3.1.7 \(\int \frac {1}{\sqrt {d+e x+f x^2} (a e+b e x+b f x^2)^2} \, dx\) [7]

Optimal. Leaf size=162 \[ -\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}} \]

[Out]

-(8*a*e*f-b*(4*d*f+e^2))*arctanh((2*f*x+e)*(-a*e+b*d)^(1/2)/e^(1/2)/(-4*a*f+b*e)^(1/2)/(f*x^2+e*x+d)^(1/2))/e^
(3/2)/(-a*e+b*d)^(3/2)/(-4*a*f+b*e)^(3/2)-b*(2*f*x+e)*(f*x^2+e*x+d)^(1/2)/e/(-a*e+b*d)/(-4*a*f+b*e)/(b*f*x^2+b
*e*x+a*e)

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Rubi [A]
time = 0.20, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {988, 12, 996, 214} \begin {gather*} -\frac {\left (8 a e f-b \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(e + 2*f*x)*Sqrt[d + e*x + f*x^2])/(e*(b*d - a*e)*(b*e - 4*a*f)*(a*e + b*e*x + b*f*x^2))) - ((8*a*e*f - b
*(e^2 + 4*d*f))*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a*f]*Sqrt[d + e*x + f*x^2])])/(e^(
3/2)*(b*d - a*e)^(3/2)*(b*e - 4*a*f)^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 214

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

Rule 988

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(2*a*
c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 996

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {\int \frac {b (b d-a e) f^2 \left (8 a e f-b \left (e^2+4 d f\right )\right )}{2 \sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{b e (b d-a e)^2 f^2 (b e-4 a f)}\\ &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{2 e (b d-a e) (b e-4 a f)}\\ &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \text {Subst}\left (\int \frac {1}{e \left (b e^2-4 a e f\right )-\left (b d e-a e^2\right ) x^2} \, dx,x,\frac {e+2 f x}{\sqrt {d+e x+f x^2}}\right )}{(b d-a e) (b e-4 a f)}\\ &=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.01, size = 241, normalized size = 1.49 \begin {gather*} \frac {\frac {2 b (e+2 f x) \sqrt {d+x (e+f x)}}{a e+b x (e+f x)}+\left (-8 a e f+b \left (e^2+4 d f\right )\right ) \text {RootSum}\left [-b d e^2+a e^3+b d^2 f+2 b d e \sqrt {f} \text {$\#$1}-4 a e^2 \sqrt {f} \text {$\#$1}+b e^2 \text {$\#$1}^2-2 b d f \text {$\#$1}^2+4 a e f \text {$\#$1}^2-2 b e \sqrt {f} \text {$\#$1}^3+b f \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )}{b d \sqrt {f}-2 a e \sqrt {f}+b e \text {$\#$1}-b \sqrt {f} \text {$\#$1}^2}\&\right ]}{2 e (-b d+a e) (b e-4 a f)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*(e + 2*f*x)*Sqrt[d + x*(e + f*x)])/(a*e + b*x*(e + f*x)) + (-8*a*e*f + b*(e^2 + 4*d*f))*RootSum[-(b*d*e^
2) + a*e^3 + b*d^2*f + 2*b*d*e*Sqrt[f]*#1 - 4*a*e^2*Sqrt[f]*#1 + b*e^2*#1^2 - 2*b*d*f*#1^2 + 4*a*e*f*#1^2 - 2*
b*e*Sqrt[f]*#1^3 + b*f*#1^4 & , Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]/(b*d*Sqrt[f] - 2*a*e*Sqrt[f] +
b*e*#1 - b*Sqrt[f]*#1^2) & ])/(2*e*(-(b*d) + a*e)*(b*e - 4*a*f))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1351\) vs. \(2(146)=292\).
time = 0.15, size = 1352, normalized size = 8.35

method	result	size
default	\(\frac {2 f \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{e \left (4 f a -e b \right ) \sqrt {-e b \left (4 f a -e b \right )}\, \sqrt {-\frac {a e -b d}{b}}}-\frac {\frac {b \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{\left (a e -b d \right ) \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{2 \left (a e -b d \right ) \sqrt {-\frac {a e -b d}{b}}}}{e \left (4 f a -e b \right ) b}-\frac {\frac {b \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{\left (a e -b d \right ) \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{2 \left (a e -b d \right ) \sqrt {-\frac {a e -b d}{b}}}}{e \left (4 f a -e b \right ) b}-\frac {2 f \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{e \left (4 f a -e b \right ) \sqrt {-e b \left (4 f a -e b \right )}\, \sqrt {-\frac {a e -b d}{b}}}\)	\(1352\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (155) = 310\).
time = 4.54, size = 1919, normalized size = 11.85 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((4*b^2*d*f^2*x^2 + (b^2*x + a*b)*e^3 + (b^2*f*x^2 - 8*a*b*f*x - 8*a^2*f)*e^2 - 4*(2*a*b*f^2*x^2 - b^2*d
*f*x - a*b*d*f)*e)*sqrt(-4*a*b*d*f*e - a*b*e^3 + (b^2*d + 4*a^2*f)*e^2)*log((16*b^2*d^2*f^4*x^4 - 4*(8*b*d*f^3
*x^3 + (b*x - a)*e^4 + (3*b*f*x^2 - 10*a*f*x + 2*b*d)*e^3 + 2*(b*f^2*x^3 - 12*a*f^2*x^2 + 4*b*d*f*x - 2*a*d*f)
*e^2 - 4*(4*a*f^3*x^3 - 3*b*d*f^2*x^2 + 2*a*d*f^2*x)*e)*sqrt(-4*a*b*d*f*e - a*b*e^3 + (b^2*d + 4*a^2*f)*e^2)*s
qrt(f*x^2 + x*e + d) + (b^2*x^2 - 6*a*b*x + a^2)*e^6 + 2*(b^2*f*x^3 - 19*a*b*f*x^2 - 4*a*b*d + 4*(b^2*d + 4*a^
2*f)*x)*e^5 + (b^2*f^2*x^4 - 64*a*b*f^2*x^3 - 80*a*b*d*f*x + 8*b^2*d^2 + 24*a^2*d*f + 32*(b^2*d*f + 5*a^2*f^2)
*x^2)*e^4 - 16*(2*a*b*f^3*x^4 + 13*a*b*d*f^2*x^2 + 2*a*b*d^2*f - (3*b^2*d*f^2 + 16*a^2*f^3)*x^3 - 2*(b^2*d^2*f
+ 4*a^2*d*f^2)*x)*e^3 - 8*(32*a*b*d*f^3*x^3 + 12*a*b*d^2*f^2*x - 2*a^2*d^2*f^2 - (3*b^2*d*f^3 + 16*a^2*f^4)*x
^4 - 2*(3*b^2*d^2*f^2 + 8*a^2*d*f^3)*x^2)*e^2 - 32*(4*a*b*d*f^4*x^4 - b^2*d^2*f^3*x^3 + 3*a*b*d^2*f^3*x^2)*e)/
(b^2*f^2*x^4 + (b^2*x^2 + 2*a*b*x + a^2)*e^2 + 2*(b^2*f*x^3 + a*b*f*x^2)*e)) - 4*(8*a*b^2*d*f^2*x*e + a*b^2*e^
4 + (2*a*b^2*f*x - b^3*d - 4*a^2*b*f)*e^3 + 2*(2*a*b^2*d*f - (b^3*d*f + 4*a^2*b*f^2)*x)*e^2)*sqrt(f*x^2 + x*e
+ d))/(16*a^2*b^3*d^2*f^3*x^2*e^2 + (a^2*b^3*x + a^3*b^2)*e^7 + (a^2*b^3*f*x^2 - 2*a^2*b^3*d - 8*a^4*b*f - 2*(
a*b^4*d + 4*a^3*b^2*f)*x)*e^6 + (a*b^4*d^2 + 16*a^3*b^2*d*f + 16*a^5*f^2 - 2*(a*b^4*d*f + 4*a^3*b^2*f^2)*x^2 +
(b^5*d^2 + 16*a^2*b^3*d*f + 16*a^4*b*f^2)*x)*e^5 - (8*a^2*b^3*d^2*f + 32*a^4*b*d*f^2 - (b^5*d^2*f + 16*a^2*b^
3*d*f^2 + 16*a^4*b*f^3)*x^2 + 8*(a*b^4*d^2*f + 4*a^3*b^2*d*f^2)*x)*e^4 + 8*(2*a^2*b^3*d^2*f^2*x + 2*a^3*b^2*d^
2*f^2 - (a*b^4*d^2*f^2 + 4*a^3*b^2*d*f^3)*x^2)*e^3), 1/2*((4*b^2*d*f^2*x^2 + (b^2*x + a*b)*e^3 + (b^2*f*x^2 -
8*a*b*f*x - 8*a^2*f)*e^2 - 4*(2*a*b*f^2*x^2 - b^2*d*f*x - a*b*d*f)*e)*sqrt(4*a*b*d*f*e + a*b*e^3 - (b^2*d + 4*
a^2*f)*e^2)*arctan(1/2*(4*b*d*f^2*x^2 + (b*x - a)*e^3 + (b*f*x^2 - 8*a*f*x + 2*b*d)*e^2 - 4*(2*a*f^2*x^2 - b*d
*f*x + a*d*f)*e)*sqrt(4*a*b*d*f*e + a*b*e^3 - (b^2*d + 4*a^2*f)*e^2)*sqrt(f*x^2 + x*e + d)/(a*b*x*e^5 + (3*a*b
*f*x^2 + a*b*d - (b^2*d + 4*a^2*f)*x)*e^4 + (2*a*b*f^2*x^3 + 6*a*b*d*f*x - b^2*d^2 - 4*a^2*d*f - 3*(b^2*d*f +
4*a^2*f^2)*x^2)*e^3 + 2*(6*a*b*d*f^2*x^2 + 2*a*b*d^2*f - (b^2*d*f^2 + 4*a^2*f^3)*x^3 - (b^2*d^2*f + 4*a^2*d*f^
2)*x)*e^2 + 8*(a*b*d*f^3*x^3 + a*b*d^2*f^2*x)*e)) + 2*(8*a*b^2*d*f^2*x*e + a*b^2*e^4 + (2*a*b^2*f*x - b^3*d -
4*a^2*b*f)*e^3 + 2*(2*a*b^2*d*f - (b^3*d*f + 4*a^2*b*f^2)*x)*e^2)*sqrt(f*x^2 + x*e + d))/(16*a^2*b^3*d^2*f^3*x
^2*e^2 + (a^2*b^3*x + a^3*b^2)*e^7 + (a^2*b^3*f*x^2 - 2*a^2*b^3*d - 8*a^4*b*f - 2*(a*b^4*d + 4*a^3*b^2*f)*x)*e
^6 + (a*b^4*d^2 + 16*a^3*b^2*d*f + 16*a^5*f^2 - 2*(a*b^4*d*f + 4*a^3*b^2*f^2)*x^2 + (b^5*d^2 + 16*a^2*b^3*d*f
+ 16*a^4*b*f^2)*x)*e^5 - (8*a^2*b^3*d^2*f + 32*a^4*b*d*f^2 - (b^5*d^2*f + 16*a^2*b^3*d*f^2 + 16*a^4*b*f^3)*x^2
+ 8*(a*b^4*d^2*f + 4*a^3*b^2*d*f^2)*x)*e^4 + 8*(2*a^2*b^3*d^2*f^2*x + 2*a^3*b^2*d^2*f^2 - (a*b^4*d^2*f^2 + 4*
a^3*b^2*d*f^3)*x^2)*e^3)]

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Sympy [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/(b*f*x**2+b*e*x+a*e)**2/(f*x**2+e*x+d)**(1/2),x)

[Out]

Timed out

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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(1/(b*f*x^2+b*e*x+a*e)^2/(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%%%{%%%{1,[2]%%%},[8,2,0,0,0]%%%}+%%%{%%{[%%%{-4,[1]%%%},0]:
[1,0,%%%{-1

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,f\,x^2+b\,e\,x+a\,e\right )}^2\,\sqrt {f\,x^2+e\,x+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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