\(\int \frac {A+B x+C x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [127]

Optimal result

Integrand size = 29, antiderivative size = 168 \[ \int \frac {A+B x+C x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {\left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d \left (b c^2+a d^2\right ) (c+d x)}+\frac {C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^2}+\frac {\left (a d^2 (2 c C-B d)+b \left (c^3 C-A c d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^2 \left (b c^2+a d^2\right )^{3/2}} \] Output:

-(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^(1/2)/d/(a*d^2+b*c^2)/(d*x+c)+C*arctanh(b^( 
1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^2+(a*d^2*(-B*d+2*C*c)+b*(-A*c*d^2+C*c^3) 
)*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 = 1.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x+C x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {\frac {d \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{\left (b c^2+a d^2\right ) (c+d x)}+\frac {2 \left (a d^2 (2 c C-B d)+b \left (c^3 C-A c d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {C \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2182, 25, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-(((c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2])/(d*(b*c^2 + a*d^2)*(c + d*x))) 
 + ((C*(b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d^2) 
 - ((A*b*c - 2*a*c*C - (b*c^3*C)/d^2 + a*B*d)*ArcTanh[(a*d - b*c*x)/(Sqrt[ 
b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/Sqrt[b*c^2 + a*d^2])/(b*c^2 + a*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(389\) vs. \(2(154)=308\).

Time = 0.19 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.32

method	result	size
default	\(\frac {C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{2} \sqrt {b}}-\frac {\left (B d -2 C c \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}\)	\(390\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (155) = 310\).

Time = 0.08 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.49 \[ \int \frac {A+B x+C x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {b x^{2} + a} C c^{2}}{b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}} + \frac {\sqrt {b x^{2} + a} B c}{b c^{2} d x + a d^{3} x + b c^{3} + a c d^{2}} - \frac {\sqrt {b x^{2} + a} A}{b c^{2} x + a d^{2} x + \frac {b c^{3}}{d} + a c d} + \frac {C \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{2}} + \frac {C b c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{5}} - \frac {B b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{4}} - \frac {2 \, C c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} + \frac {A b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} + \frac {B \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{2}} \] Input:

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

Output:

-sqrt(b*x^2 + a)*C*c^2/(b*c^2*d^2*x + a*d^4*x + b*c^3*d + a*c*d^3) + sqrt( 
b*x^2 + a)*B*c/(b*c^2*d*x + a*d^3*x + b*c^3 + a*c*d^2) - sqrt(b*x^2 + a)*A 
/(b*c^2*x + a*d^2*x + b*c^3/d + a*c*d) + C*arcsinh(b*x/sqrt(a*b))/(sqrt(b) 
*d^2) + C*b*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*ab 
s(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d^5) - B*b*c^2*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^4 
) - 2*C*c*arcsinh(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) + A*b*c*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^3) + B*arcs 
inh(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^2)
 

Giac [F(-2)]

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

integrate((C*x^2+B*x+A)/(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. \[ \int \frac {A+B x+C x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{\sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1148, normalized size of antiderivative = 6.83 \[ \int \frac {A+B x+C x^2}{(c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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)*a*b**2*c**2*d**2 - 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)*a*b**2*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)* 
a*b**2*c*d**3 - 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)*a*b**2*d**4*x + 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**3*d**2 + 4*sq 
rt(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**3*x + 2*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c**5 + 2*sqrt(a*d**2 + b*c**2)*lo 
g( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c**4*d*x + 
 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**2*d**2 + 2*sqrt(a*d**2 + b 
*c**2)*log(c + d*x)*a*b**2*c*d**3*x + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x) 
*a*b**2*c*d**3 + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*d**4*x - 4*sq 
rt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**3*d**2 - 4*sqrt(a*d**2 + b*c**2)*l 
og(c + d*x)*a*b*c**2*d**3*x - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c* 
*5 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**4*d*x - 2*sqrt(a + b*x** 
2)*a**2*b*d**5 - 2*sqrt(a + b*x**2)*a*b**2*c**2*d**3 + 2*sqrt(a + b*x**2)* 
a*b**2*c*d**4 - 2*sqrt(a + b*x**2)*a*b*c**3*d**3 + 2*sqrt(a + b*x**2)*b**3 
*c**3*d**2 - 2*sqrt(a + b*x**2)*b**2*c**5*d - sqrt(b)*log(sqrt(a + b*x*...