c +B \sqrt {-4 a c +b^{2}}-b B \right ) \ln \left (\frac {-\frac {b f \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d}{c^{2}}-\frac {\left (-f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}+\frac {\sqrt {-\frac {2 \left (b f \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d \right )}{c^{2}}}\, \sqrt {4 f \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\frac {4 \left (-f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}-\frac {2 \left (b f \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d \right )}{c^{2}}}}{2}}{x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}\right )}{\sqrt {-4 a c +b^{2}}\, c \sqrt {-\frac {2 \left (b f \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d \right )}{c^{2}}}}-\frac {\left (-2 A c +B \sqrt {-4 a c +b^{2}}+b B \right ) \ln \left (\frac {-\frac {-b f \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d}{c^{2}}-\frac {\left (f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}+\frac {\sqrt {-\frac {2 \left (-b f \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d \right )}{c^{2}}}\, \sqrt {4 f \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\frac {4 \left (f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{c}-\frac {2 \left (-b f \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d \right )}{c^{2}}}}{2}}{x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}\right )}{\sqrt {-4 a c +b^{2}}\, c \sqrt {-\frac {2 \left (-b f \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, c e +2 a c f -b^{2} f +b c e -2 c^{2} d \right )}{c^{2}}}}\)	\(805\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x+a)/(f*x^2+e*x+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)-(-4*a*c+b^2)^(1/2)*c*e+2*a*
c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*
c^2*d)/c^2-(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/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)
-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-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
*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*
c*e+2*a*c*f-b^2*f+b*c*e-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)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)
^(1/2)*ln((-(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(f*(-4*a*c+b^2)^(
1/2)+b*f-c*e)/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)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c
*f-b^2*f+b*c*e-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*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*
(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-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+e*x+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. 21959 vs. \(2 (383) = 766\).
time = 169.42, size = 21959, normalized size = 52.79 \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+e*x+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*a*b - (4*A*B*a - A^2*b)*c)*e + ((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 + (a*b^2*c - 4*a^2*c^2)*e^2 - ((b^3*c - 4*a*b*c^2)*d + (a*b^3 - 4*a^2*b*c)*f)*e)*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^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 2*((B^4*a*b + 2*A^3*B*c^
2 - (2*A*B^3*a + A^2*B^2*b)*c)*d + (2*A*B^3*a^2 - A^2*B^2*a*b - (2*A^3*B*a - A^4*b)*c)*f)*e)/((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 + (a^2*b^2*c^2 - 4*a^3*c^3)*e^4 - 2*
((a*b^3*c^2 - 4*a^2*b*c^3)*d + (a^2*b^3*c - 4*a^3*b*c^2)*f)*e^3 + ((b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^2 + 4
*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d*f + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*f^2)*e^2 - 2*((b^3*c^3 - 4*a*
b*c^4)*d^3 + (b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d^2*f + (a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d*f^2 + (a^3*b^3
- 4*a^4*b*c)*f^3)*e)))/((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
+ (a*b^2*c - 4*a^2*c^2)*e^2 - ((b^3*c - 4*a*b*c^2)*d + (a*b^3 - 4*a^2*b*c)*f)*e))*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^3*a^2*b^2 + 4*A^2*B*a^2
*c^2 - (4*B^3*a^3 + A^2*B*a*b^2)*c)*e^2 - ((2*B^3*a*b^3 - 4*A^3*a*c^3 + (12*A*B^2*a^2 + 4*A^2*B*a*b + A^3*b^2)
*c^2 - (8*B^3*a^2*b + 3*A*B^2*a*b^2 + A^2*B*b^3)*c)*d + (3*A*B^2*a^2*b^2 - A^2*B*a*b^3 + 4*A^3*a^2*c^2 - (12*A
*B^2*a^3 - 4*A^2*B*a^2*b + A^3*a*b^2)*c)*f)*e - ((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 - (B*a^2*b^3*c + 8*A*a^3*c^3 - 2*(2*B*a^3*b + A*a^2*b^2)*c^2)*e^3 + ((2
*B*a*b^4*c + 4*(2*B*a^3 + 3*A*a^2*b)*c^3 - (10*B*a^2*b^2 + 3*A*a*b^3)*c^2)*d + (B*a^2*b^4 - 4*(2*B*a^4 - 3*A*a
^3*b)*c^2 - (2*B*a^3*b^2 + 3*A*a^2*b^3)*c)*f)*e^2 - ((B*b^5*c + 8*A*a^2*c^4 + 2*(2*B*a^2*b + A*a*b^2)*c^3 - (5
*B*a*b^3 + A*b^4)*c^2)*d^2 + 2*(B*a*b^5 - 8*A*a^3*c^3 + 2*(2*B*a^3*b + 5*A*a^2*b^2)*c^2 - (5*B*a^2*b^3 + 2*A*a
*b^4)*c)*d*f + (3*B*a^3*b^3 - A*a^2*b^4 + 8*A*a^4*c^2 - 2*(6*B*a^4*b - A*a^3*b^2)*c)*f^2)*e)*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^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 2*((B^4*a*b + 2*A^3*B*c^2 - (2*
A*B^3*a + A^2*B^2*b)*c)*d + (2*A*B^3*a^2 - A^2*B^2*a*b - (2*A^3*B*a - A^4*b)*c)*f)*e)/((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 + (a^2*b^2*c^2 - 4*a^3*c^3)*e^4 - 2*((a*b^3
*c^2 - 4*a^2*b*c^3)*d + (a^2*b^3*c - 4*a^3*b*c^2)*f)*e^3 + ((b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^2 + 4*(a*b^4
*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d*f + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*f^2)*e^2 - 2*((b^3*c^3 - 4*a*b*c^4)*
d^3 + (b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d^2*f + (a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d*f^2 + (a^3*b^3 - 4*a^4
*b*c)*f^3)*e)))*sqrt(f*x^2 + x*e + 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*a*b - (4*A*B*a - A^2*b)*c)*e + ((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 + (a*b^2*c - 4*a^2*c^2)*e^2 - ((b^3*c - 4*a*b*c^2)*d + (a*b^3 - 4
*a^2*b*c)*f)*e)*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^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2
- 2*((B^4*a*b + 2*A^3*B*c^2 - (2*A*B^3*a + A^2*B^2*b)*c)*d + (2*A*B^3*a^2 - A^2*B^2*a*b - (2*A^3*B*a - A^4*b)*
c)*f)*e)/((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 + (a^2*b^2
*c^2 - 4*a^3*c^3)*e^4 - 2*((a*b^3*c^2 - 4*a^2*b*c^3)*d + (a^2*b^3*c - 4*a^3*b*c^2)*f)*e^3 + ((b^4*c^2 - 2*a*b^
2*c^3 - 8*a^2*c^4)*d^2 + 4*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d*f + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*f^2
)*e^2 - 2*((b^3*c^3 - 4*a*b*c^4)*d^3 + (b^5*c -...

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

[Out]

Integral((A + B*x)/((a + b*x + c*x**2)*sqrt(d + e*x + f*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+e*x+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}{\left (c\,x^2+b\,x+a\right )\,\sqrt {f\,x^2+e\,x+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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