3.96 \(\int \frac{x^2}{\sqrt{a+b x+c x^2} (d-f x^2)} \, dx\)

Optimal. Leaf size=266 \[ \frac{\sqrt{d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} f} \]

[Out]

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

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Rubi [A]  time = 0.216425, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1079, 621, 206, 984, 724} \[ \frac{\sqrt{d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1079

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

Rule 621

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 984

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 724

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]

Rubi steps

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

Mathematica [A]  time = 0.456077, size = 250, normalized size = 0.94 \[ \frac{\sqrt{d} \left (\frac{\tanh ^{-1}\left (\frac{-2 a \sqrt{f}+b \sqrt{d}-b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\tanh ^{-1}\left (\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{\sqrt{c}}}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.279, size = 399, normalized size = 1.5 \begin{align*} -{\frac{1}{f}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{d}{2\,f}\ln \left ({ \left ( 2\,{\frac{-b\sqrt{df}+af+cd}{f}}+{\frac{1}{f} \left ( -2\,c\sqrt{df}+bf \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{-b\sqrt{df}+af+cd}{f}}}\sqrt{ \left ( x+{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{-2\,c\sqrt{df}+bf}{f} \left ( x+{\frac{\sqrt{df}}{f}} \right ) }+{\frac{-b\sqrt{df}+af+cd}{f}}} \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{df}}}{\frac{1}{\sqrt{{\frac{1}{f} \left ( -b\sqrt{df}+af+cd \right ) }}}}}+{\frac{d}{2\,f}\ln \left ({ \left ( 2\,{\frac{b\sqrt{df}+af+cd}{f}}+{\frac{1}{f} \left ( 2\,c\sqrt{df}+bf \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{b\sqrt{df}+af+cd}{f}}}\sqrt{ \left ( x-{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{2\,c\sqrt{df}+bf}{f} \left ( x-{\frac{\sqrt{df}}{f}} \right ) }+{\frac{b\sqrt{df}+af+cd}{f}}} \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{df}}}{\frac{1}{\sqrt{{\frac{1}{f} \left ( b\sqrt{df}+af+cd \right ) }}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x**2/(-d*sqrt(a + b*x + c*x**2) + f*x**2*sqrt(a + b*x + c*x**2)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError