\(\int \frac {x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [1193]

Optimal result

Integrand size = 22, antiderivative size = 137 \[ \int \frac {x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {c^2 \sqrt {a+b x^2}}{d \left (b c^2+a d^2\right ) (c+d x)}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^2}+\frac {c \left (b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^2 \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {c \left (-\frac {c d \sqrt {a+b x^2}}{\left (b c^2+a d^2\right ) (c+d x)}-\frac {2 \left (b c^2+2 a d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}\right )+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{\sqrt {b}}}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {603, 27, 719, 224, 219, 488, 219}

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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((c^2*Sqrt[a + b*x^2])/(d*(b*c^2 + a*d^2)*(c + d*x))) - (-(((b*c^2 + a*d^ 
2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (c*(b*c^2 + 2*a*d^ 
2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*Sqrt[b 
*c^2 + a*d^2]))/(d*(b*c^2 + a*d^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 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 488

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

rule 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(368\) vs. \(2(123)=246\).

Time = 0.36 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.69

method	result	size
default	\(\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{2} \sqrt {b}}+\frac {c^{2} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}+\frac {2 c \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(369\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (124) = 248\).

Time = 3.79 (sec) , antiderivative size = 1260, normalized size of antiderivative = 9.20 \[ \int \frac {x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*((b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4 + (b^2*c^4*d + 2*a*b*c^2*d^3 + 
a^2*d^5)*x)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (b^2 
*c^4 + 2*a*b*c^2*d^2 + (b^2*c^3*d + 2*a*b*c*d^3)*x)*sqrt(b*c^2 + a*d^2)*lo 
g((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt( 
b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 
 2*(b^2*c^4*d + a*b*c^2*d^3)*sqrt(b*x^2 + a))/(b^3*c^5*d^2 + 2*a*b^2*c^3*d 
^4 + a^2*b*c*d^6 + (b^3*c^4*d^3 + 2*a*b^2*c^2*d^5 + a^2*b*d^7)*x), 1/2*(2* 
(b^2*c^4 + 2*a*b*c^2*d^2 + (b^2*c^3*d + 2*a*b*c*d^3)*x)*sqrt(-b*c^2 - a*d^ 
2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^ 
2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + (b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4 + 
 (b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5)*x)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b* 
x^2 + a)*sqrt(b)*x - a) - 2*(b^2*c^4*d + a*b*c^2*d^3)*sqrt(b*x^2 + a))/(b^ 
3*c^5*d^2 + 2*a*b^2*c^3*d^4 + a^2*b*c*d^6 + (b^3*c^4*d^3 + 2*a*b^2*c^2*d^5 
 + a^2*b*d^7)*x), -1/2*(2*(b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4 + (b^2*c^4* 
d + 2*a*b*c^2*d^3 + a^2*d^5)*x)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a) 
) - (b^2*c^4 + 2*a*b*c^2*d^2 + (b^2*c^3*d + 2*a*b*c*d^3)*x)*sqrt(b*c^2 + a 
*d^2)*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 
 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + 
 c^2)) + 2*(b^2*c^4*d + a*b*c^2*d^3)*sqrt(b*x^2 + a))/(b^3*c^5*d^2 + 2*a*b 
^2*c^3*d^4 + a^2*b*c*d^6 + (b^3*c^4*d^3 + 2*a*b^2*c^2*d^5 + a^2*b*d^7)*...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.25 \[ \int \frac {x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {b x^{2} + a} c^{2}}{b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}} + \frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{2}} + \frac {b c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{5}} - \frac {2 \, c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} \] Input:

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

Output:

-sqrt(b*x^2 + a)*c^2/(b*c^2*d^2*x + a*d^4*x + b*c^3*d + a*c*d^3) + arcsinh 
(b*x/sqrt(a*b))/(sqrt(b)*d^2) + b*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c 
)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d^5) - 2*c*arcsi 
nh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a 
+ b*c^2/d^2)*d^3)
 

Giac [F(-2)]

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

integrate(x^2/(d*x+c)^2/(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:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 730, normalized size of antiderivative = 5.33 \[ \int \frac {x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a 
*d + b*c*x)*a*b*c**2*d**2 + 4*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2 
)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c*d**3*x + 2*sqrt(a*d**2 + b*c* 
*2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c**4 
 + 2*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
 a*d + b*c*x)*b**2*c**3*d*x - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c** 
2*d**2 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c*d**3*x - 2*sqrt(a*d**2 
 + b*c**2)*log(c + d*x)*b**2*c**4 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b 
**2*c**3*d*x - 2*sqrt(a + b*x**2)*a*b*c**2*d**3 - 2*sqrt(a + b*x**2)*b**2* 
c**4*d - sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*c*d**4 - sqrt(b)*l 
og(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*d**5*x - 2*sqrt(b)*log(sqrt(a + b*x* 
*2) - sqrt(b)*x)*a*b*c**3*d**2 - 2*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)* 
x)*a*b*c**2*d**3*x - sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c**5 - 
 sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c**4*d*x + sqrt(b)*log(sqr 
t(a + b*x**2) + sqrt(b)*x)*a**2*c*d**4 + sqrt(b)*log(sqrt(a + b*x**2) + sq 
rt(b)*x)*a**2*d**5*x + 2*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a*b*c** 
3*d**2 + 2*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a*b*c**2*d**3*x + sqr 
t(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*b**2*c**5 + sqrt(b)*log(sqrt(a + b* 
x**2) + sqrt(b)*x)*b**2*c**4*d*x)/(2*b*d**2*(a**2*c*d**4 + a**2*d**5*x + 2 
*a*b*c**3*d**2 + 2*a*b*c**2*d**3*x + b**2*c**5 + b**2*c**4*d*x))