3.1.100 \(\int \frac {1}{x^2 \sqrt {a+b x+c x^2} (d-f x^2)} \, dx\) [100]

Optimal. Leaf size=291 \[ -\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}} \]

[Out]

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

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Rubi [A]
time = 0.42, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6857, 744, 738, 212, 998} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {f \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\sqrt {a+b x+c x^2}}{a d x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x + c*x^2]/(a*d*x)) + (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d) +
(f*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt
[a + b*x + c*x^2])])/(2*d^(3/2)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (f*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (
2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*Sqrt[c*
d + b*Sqrt[d]*Sqrt[f] + a*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 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]

Rule 998

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

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx &=\int \left (\frac {1}{d x^2 \sqrt {a+b x+c x^2}}+\frac {f}{d \sqrt {a+b x+c x^2} \left (d-f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{x^2 \sqrt {a+b x+c x^2}} \, dx}{d}+\frac {f \int \frac {1}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{d}\\ &=-\frac {\sqrt {a+b x+c x^2}}{a d x}-\frac {b \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 a d}+\frac {f \int \frac {1}{\left (d-\sqrt {d} \sqrt {f} x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d}+\frac {f \int \frac {1}{\left (d+\sqrt {d} \sqrt {f} x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d}\\ &=-\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {b \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{a d}-\frac {f \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d^{3/2} \sqrt {f}+4 a d f-x^2} \, dx,x,\frac {-b d+2 a \sqrt {d} \sqrt {f}-\left (2 c d-b \sqrt {d} \sqrt {f}\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d}-\frac {f \text {Subst}\left (\int \frac {1}{4 c d^2+4 b d^{3/2} \sqrt {f}+4 a d f-x^2} \, dx,x,\frac {-b d-2 a \sqrt {d} \sqrt {f}-\left (2 c d+b \sqrt {d} \sqrt {f}\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d}\\ &=-\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a 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 = 0.47, size = 216, normalized size = 0.74 \begin {gather*} -\frac {\frac {2 \sqrt {a+x (b+c x)}}{a x}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{3/2}}+f \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 426, normalized size = 1.46

method	result	size
default	\(\frac {-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}+\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}\)	\(426\)
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}}{a d x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}} d}+\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 d \sqrt {d f}\, \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}\)	\(427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/a/x*(c*x^2+b*x+a)^(1/2)+1/2*b/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))-1/2*f/d/(d*f)^(1/
2)/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+f*a+c*d)+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(
1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2
)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2)/f))+1/2*f/d/(d*f)^(1/2)/((b*(d*f)^(1/2)+f*a+c*d)/f)^(
1/2)*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(
1/2)*((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*
f)^(1/2)/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/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3005 vs. \(2 (223) = 446\).
time = 170.98, size = 6018, normalized size = 20.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(a^2*d*x*sqrt((c*d*f^2 + a*f^3 + (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c^4*d^9 + a^
4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^
3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f))*log((2*b*c*f^3*x + b^2*f^3 + 2*(b*c*d^2*f^2 + a*b*d*f^3 -
(b*c^2*d^6 + a^2*b*d^4*f^2 - (b^3 - 2*a*b*c)*d^5*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3
)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))*sqrt(c*x^2 + b*x + a)*sqrt(
(c*d*f^2 + a*f^3 + (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*
c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/(c^2*d^5 + a^2
*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)) - (2*a*c^2*d^4*f + 2*a^3*d^2*f^3 - 2*(a*b^2 - 2*a^2*c)*d^3*f^2 + (b*c^2*d^4*f
+ a^2*b*d^2*f^3 - (b^3 - 2*a*b*c)*d^3*f^2)*x)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8
*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/x) - a^2*d*x*sqrt((c*d*f^2 + a*f
^3 + (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)
*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^
2 - 2*a*c)*d^4*f))*log((2*b*c*f^3*x + b^2*f^3 - 2*(b*c*d^2*f^2 + a*b*d*f^3 - (b*c^2*d^6 + a^2*b*d^4*f^2 - (b^3
- 2*a*b*c)*d^5*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^
2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))*sqrt(c*x^2 + b*x + a)*sqrt((c*d*f^2 + a*f^3 + (c^2*d^5 + a^2
*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a
*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f))
- (2*a*c^2*d^4*f + 2*a^3*d^2*f^3 - 2*(a*b^2 - 2*a^2*c)*d^3*f^2 + (b*c^2*d^4*f + a^2*b*d^2*f^3 - (b^3 - 2*a*b*c
)*d^3*f^2)*x)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2
)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/x) + a^2*d*x*sqrt((c*d*f^2 + a*f^3 - (c^2*d^5 + a^2*d^3*f^2 - (b^
2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2
*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f))*log((2*b*c*f^3
*x + b^2*f^3 + 2*(b*c*d^2*f^2 + a*b*d*f^3 + (b*c^2*d^6 + a^2*b*d^4*f^2 - (b^3 - 2*a*b*c)*d^5*f)*sqrt(b^2*f^5/(
c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a
^3*c)*d^6*f^3)))*sqrt(c*x^2 + b*x + a)*sqrt((c*d*f^2 + a*f^3 - (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*s
qrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(
a^2*b^2 - 2*a^3*c)*d^6*f^3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)) + (2*a*c^2*d^4*f + 2*a^3*d^2*f^3
- 2*(a*b^2 - 2*a^2*c)*d^3*f^2 + (b*c^2*d^4*f + a^2*b*d^2*f^3 - (b^3 - 2*a*b*c)*d^3*f^2)*x)*sqrt(b^2*f^5/(c^4*d
^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)
*d^6*f^3)))/x) - a^2*d*x*sqrt((c*d*f^2 + a*f^3 - (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c
^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^
3*c)*d^6*f^3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f))*log((2*b*c*f^3*x + b^2*f^3 - 2*(b*c*d^2*f^2 + a
*b*d*f^3 + (b*c^2*d^6 + a^2*b*d^4*f^2 - (b^3 - 2*a*b*c)*d^5*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^
2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))*sqrt(c*x^2 + b*x
+ a)*sqrt((c*d*f^2 + a*f^3 - (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^
4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/(c^
2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)) + (2*a*c^2*d^4*f + 2*a^3*d^2*f^3 - 2*(a*b^2 - 2*a^2*c)*d^3*f^2 + (
b*c^2*d^4*f + a^2*b*d^2*f^3 - (b^3 - 2*a*b*c)*d^3*f^2)*x)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2
*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))/x) + sqrt(a)*b*x*log(
-(8*a*b*x + (b^2 + 4*a*c)*x^2 + 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*sqrt(c*x^2 + b*x
+ a)*a)/(a^2*d*x), 1/4*(a^2*d*x*sqrt((c*d*f^2 + a*f^3 + (c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f)*sqrt(b^
2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^
2 - 2*a^3*c)*d^6*f^3)))/(c^2*d^5 + a^2*d^3*f^2 - (b^2 - 2*a*c)*d^4*f))*log((2*b*c*f^3*x + b^2*f^3 + 2*(b*c*d^2
*f^2 + a*b*d*f^3 - (b*c^2*d^6 + a^2*b*d^4*f^2 - (b^3 - 2*a*b*c)*d^5*f)*sqrt(b^2*f^5/(c^4*d^9 + a^4*d^5*f^4 - 2
*(b^2*c^2 - 2*a*c^3)*d^8*f + (b^4 - 4*a*b^2*c + 6*a^2*c^2)*d^7*f^2 - 2*(a^2*b^2 - 2*a^3*c)*d^6*f^3)))*sqrt(c*x
^2 + b*x + a)*sqrt((c*d*f^2 + a*f^3 + (c^2*d^5 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- d x^{2} \sqrt {a + b x + c x^{2}} + f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(-d*x**2*sqrt(a + b*x + c*x**2) + f*x**4*sqrt(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(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time:
0.77Done

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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