3.6.49 \(\int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx\) [549]

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

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 147, normalized size of antiderivative = 1.00, 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 = {2187, 327, 211, 494, 223, 212, 385} \begin {gather*} \frac {\sqrt {a c^2-d^2} \text {ArcTan}\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} \text {ArcTan}\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} \end {gather*}

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 211

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

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 223

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 327

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 385

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]

Rule 494

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 2187

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]

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 \text {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 ) \text {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*}

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Mathematica [A]
time = 0.26, size = 99, normalized size = 0.67 \begin {gather*} \frac {\sqrt {b} c x+2 \sqrt {a c^2-d^2} \tan ^{-1}\left (\frac {d+c \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {a c^2-d^2}}\right )+d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3484\) vs. \(2(123)=246\).
time = 0.04, size = 3485, normalized size = 23.71

method	result	size
default	\(\text {Expression too large to display}\)	\(3485\)

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,method=_RETURNVERBOSE)

[Out]

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

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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(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 = 0.49, size = 1168, normalized size = 7.95 \begin {gather*} \left [\frac {4 \, b c x + 2 \, \sqrt {b} d \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + b \sqrt {-\frac {a c^{2} - d^{2}}{b}} \log \left (\frac {a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} + {\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2} - 4 \, {\left ({\left (a b^{2} c^{2} d - 2 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} b c^{2} d - a b d^{3}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {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 \, {\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right ) + 2 \, b \sqrt {-\frac {a c^{2} - d^{2}}{b}} \log \left (\frac {b c^{2} x^{2} - 2 \, b c x \sqrt {-\frac {a c^{2} - d^{2}}{b}} - a c^{2} + d^{2}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right )}{4 \, b^{2} c^{2}}, \frac {4 \, b c x + 4 \, \sqrt {-b} d \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + b \sqrt {-\frac {a c^{2} - d^{2}}{b}} \log \left (\frac {a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} + {\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2} - 4 \, {\left ({\left (a b^{2} c^{2} d - 2 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} b c^{2} d - a b d^{3}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {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 \, {\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right ) + 2 \, b \sqrt {-\frac {a c^{2} - d^{2}}{b}} \log \left (\frac {b c^{2} x^{2} - 2 \, b c x \sqrt {-\frac {a c^{2} - d^{2}}{b}} - a c^{2} + d^{2}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right )}{4 \, b^{2} c^{2}}, \frac {2 \, b c x + 2 \, b \sqrt {\frac {a c^{2} - d^{2}}{b}} \arctan \left (-\frac {b c x \sqrt {\frac {a c^{2} - d^{2}}{b}}}{a c^{2} - d^{2}}\right ) - b \sqrt {\frac {a c^{2} - d^{2}}{b}} \arctan \left (\frac {{\left (a^{2} c^{2} - a d^{2} + {\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {a c^{2} - d^{2}}{b}}}{2 \, {\left ({\left (a b c^{2} d - b d^{3}\right )} x^{3} + {\left (a^{2} c^{2} d - a d^{3}\right )} x\right )}}\right ) + \sqrt {b} d \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, b^{2} c^{2}}, \frac {2 \, b c x + 2 \, b \sqrt {\frac {a c^{2} - d^{2}}{b}} \arctan \left (-\frac {b c x \sqrt {\frac {a c^{2} - d^{2}}{b}}}{a c^{2} - d^{2}}\right ) + 2 \, \sqrt {-b} d \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - b \sqrt {\frac {a c^{2} - d^{2}}{b}} \arctan \left (\frac {{\left (a^{2} c^{2} - a d^{2} + {\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {a c^{2} - d^{2}}{b}}}{2 \, {\left ({\left (a b c^{2} d - b d^{3}\right )} x^{3} + {\left (a^{2} c^{2} d - a d^{3}\right )} x\right )}}\right )}{2 \, b^{2} c^{2}}\right ] \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{a c + b c x^{2} + d \sqrt {a + b x^{2}}}\, dx \end {gather*}

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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