\(\int \frac {A+B x}{(c+d x) \sqrt {a+b x^2}} \, dx\) [174]

Optimal result

Integrand size = 24, antiderivative size = 94 \[ \int \frac {A+B x}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d}+\frac {(B c-A d) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {\frac {2 (B c-A d) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}-\frac {B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {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.

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}-\frac {\left (A d -B c \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(160\)

Input:

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

Output:

B/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)-(A*d-B*c)/d^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 [A] (verification not implemented)

Time = 5.46 (sec) , antiderivative size = 672, normalized size of antiderivative = 7.15 \[ \int \frac {A+B x}{(c+d x) \sqrt {a+b x^2}} \, dx=\left [\frac {{\left (B b c^{2} + B a d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (B b c - A b d\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (b^{2} c^{2} d + a b d^{3}\right )}}, -\frac {2 \, {\left (B b c^{2} + B a d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (B b c - A b d\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (b^{2} c^{2} d + a b d^{3}\right )}}, \frac {2 \, {\left (B b c - A b d\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) + {\left (B b c^{2} + B a d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, {\left (b^{2} c^{2} d + a b d^{3}\right )}}, \frac {{\left (B b c - A b d\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - {\left (B b c^{2} + B a d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{b^{2} c^{2} d + a b d^{3}}\right ] \] Input:

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

Output:

[1/2*((B*b*c^2 + B*a*d^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b) 
*x - a) - (B*b*c - A*b*d)*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)))/(b^2*c^2*d + a*b*d^3), -1/ 
2*(2*(B*b*c^2 + B*a*d^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (B* 
b*c - A*b*d)*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)))/(b^2*c^2*d + a*b*d^3), 1/2*(2*(B*b*c - 
A*b*d)*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*b*c^2 + B* 
a*d^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a))/(b^2*c^2*d 
 + a*b*d^3), ((B*b*c - A*b*d)*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*b*c^2 + B*a*d^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a))) 
/(b^2*c^2*d + a*b*d^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d} - \frac {B c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{2}} + \frac {A \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d} \] Input:

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

Output:

B*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d) - B*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^2) + A*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)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 2126, normalized size of antiderivative = 22.62 \[ \int \frac {A+B x}{(c+d x) \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)* 
sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt( 
b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*c*d + 2*sqrt(b)*sqrt(2* 
sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2) 
*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c* 
*2)*c - a*d**2 - 2*b*c**2))*b*c**2 - 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2 
)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sq 
rt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*d**3 - 2*sqrt(2*s 
qrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2) 
*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c* 
*2))*a*b*c**2*d + 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b* 
c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + 
 b*c**2)*c - a*d**2 - 2*b*c**2))*a*b*c*d**2 + 2*sqrt(2*sqrt(b)*sqrt(a*d**2 
 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/ 
sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*b**2*c**3 - s 
qrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2)*sqrt(a* 
d**2 + b*c**2)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2* 
b*c**2) + sqrt(a + b*x**2)*d + sqrt(b)*d*x)*a*c*d + sqrt(b)*sqrt(2*sqrt(b) 
*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2)*sqrt(a*d**2 + b*c**2)*log( - 
 sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + sqrt(a +...