\(\int \frac {A+B x}{(c+d x) (a+b x^2)^{3/2}} \, dx\) [183]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a (B c-A d)-(A b c+a B d) x}{a \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {d (B c-A d) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b c x+a (-B c+A d+B d x)}{a \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {2 d (B c-A d) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}} \] Input:

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

Output:

(A*b*c*x + a*(-(B*c) + A*d + B*d*x))/(a*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) - 
 (2*d*(B*c - A*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 = 0.25 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {686, 27, 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.

\(\displaystyle \int \frac {A+B x}{\left (a+b x^2\right )^{3/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((a*(B*c - A*d) - (A*b*c + a*B*d)*x)/(a*(b*c^2 + a*d^2)*Sqrt[a + b*x^2])) 
 + (d*(B*c - A*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 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 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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(341\) vs. \(2(107)=214\).

Time = 1.28 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.97

method	result	size
default	\(\frac {B x}{d \sqrt {b \,x^{2}+a}\, a}+\frac {\left (A d -B c \right ) \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\)	\(342\)

Input:

int((B*x+A)/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

B/d/(b*x^2+a)^(1/2)/a*x+(A*d-B*c)/d^2*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2* 
b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)- 
2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d 
)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*l 
n((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. 272 vs. \(2 (108) = 216\).

Time = 0.32 (sec) , antiderivative size = 571, normalized size of antiderivative = 4.97 \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (B a^{2} c d - A a^{2} d^{2} + {\left (B a b c d - A a b d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (B a b c^{3} - A a b c^{2} d + B a^{2} c d^{2} - A a^{2} d^{3} - {\left (A b^{2} c^{3} + B a b c^{2} d + A a b c d^{2} + B a^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4} + {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}\right )} x^{2}\right )}}, \frac {{\left (B a^{2} c d - A a^{2} d^{2} + {\left (B a b c d - A a b d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - {\left (B a b c^{3} - A a b c^{2} d + B a^{2} c d^{2} - A a^{2} d^{3} - {\left (A b^{2} c^{3} + B a b c^{2} d + A a b c d^{2} + B a^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4} + {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}\right )} x^{2}}\right ] \] Input:

integrate((B*x+A)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/2*((B*a^2*c*d - A*a^2*d^2 + (B*a*b*c*d - A*a*b*d^2)*x^2)*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*a*b*c^3 - A*a*b*c^2*d + B*a^2*c*d^2 - A*a^2*d^3 - (A*b^2*c^ 
3 + B*a*b*c^2*d + A*a*b*c*d^2 + B*a^2*d^3)*x)*sqrt(b*x^2 + a))/(a^2*b^2*c^ 
4 + 2*a^3*b*c^2*d^2 + a^4*d^4 + (a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4 
)*x^2), ((B*a^2*c*d - A*a^2*d^2 + (B*a*b*c*d - A*a*b*d^2)*x^2)*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*a*b*c^3 - A*a*b*c^2*d + B*a^ 
2*c*d^2 - A*a^2*d^3 - (A*b^2*c^3 + B*a*b*c^2*d + A*a*b*c*d^2 + B*a^2*d^3)* 
x)*sqrt(b*x^2 + a))/(a^2*b^2*c^4 + 2*a^3*b*c^2*d^2 + a^4*d^4 + (a*b^3*c^4 
+ 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)*x^2)]
 

Sympy [F]

\[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:

integrate((B*x+A)/(d*x+c)/(b*x**2+a)**(3/2),x)
 

Output:

Integral((A + B*x)/((a + b*x**2)**(3/2)*(c + d*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (108) = 216\).

Time = 0.07 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.42 \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {B b c^{2} x}{\sqrt {b x^{2} + a} a b c^{2} d + \sqrt {b x^{2} + a} a^{2} d^{3}} + \frac {A b c x}{\sqrt {b x^{2} + a} a b c^{2} + \sqrt {b x^{2} + a} a^{2} d^{2}} - \frac {B c}{\sqrt {b x^{2} + a} b c^{2} + \sqrt {b x^{2} + a} a d^{2}} + \frac {A}{\frac {\sqrt {b x^{2} + a} b c^{2}}{d} + \sqrt {b x^{2} + a} a d} + \frac {B x}{\sqrt {b x^{2} + a} a d} - \frac {B c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{2}} + \frac {A \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d} \] Input:

integrate((B*x+A)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (108) = 216\).

Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.97 \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (A b^{2} c^{3} + B a b c^{2} d + A a b c d^{2} + B a^{2} d^{3}\right )} x}{a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}} - \frac {B a b c^{3} - A a b c^{2} d + B a^{2} c d^{2} - A a^{2} d^{3}}{a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}}}{\sqrt {b x^{2} + a}} + \frac {2 \, {\left (B c d - A d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt {-b c^{2} - a d^{2}}} \] Input:

integrate((B*x+A)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

((A*b^2*c^3 + B*a*b*c^2*d + A*a*b*c*d^2 + B*a^2*d^3)*x/(a*b^2*c^4 + 2*a^2* 
b*c^2*d^2 + a^3*d^4) - (B*a*b*c^3 - A*a*b*c^2*d + B*a^2*c*d^2 - A*a^2*d^3) 
/(a*b^2*c^4 + 2*a^2*b*c^2*d^2 + a^3*d^4))/sqrt(b*x^2 + a) + 2*(B*c*d - A*d 
^2)*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d 
^2))/((b*c^2 + a*d^2)*sqrt(-b*c^2 - a*d^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:

int((A + B*x)/((a + b*x^2)^(3/2)*(c + d*x)),x)
 

Output:

int((A + B*x)/((a + b*x^2)^(3/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 4555, normalized size of antiderivative = 39.61 \[ \int \frac {A+B x}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(d*x+c)/(b*x^2+a)^(3/2),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**2*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))*a*b*c**2 - 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*b*c*d*x**2 + 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 + s 
qrt(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**2*x**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))*a**3*d**3 - 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)/sqr 
t(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*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**2*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(...