3.2.17 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+e x+f x^2)} \, dx\) [117]

Optimal. Leaf size=374 \[ -\frac {\sqrt {2} f \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\sqrt {2} f \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {997, 738, 212} \begin {gather*} \frac {\sqrt {2} f \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\sqrt {2} f \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]

[Out]

-((Sqrt[2]*f*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sq
rt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[e^2 - 4*d
*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]])) + (Sqrt[2]*f*ArcTanh[(4*a*f - b*
(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a
*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f
+ 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*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 997

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx &=\frac {(2 f) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{\sqrt {e^2-4 d f}}\\ &=-\frac {(4 f) \text {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e-\sqrt {e^2-4 d f}\right )+4 c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {e^2-4 d f}}+\frac {(4 f) \text {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e+\sqrt {e^2-4 d f}\right )+4 c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {e^2-4 d f}}\\ &=-\frac {\sqrt {2} f \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\sqrt {2} f \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d 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 = 0.01, size = 211, normalized size = 0.56 \begin {gather*} -\text {RootSum}\left [b^2 d-a b e+a^2 f-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}}{2 b \sqrt {c} d-a \sqrt {c} e-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+3 \sqrt {c} e \text {$\#$1}^2-2 f \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(760\) vs. \(2(330)=660\).
time = 0.00, size = 761, normalized size = 2.03

method	result	size
default	\(\frac {\sqrt {2}\, \ln \left (\frac {\frac {-b f \sqrt {-4 d f +e^{2}}+\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-b e f -2 c d f +c \,e^{2}}{f^{2}}+\frac {\left (-c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-b f \sqrt {-4 d f +e^{2}}+\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-b e f -2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )^{2} c +\frac {4 \left (-c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 b f \sqrt {-4 d f +e^{2}}+2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-2 b e f -4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \sqrt {\frac {-b f \sqrt {-4 d f +e^{2}}+\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-b e f -2 c d f +c \,e^{2}}{f^{2}}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {b f \sqrt {-4 d f +e^{2}}-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-b e f -2 c d f +c \,e^{2}}{f^{2}}+\frac {\left (c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {b f \sqrt {-4 d f +e^{2}}-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-b e f -2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )^{2} c +\frac {4 \left (c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 b f \sqrt {-4 d f +e^{2}}-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-2 b e f -4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \sqrt {\frac {b f \sqrt {-4 d f +e^{2}}-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-b e f -2 c d f +c \,e^{2}}{f^{2}}}}\)	\(761\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)

[Out]

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

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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(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d),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*d*f-%e^2>0)', see `assume?`
for more det

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11127 vs. \(2 (326) = 652\).
time = 4.97, size = 11127, normalized size = 29.75 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*sqrt((2*c*d*f - 2*a*f^2 + b*f*e - c*e^2 + (4*c^2*d^3*f + 4*a^2*d*f^3 + 4*(b^2 - 2*a*c)*d^2*f^2 - a
*c*e^4 + (b*c*d + a*b*f)*e^3 - (c^2*d^2 + a^2*f^2 + (b^2 - 6*a*c)*d*f)*e^2 - 4*(b*c*d^2*f + a*b*d*f^2)*e)*sqrt
(-(b^2*f^2 - 2*b*c*f*e + c^2*e^2)/(4*c^4*d^5*f + 4*a^4*d*f^5 + 8*(b^2*c^2 - 2*a*c^3)*d^4*f^2 + 4*(b^4 - 4*a*b^
2*c + 6*a^2*c^2)*d^3*f^3 + 8*(a^2*b^2 - 2*a^3*c)*d^2*f^4 - a^2*c^2*e^6 + 2*(a*b*c^2*d + a^2*b*c*f)*e^5 - ((b^2
*c^2 + 2*a*c^3)*d^2 + 4*(a*b^2*c - 2*a^2*c^2)*d*f + (a^2*b^2 + 2*a^3*c)*f^2)*e^4 + 2*(b*c^3*d^3 + a^3*b*f^3 +
(b^3*c - 5*a*b*c^2)*d^2*f + (a*b^3 - 5*a^2*b*c)*d*f^2)*e^3 - (c^4*d^4 + a^4*f^4 - 2*(b^2*c^2 + 6*a*c^3)*d^3*f
+ (b^4 - 20*a*b^2*c + 22*a^2*c^2)*d^2*f^2 - 2*(a^2*b^2 + 6*a^3*c)*d*f^3)*e^2 - 8*(b*c^3*d^4*f + a^3*b*d*f^4 +
(b^3*c - a*b*c^2)*d^3*f^2 + (a*b^3 - a^2*b*c)*d^2*f^3)*e)))/(4*c^2*d^3*f + 4*a^2*d*f^3 + 4*(b^2 - 2*a*c)*d^2*f
^2 - a*c*e^4 + (b*c*d + a*b*f)*e^3 - (c^2*d^2 + a^2*f^2 + (b^2 - 6*a*c)*d*f)*e^2 - 4*(b*c*d^2*f + a*b*d*f^2)*e
))*log(-(4*b*c*d*f^2*x + 2*b^2*d*f^2 + sqrt(2)*(4*b*c*d^2*f^2 - 4*a*b*d*f^3 + (c^2*d - a*c*f)*e^3 - (b*c*d*f -
a*b*f^2)*e^2 - 4*(c^2*d^2*f - a*c*d*f^2)*e - (8*b*c^2*d^4*f^2 + 8*a^2*b*d^2*f^4 + 8*(b^3 - 2*a*b*c)*d^3*f^3 +
(a*c^2*d + a^2*c*f)*e^5 - (b*c^2*d^2 + 4*a*b*c*d*f + a^2*b*f^2)*e^4 + (c^3*d^3 + a^3*f^3 + (3*b^2*c - 5*a*c^2
)*d^2*f + (3*a*b^2 - 5*a^2*c)*d*f^2)*e^3 + 2*(b*c^2*d^3*f + a^2*b*d*f^3 - (b^3 - 10*a*b*c)*d^2*f^2)*e^2 - 4*(c
^3*d^4*f + a^3*d*f^4 + (3*b^2*c - a*c^2)*d^3*f^2 + (3*a*b^2 - a^2*c)*d^2*f^3)*e)*sqrt(-(b^2*f^2 - 2*b*c*f*e +
c^2*e^2)/(4*c^4*d^5*f + 4*a^4*d*f^5 + 8*(b^2*c^2 - 2*a*c^3)*d^4*f^2 + 4*(b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^3*f^3
+ 8*(a^2*b^2 - 2*a^3*c)*d^2*f^4 - a^2*c^2*e^6 + 2*(a*b*c^2*d + a^2*b*c*f)*e^5 - ((b^2*c^2 + 2*a*c^3)*d^2 + 4*(
a*b^2*c - 2*a^2*c^2)*d*f + (a^2*b^2 + 2*a^3*c)*f^2)*e^4 + 2*(b*c^3*d^3 + a^3*b*f^3 + (b^3*c - 5*a*b*c^2)*d^2*f
+ (a*b^3 - 5*a^2*b*c)*d*f^2)*e^3 - (c^4*d^4 + a^4*f^4 - 2*(b^2*c^2 + 6*a*c^3)*d^3*f + (b^4 - 20*a*b^2*c + 22*
a^2*c^2)*d^2*f^2 - 2*(a^2*b^2 + 6*a^3*c)*d*f^3)*e^2 - 8*(b*c^3*d^4*f + a^3*b*d*f^4 + (b^3*c - a*b*c^2)*d^3*f^2
+ (a*b^3 - a^2*b*c)*d^2*f^3)*e)))*sqrt(c*x^2 + b*x + a)*sqrt((2*c*d*f - 2*a*f^2 + b*f*e - c*e^2 + (4*c^2*d^3*
f + 4*a^2*d*f^3 + 4*(b^2 - 2*a*c)*d^2*f^2 - a*c*e^4 + (b*c*d + a*b*f)*e^3 - (c^2*d^2 + a^2*f^2 + (b^2 - 6*a*c)
*d*f)*e^2 - 4*(b*c*d^2*f + a*b*d*f^2)*e)*sqrt(-(b^2*f^2 - 2*b*c*f*e + c^2*e^2)/(4*c^4*d^5*f + 4*a^4*d*f^5 + 8*
(b^2*c^2 - 2*a*c^3)*d^4*f^2 + 4*(b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^3*f^3 + 8*(a^2*b^2 - 2*a^3*c)*d^2*f^4 - a^2*c^
2*e^6 + 2*(a*b*c^2*d + a^2*b*c*f)*e^5 - ((b^2*c^2 + 2*a*c^3)*d^2 + 4*(a*b^2*c - 2*a^2*c^2)*d*f + (a^2*b^2 + 2*
a^3*c)*f^2)*e^4 + 2*(b*c^3*d^3 + a^3*b*f^3 + (b^3*c - 5*a*b*c^2)*d^2*f + (a*b^3 - 5*a^2*b*c)*d*f^2)*e^3 - (c^4
*d^4 + a^4*f^4 - 2*(b^2*c^2 + 6*a*c^3)*d^3*f + (b^4 - 20*a*b^2*c + 22*a^2*c^2)*d^2*f^2 - 2*(a^2*b^2 + 6*a^3*c)
*d*f^3)*e^2 - 8*(b*c^3*d^4*f + a^3*b*d*f^4 + (b^3*c - a*b*c^2)*d^3*f^2 + (a*b^3 - a^2*b*c)*d^2*f^3)*e)))/(4*c^
2*d^3*f + 4*a^2*d*f^3 + 4*(b^2 - 2*a*c)*d^2*f^2 - a*c*e^4 + (b*c*d + a*b*f)*e^3 - (c^2*d^2 + a^2*f^2 + (b^2 -
6*a*c)*d*f)*e^2 - 4*(b*c*d^2*f + a*b*d*f^2)*e)) + (b*c*f*x + 2*a*c*f)*e^2 - (2*b*c*d*f + 2*a*b*f^2 + (4*c^2*d*
f + b^2*f^2)*x)*e - (8*a*c^2*d^3*f^2 + 8*a^3*d*f^4 + 8*(a*b^2 - 2*a^2*c)*d^2*f^3 + 4*(b*c^2*d^3*f^2 + a^2*b*d*
f^4 + (b^3 - 2*a*b*c)*d^2*f^3)*x - (a*b*c*f*x + 2*a^2*c*f)*e^4 + (2*a*b*c*d*f + 2*a^2*b*f^2 + (b^2*c*d*f + a*b
^2*f^2)*x)*e^3 - (2*a*c^2*d^2*f + 2*a^3*f^3 + 2*(a*b^2 - 6*a^2*c)*d*f^2 + (b*c^2*d^2*f + a^2*b*f^3 + (b^3 - 6*
a*b*c)*d*f^2)*x)*e^2 - 4*(2*a*b*c*d^2*f^2 + 2*a^2*b*d*f^3 + (b^2*c*d^2*f^2 + a*b^2*d*f^3)*x)*e)*sqrt(-(b^2*f^2
- 2*b*c*f*e + c^2*e^2)/(4*c^4*d^5*f + 4*a^4*d*f^5 + 8*(b^2*c^2 - 2*a*c^3)*d^4*f^2 + 4*(b^4 - 4*a*b^2*c + 6*a^
2*c^2)*d^3*f^3 + 8*(a^2*b^2 - 2*a^3*c)*d^2*f^4 - a^2*c^2*e^6 + 2*(a*b*c^2*d + a^2*b*c*f)*e^5 - ((b^2*c^2 + 2*a
*c^3)*d^2 + 4*(a*b^2*c - 2*a^2*c^2)*d*f + (a^2*b^2 + 2*a^3*c)*f^2)*e^4 + 2*(b*c^3*d^3 + a^3*b*f^3 + (b^3*c - 5
*a*b*c^2)*d^2*f + (a*b^3 - 5*a^2*b*c)*d*f^2)*e^3 - (c^4*d^4 + a^4*f^4 - 2*(b^2*c^2 + 6*a*c^3)*d^3*f + (b^4 - 2
0*a*b^2*c + 22*a^2*c^2)*d^2*f^2 - 2*(a^2*b^2 + 6*a^3*c)*d*f^3)*e^2 - 8*(b*c^3*d^4*f + a^3*b*d*f^4 + (b^3*c - a
*b*c^2)*d^3*f^2 + (a*b^3 - a^2*b*c)*d^2*f^3)*e)))/x) - 1/4*sqrt(2)*sqrt((2*c*d*f - 2*a*f^2 + b*f*e - c*e^2 + (
4*c^2*d^3*f + 4*a^2*d*f^3 + 4*(b^2 - 2*a*c)*d^2*f^2 - a*c*e^4 + (b*c*d + a*b*f)*e^3 - (c^2*d^2 + a^2*f^2 + (b^
2 - 6*a*c)*d*f)*e^2 - 4*(b*c*d^2*f + a*b*d*f^2)*e)*sqrt(-(b^2*f^2 - 2*b*c*f*e + c^2*e^2)/(4*c^4*d^5*f + 4*a^4*
d*f^5 + 8*(b^2*c^2 - 2*a*c^3)*d^4*f^2 + 4*(b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^3*f^3 + 8*(a^2*b^2 - 2*a^3*c)*d^2*f^
4 - a^2*c^2*e^6 + 2*(a*b*c^2*d + a^2*b*c*f)*e^5 - ((b^2*c^2 + 2*a*c^3)*d^2 + 4*(a*b^2*c - 2*a^2*c^2)*d*f + (a^
2*b^2 + 2*a^3*c)*f^2)*e^4 + 2*(b*c^3*d^3 + a^3*b*f^3 + (b^3*c - 5*a*b*c^2)*d^2*f + (a*b^3 - 5*a^2*b*c)*d*f^2)*
e^3 - (c^4*d^4 + a^4*f^4 - 2*(b^2*c^2 + 6*a*c^3)*d^3*f + (b^4 - 20*a*b^2*c + 22*a^2*c^2)*d^2*f^2 - 2*(a^2*b^2
+ 6*a^3*c)*d*f^3)*e^2 - 8*(b*c^3*d^4*f + a^3*b*...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a + b*x + c*x**2)*(d + e*x + f*x**2)), 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(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d),x, algorithm="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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