\(\int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx\) [54]

Optimal result

Integrand size = 29, antiderivative size = 106 \[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\frac {x}{b c}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} \sqrt {a c^2-d^2} x}{a c+d \sqrt {a+b x^2}}\right )}{b^{3/2} c^2}-\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b} c x+2 \sqrt {a c^2-d^2} \arctan \left (\frac {d+c \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {a c^2-d^2}}\right )+d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2} c^2} \] Input:

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

Output:

(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)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.47, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2587, 27, 262, 218, 385, 224, 219, 291, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2587

\(\displaystyle a c \int \frac {x^2}{a \left (b x^2 c^2+a c^2-d^2\right )}dx-a d \int \frac {x^2}{a \sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle c \int \frac {x^2}{b x^2 c^2+a c^2-d^2}dx-d \int \frac {x^2}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 262

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \int \frac {1}{b x^2 c^2+a c^2-d^2}dx}{b c^2}\right )-d \int \frac {x^2}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 218

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{b^{3/2} c^3}\right )-d \int \frac {x^2}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 385

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{b^{3/2} c^3}\right )-d \left (\frac {\int \frac {1}{\sqrt {b x^2+a}}dx}{b c^2}-\frac {\left (a c^2-d^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx}{b c^2}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{b^{3/2} c^3}\right )-d \left (\frac {\int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b c^2}-\frac {\left (a c^2-d^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx}{b c^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{b^{3/2} c^3}\right )-d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} c^2}-\frac {\left (a c^2-d^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx}{b c^2}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{b^{3/2} c^3}\right )-d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} c^2}-\frac {\left (a c^2-d^2\right ) \int \frac {1}{a c^2-d^2-\frac {\left (b \left (a c^2-d^2\right )-a b c^2\right ) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b c^2}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{b^{3/2} c^3}\right )-d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} c^2}-\frac {\sqrt {a c^2-d^2} \arctan \left (\frac {\sqrt {b} d x}{\sqrt {a+b x^2} \sqrt {a c^2-d^2}}\right )}{b^{3/2} c^2 d}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 291

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

rule 385

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

rule 2587

Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_ 
Symbol] :> Simp[c   Int[u/(c^2 - a*e^2 + c*d*x^n), x], x] - Simp[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1779\) vs. \(2(90)=180\).

Time = 0.03 (sec) , antiderivative size = 1780, normalized size of antiderivative = 16.79

method	result	size
default	\(\text {Expression too large to display}\)	\(1780\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (90) = 180\).

Time = 0.17 (sec) , antiderivative size = 1168, normalized size of antiderivative = 11.02 \[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[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)*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), 1/2*(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)*arctan(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...
 

Sympy [F]

\[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\int \frac {x^{2}}{a c + b c x^{2} + d \sqrt {a + b x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\int { \frac {x^{2}}{b c x^{2} + a c + \sqrt {b x^{2} + a} d} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\int \frac {x^2}{a\,c+d\,\sqrt {b\,x^2+a}+b\,c\,x^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a \,c^{2}-d^{2}}\, \mathit {atan} \left (\frac {\sqrt {b \,x^{2}+a}\, c +\sqrt {b}\, c x +d}{\sqrt {a \,c^{2}-d^{2}}}\right )-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) d +b c x}{b^{2} c^{2}} \] Input:

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

Output:

( - 2*sqrt(b)*sqrt(a*c**2 - d**2)*atan((sqrt(a + b*x**2)*c + sqrt(b)*c*x + 
 d)/sqrt(a*c**2 - d**2)) - sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt 
(a))*d + b*c*x)/(b**2*c**2)