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

Optimal. Leaf size=147 \[ \frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2} c^2}+\frac{x}{b c} \]

[Out]

x/(b*c) - (Sqrt[a*c^2 - d^2]*ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]])/(b^(3/2)*c^2) + (Sqrt[a*c^2 - d^2]*ArcTa
n[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])])/(b^(3/2)*c^2) - (d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])
/(b^(3/2)*c^2)

________________________________________________________________________________________

Rubi [A]  time = 0.239423, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2156, 321, 205, 483, 217, 206, 377} \[ \frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{\sqrt{a c^2-d^2} \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2} c^2}+\frac{x}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(b*c) - (Sqrt[a*c^2 - d^2]*ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]])/(b^(3/2)*c^2) + (Sqrt[a*c^2 - d^2]*ArcTa
n[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])])/(b^(3/2)*c^2) - (d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])
/(b^(3/2)*c^2)

Rule 2156

Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[c, Int[u/(c^2 - a*e
^2 + c*d*x^n), x], x] - Dist[a*e, Int[u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d
, e, n}, x] && EqQ[b*c - a*d, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

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

Rule 483

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[(a*e^n)/b, Int[((e*x)^(m - n)*(c + d*x^n)^q)/(a + b*x^n), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.257765, size = 157, normalized size = 1.07 \[ \frac{\sqrt{a c^2-d^2} \left (\sqrt{b} c x-d \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )\right )+\left (a c^2-d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )+\left (d^2-a c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{b^{3/2} c^2 \sqrt{a c^2-d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-(a*c^2) + d^2)*ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]] + (a*c^2 - d^2)*ArcTan[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d
^2]*Sqrt[a + b*x^2])] + Sqrt[a*c^2 - d^2]*(Sqrt[b]*c*x - d*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]))/(b^(3/2)*c^2*S
qrt[a*c^2 - d^2])

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Maple [B]  time = 0.029, size = 3485, normalized size = 23.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*c^2*a/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))
^(1/2))*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)-1/2*d*c^2*a/((-a*b)^(1/2)*c^2+(-c^2*b
*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((b*(x+(-a*b)^(1/2)/b)-(-a*b)^(1/2))/b^(
1/2)+((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2))/b^(1/2)+1/2*d*c^4/((-a*b)^(1/2)*c^2+(-c
^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*((x-(-c^2*b
*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2
)^(1/2)*a-1/2*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))
/(-c^2*b*(a*c^2-d^2))^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^
2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2)*d^3+1/2*d*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(
-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+b*(x-(-c^2*b*(a*c^2-d^2))^(1/2)
/c^2/b))/b^(1/2)+((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-
d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/b^(1/2)*a-1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2
)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+b*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))/b
^(1/2)+((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2
)/c^2/b)+1/c^2*d^2)^(1/2))/b^(1/2)*d^3-1/2*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^
2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^3/(1/c^2*d^2)^(1/2)*ln((2/c^2*d^2+2*(-c^2*b*(a*c^2-
d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(1/c^2*d^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)
^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/(x-(-c^2*b*(a*c^2
-d^2))^(1/2)/c^2/b))*a+1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2
))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^5/(1/c^2*d^2)^(1/2)*ln((2/c^2*d^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(
-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(1/c^2*d^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^
2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))
-1/2*d*c^4/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*
b*(a*c^2-d^2))^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*
c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2)*a+1/2*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)
*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2)*d^3+1/2*d*c^2/((-a*b)^(1/2)*c^2
+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^
2+b*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d
^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/b^(1/2)*a-1/2/((-a*b)^(1/2)*c^2+(-c^2*b*
(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+b*(x+(-
c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2
)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/b^(1/2)*d^3+1/2*c^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*
c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^3/(1/c^2*d^2)^(1/
2)*ln((2/c^2*d^2-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(1/c^2*d^2)^(1/2)*((x
+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1
/c^2*d^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))*a-1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(
-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^5/(1/c^2*d^2)^(1/2)*ln((2/c^2*d^2-2*(
-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(1/c^2*d^2)^(1/2)*((x+(-c^2*b*(a*c^2-d^2)
)^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/(x+
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))-1/2*d*c^2*a/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-
a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2)-1/
2*d*c^2*a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((b*(
x-(-a*b)^(1/2)/b)+(-a*b)^(1/2))/b^(1/2)+((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2))/b^(1
/2)-a/b/(b*(a*c^2-d^2))^(1/2)*arctan(x*c*b/(b*(a*c^2-d^2))^(1/2))+x/b/c+1/b/c^2*d^2/(b*(a*c^2-d^2))^(1/2)*arct
an(x*c*b/(b*(a*c^2-d^2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.30825, size = 2371, normalized size = 16.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*b*c*x + 2*sqrt(b)*d*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + b*sqrt(-(a*c^2 - d^2)/b)*log((a^
4*c^4 - 2*a^3*c^2*d^2 + a^2*d^4 + (a^2*b^2*c^4 - 8*a*b^2*c^2*d^2 + 8*b^2*d^4)*x^4 + 2*(a^3*b*c^4 - 5*a^2*b*c^2
*d^2 + 4*a*b*d^4)*x^2 - 4*((a*b^2*c^2*d - 2*b^2*d^3)*x^3 + (a^2*b*c^2*d - a*b*d^3)*x)*sqrt(b*x^2 + a)*sqrt(-(a
*c^2 - d^2)/b))/(b^2*c^4*x^4 + a^2*c^4 - 2*a*c^2*d^2 + d^4 + 2*(a*b*c^4 - b*c^2*d^2)*x^2)) + 2*b*sqrt(-(a*c^2
- d^2)/b)*log((b*c^2*x^2 - 2*b*c*x*sqrt(-(a*c^2 - d^2)/b) - a*c^2 + d^2)/(b*c^2*x^2 + a*c^2 - d^2)))/(b^2*c^2)
, 1/4*(4*b*c*x + 4*sqrt(-b)*d*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + b*sqrt(-(a*c^2 - d^2)/b)*log((a^4*c^4 - 2*a
^3*c^2*d^2 + a^2*d^4 + (a^2*b^2*c^4 - 8*a*b^2*c^2*d^2 + 8*b^2*d^4)*x^4 + 2*(a^3*b*c^4 - 5*a^2*b*c^2*d^2 + 4*a*
b*d^4)*x^2 - 4*((a*b^2*c^2*d - 2*b^2*d^3)*x^3 + (a^2*b*c^2*d - a*b*d^3)*x)*sqrt(b*x^2 + a)*sqrt(-(a*c^2 - d^2)
/b))/(b^2*c^4*x^4 + a^2*c^4 - 2*a*c^2*d^2 + d^4 + 2*(a*b*c^4 - b*c^2*d^2)*x^2)) + 2*b*sqrt(-(a*c^2 - d^2)/b)*l
og((b*c^2*x^2 - 2*b*c*x*sqrt(-(a*c^2 - d^2)/b) - a*c^2 + d^2)/(b*c^2*x^2 + a*c^2 - d^2)))/(b^2*c^2), 1/2*(2*b*
c*x + 2*b*sqrt((a*c^2 - d^2)/b)*arctan(-b*c*x*sqrt((a*c^2 - d^2)/b)/(a*c^2 - d^2)) - b*sqrt((a*c^2 - d^2)/b)*a
rctan(1/2*(a^2*c^2 - a*d^2 + (a*b*c^2 - 2*b*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt((a*c^2 - d^2)/b)/((a*b*c^2*d - b*d^
3)*x^3 + (a^2*c^2*d - a*d^3)*x)) + sqrt(b)*d*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a))/(b^2*c^2), 1/2*(
2*b*c*x + 2*b*sqrt((a*c^2 - d^2)/b)*arctan(-b*c*x*sqrt((a*c^2 - d^2)/b)/(a*c^2 - d^2)) + 2*sqrt(-b)*d*arctan(s
qrt(-b)*x/sqrt(b*x^2 + a)) - b*sqrt((a*c^2 - d^2)/b)*arctan(1/2*(a^2*c^2 - a*d^2 + (a*b*c^2 - 2*b*d^2)*x^2)*sq
rt(b*x^2 + a)*sqrt((a*c^2 - d^2)/b)/((a*b*c^2*d - b*d^3)*x^3 + (a^2*c^2*d - a*d^3)*x)))/(b^2*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18521, size = 244, normalized size = 1.66 \begin{align*} \frac{x}{b c} - \frac{{\left (a c^{2} - d^{2}\right )} \arctan \left (\frac{b c x}{\sqrt{a b c^{2} - b d^{2}}}\right )}{\sqrt{a b c^{2} - b d^{2}} b c^{2}} + \frac{d \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, b^{\frac{3}{2}} c^{2}} - \frac{{\left (a c^{2} d - d^{3}\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt{a c^{2} - d^{2}} d}\right )}{\sqrt{a c^{2} - d^{2}} b^{\frac{3}{2}} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(b*c) - (a*c^2 - d^2)*arctan(b*c*x/sqrt(a*b*c^2 - b*d^2))/(sqrt(a*b*c^2 - b*d^2)*b*c^2) + 1/2*d*log((sqrt(b)
*x - sqrt(b*x^2 + a))^2)/(b^(3/2)*c^2) - (a*c^2*d - d^3)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*c^2 + a*c
^2 - 2*d^2)/(sqrt(a*c^2 - d^2)*d))/(sqrt(a*c^2 - d^2)*b^(3/2)*c^2*d)