\(\int \frac {x^2 \sqrt {c+d x^2}}{(a+b x^2)^2} \, dx\) [989]

Optimal result

Integrand size = 24, antiderivative size = 120 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {(b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^2 \sqrt {b c-a d}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {-\frac {b x \sqrt {c+d x^2}}{a+b x^2}+\frac {(-b c+2 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} \sqrt {b c-a d}}-2 \sqrt {d} \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{2 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {369, 398, 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.

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

-1/2*(x*Sqrt[c + d*x^2])/(b*(a + b*x^2)) + (((b*c - 2*a*d)*ArcTan[(Sqrt[b* 
c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*b*Sqrt[b*c - a*d]) + (2*S 
qrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b)/(2*b)
 

Defintions of rubi rules used

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 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 369

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

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {x^{2} d +c}\, b x}{b \,x^{2}+a}+\frac {\left (2 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )}{2 b^{2}}\)	\(99\)
default	\(\text {Expression too large to display}\)	\(1959\)

Input:

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

Output:

-1/2/b^2*((d*x^2+c)^(1/2)*b*x/(b*x^2+a)+(2*a*d-b*c)/((a*d-b*c)*a)^(1/2)*ar 
ctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2))-2*d^(1/2)*arctanh((d*x^2+c) 
^(1/2)/x/d^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (98) = 196\).

Time = 0.17 (sec) , antiderivative size = 1069, normalized size of antiderivative = 8.91 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/8*(4*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x - 4*(a^2*b*c - a^3*d + (a*b 
^2*c - a^2*b*d)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - 
c) - (a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*sqrt(-a*b*c + a^2*d)*log((( 
b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d) 
*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c)) 
/(b^2*x^4 + 2*a*b*x^2 + a^2)))/(a^2*b^3*c - a^3*b^2*d + (a*b^4*c - a^2*b^3 
*d)*x^2), -1/8*(4*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x + 8*(a^2*b*c - a^3 
*d + (a*b^2*c - a^2*b*d)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) 
- (a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*sqrt(-a*b*c + a^2*d)*log(((b^2 
*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^ 
2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b 
^2*x^4 + 2*a*b*x^2 + a^2)))/(a^2*b^3*c - a^3*b^2*d + (a*b^4*c - a^2*b^3*d) 
*x^2), -1/4*(2*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x - sqrt(a*b*c - a^2*d) 
*(a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*arctan(1/2*sqrt(a*b*c - a^2*d)* 
((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b* 
c^2 - a^2*c*d)*x)) - 2*(a^2*b*c - a^3*d + (a*b^2*c - a^2*b*d)*x^2)*sqrt(d) 
*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c))/(a^2*b^3*c - a^3*b^2*d + 
 (a*b^4*c - a^2*b^3*d)*x^2), -1/4*(2*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x 
 - sqrt(a*b*c - a^2*d)*(a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*arctan(1/ 
2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (98) = 196\).

Time = 0.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.09 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b c \sqrt {d} - 2 \, a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{2}} - \frac {\sqrt {d} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, b^{2}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{2}} \] Input:

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

Output:

-1/2*(b*c*sqrt(d) - 2*a*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c)) 
^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*b^2) 
 - 1/2*sqrt(d)*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/b^2 + ((sqrt(d)*x - sq 
rt(d*x^2 + c))^2*b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d^(3/2) 
 - b*c^2*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqr 
t(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*b^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 919, normalized size of antiderivative = 7.66 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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