3.104 \(\int \frac{x^2}{(a+b x+c x^2)^{3/2} (d-f x^2)} \, dx\)

Optimal. Leaf size=297 \[ \frac{2 \left (c x \left (b^2 d-2 a (a f+c d)\right )+a b (c d-a f)\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

[Out]

(2*(a*b*(c*d - a*f) + c*(b^2*d - 2*a*(c*d + a*f))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x +
c*x^2]) + (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]*Sqr
t[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (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
*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))

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Rubi [A]  time = 0.453871, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1065, 1033, 724, 206} \[ \frac{2 \left (c x \left (b^2 d-2 a (a f+c d)\right )+a b (c d-a f)\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]

[Out]

(2*(a*b*(c*d - a*f) + c*(b^2*d - 2*a*(c*d + a*f))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x +
c*x^2]) + (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]*Sqr
t[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (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
*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))

Rule 1065

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

Rule 1033

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[-(a*c), 2]}, Dist[h/2 + (c*g)/(2*q), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - (c*g)
/(2*q), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, 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]

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])

Rubi steps

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

Mathematica [A]  time = 0.468073, size = 352, normalized size = 1.19 \[ \frac{2 \left (\frac{a^2 f (b+2 c x)+a c d (2 c x-b)-b^2 c d x}{\sqrt{a+x (b+c x)}}+\frac{\sqrt{d} \left (b^2-4 a c\right ) \left (a f+b \sqrt{d} \sqrt{f}+c d\right ) \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+b \left (\sqrt{d}-\sqrt{f} x\right )+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{4 \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\sqrt{d} \left (4 a c-b^2\right ) \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right ) \tanh ^{-1}\left (\frac{-2 \left (a \sqrt{f}+c \sqrt{d} x\right )-b \left (\sqrt{d}+\sqrt{f} x\right )}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{4 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\left (b^2-4 a c\right ) \left ((a f+c d)^2-b^2 d f\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]

[Out]

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

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Maple [B]  time = 0.309, size = 1427, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)

[Out]

-2/f*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+1/2*d/(d*f)^(1/2)/(-b*(d*f)^(1/2)+a*f+c*d)/((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)+2*d/f/(-b*(d*f)^(1/2)+a*f+
c*d)/(4*a*c-b^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*c^2-d/(d*f)^(1/2)/(-b*(d*f)^(1/2)+a*f+c*d)/(4*a*c-b^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*b*c+d/f/(-b*(d*f)^(1/2)+a*f+c*d)/(4*a*c-b^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)*b*c-1/2
*d/(d*f)^(1/2)/(-b*(d*f)^(1/2)+a*f+c*d)/(4*a*c-b^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)*b^2-1/2*d/(d*f)^(1/2)/(-b*(d*f)^(1/2)+a*f+c*d)/(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)/(b*(d*f)^(1/2)+a*f+c*d)/((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)+2*d/f/(b*(d*f)^(1/2)+a*f+c*d)/(4*a*c-b^
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*c^2+d/(
d*f)^(1/2)/(b*(d*f)^(1/2)+a*f+c*d)/(4*a*c-b^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*b*c+d/f/(b*(d*f)^(1/2)+a*f+c*d)/(4*a*c-b^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)*b*c+1/2*d/(d*f)^(1/2)/(b*(d*f)^(1/2)+a*f+
c*d)/(4*a*c-b^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)*b^2+1/2*d/(d*f)^(1/2)/(b*(d*f)^(1/2)+a*f+c*d)/((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)^(3/2)/(-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)^(3/2)/(-f*x^2+d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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)**(3/2)/(-f*x**2+d),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="giac")

[Out]

sage2