\(\int \frac {A+B x}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [175]

Optimal result

Integrand size = 24, antiderivative size = 104 \[ \int \frac {A+B x}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {(B c-A d) \sqrt {a+b x^2}}{\left (b c^2+a d^2\right ) (c+d x)}-\frac {(A b c+a B d) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {(B c-A d) \sqrt {a+b x^2}}{\left (b c^2+a d^2\right ) (c+d x)}+\frac {2 (A b c+a B d) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {679, 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.

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(96)=192\).

Time = 1.34 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.38

method	result	size
default	\(-\frac {B \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\left (A d -B c \right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}\)	\(351\)

Input:

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

Output:

-B/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))+(A*d-B*c)/d^3*(-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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (97) = 194\).

Time = 0.25 (sec) , antiderivative size = 446, normalized size of antiderivative = 4.29 \[ \int \frac {A+B x}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\left [\frac {{\left (A b c^{2} + B a c d + {\left (A b c d + B a d^{2}\right )} x\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (B b c^{3} - A b c^{2} d + B a c d^{2} - A a d^{3}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (b^{2} c^{5} + 2 \, a b c^{3} d^{2} + a^{2} c d^{4} + {\left (b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}\right )} x\right )}}, -\frac {{\left (A b c^{2} + B a c d + {\left (A b c d + B a d^{2}\right )} x\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - {\left (B b c^{3} - A b c^{2} d + B a c d^{2} - A a d^{3}\right )} \sqrt {b x^{2} + a}}{b^{2} c^{5} + 2 \, a b c^{3} d^{2} + a^{2} c d^{4} + {\left (b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((A + B*x)/(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. 245 vs. \(2 (97) = 194\).

Time = 0.07 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.36 \[ \int \frac {A+B x}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} B c}{b c^{2} d x + a d^{3} x + b c^{3} + a c d^{2}} - \frac {\sqrt {b x^{2} + a} A}{b c^{2} x + a d^{2} x + \frac {b c^{3}}{d} + a c d} - \frac {B b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{4}} + \frac {A b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} + \frac {B \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{2}} \] Input:

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

Output:

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) - 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) + 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*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)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 416, normalized size of antiderivative = 4.00 \[ \int \frac {A+B x}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,c^{2}+\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b c d x +\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b c d +\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,d^{2} x -\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,c^{2}-\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b c d x -\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b c d -\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,d^{2} x -\sqrt {b \,x^{2}+a}\, a^{2} d^{3}-\sqrt {b \,x^{2}+a}\, a b \,c^{2} d +\sqrt {b \,x^{2}+a}\, a b c \,d^{2}+\sqrt {b \,x^{2}+a}\, b^{2} c^{3}}{a^{2} d^{5} x +2 a b \,c^{2} d^{3} x +b^{2} c^{4} d x +a^{2} c \,d^{4}+2 a b \,c^{3} d^{2}+b^{2} c^{5}} \] Input:

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

Output:

(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 + 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*x + 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 + 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*d**2*x - sqr 
t(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2 - sqrt(a*d**2 + b*c**2)*log(c + d 
*x)*a*b*c*d*x - sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c*d - sqrt(a*d**2 + 
 b*c**2)*log(c + d*x)*a*b*d**2*x - sqrt(a + b*x**2)*a**2*d**3 - sqrt(a + b 
*x**2)*a*b*c**2*d + sqrt(a + b*x**2)*a*b*c*d**2 + sqrt(a + b*x**2)*b**2*c* 
*3)/(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)