\(\int \frac {A+B x}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [184]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {B c-A d}{\left (b c^2+a d^2\right ) (c+d x) \sqrt {a+b x^2}}+\frac {a \left (a B d^2-b c (2 B c-3 A d)\right )+b \left (A b c^2+3 a B c d-2 a A d^2\right ) x}{a \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {d \left (a B d^2-b c (2 B c-3 A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{5/2}} \] Output:

(-A*d+B*c)/(a*d^2+b*c^2)/(d*x+c)/(b*x^2+a)^(1/2)+(a*(a*B*d^2-b*c*(-3*A*d+2 
*B*c))+b*(-2*A*a*d^2+A*b*c^2+3*B*a*c*d)*x)/a/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/ 
2)-d*(a*B*d^2-b*c*(-3*A*d+2*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)^(5/2)
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b^2 c^2 x (c+d x)+a^2 d^2 (2 B c-A d+B d x)+a b \left (A d \left (2 c^2+c d x-2 d^2 x^2\right )+B c \left (-c^2+c d x+3 d^2 x^2\right )\right )}{a \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}-\frac {2 d \left (a B d^2+b c (-2 B c+3 A d)\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

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

Output:

(A*b^2*c^2*x*(c + d*x) + a^2*d^2*(2*B*c - A*d + B*d*x) + a*b*(A*d*(2*c^2 + 
 c*d*x - 2*d^2*x^2) + B*c*(-c^2 + c*d*x + 3*d^2*x^2)))/(a*(b*c^2 + a*d^2)^ 
2*(c + d*x)*Sqrt[a + b*x^2]) - (2*d*(a*B*d^2 + b*c*(-2*B*c + 3*A*d))*ArcTa 
n[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^ 
2) - a*d^2)^(5/2)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {686, 27, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

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. \(838\) vs. \(2(180)=360\).

Time = 1.30 (sec) , antiderivative size = 839, normalized size of antiderivative = 4.42

method	result	size
default	\(\frac {B \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}}+\frac {\left (A d -B c \right ) \left (-\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\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 {3 b c d \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 )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \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}}}}\right )}{d^{3}}\)	\(839\)

Input:

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

Output:

B/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)*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)+3*b*c*d/(a*d^2+b*c^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)*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)))-4*b/(a*d^2+b*c^2)*d^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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (181) = 362\).

Time = 0.75 (sec) , antiderivative size = 1297, normalized size of antiderivative = 6.83 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*((2*B*a^2*b*c^3*d - 3*A*a^2*b*c^2*d^2 - B*a^3*c*d^3 + (2*B*a*b^2*c^2 
*d^2 - 3*A*a*b^2*c*d^3 - B*a^2*b*d^4)*x^3 + (2*B*a*b^2*c^3*d - 3*A*a*b^2*c 
^2*d^2 - B*a^2*b*c*d^3)*x^2 + (2*B*a^2*b*c^2*d^2 - 3*A*a^2*b*c*d^3 - B*a^3 
*d^4)*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*a*b^2*c^5 - 2*A*a*b^2*c^4*d - B*a^2*b 
*c^3*d^2 - A*a^2*b*c^2*d^3 - 2*B*a^3*c*d^4 + A*a^3*d^5 - (A*b^3*c^4*d + 3* 
B*a*b^2*c^3*d^2 - A*a*b^2*c^2*d^3 + 3*B*a^2*b*c*d^4 - 2*A*a^2*b*d^5)*x^2 - 
 (A*b^3*c^5 + B*a*b^2*c^4*d + 2*A*a*b^2*c^3*d^2 + 2*B*a^2*b*c^2*d^3 + A*a^ 
2*b*c*d^4 + B*a^3*d^5)*x)*sqrt(b*x^2 + a))/(a^2*b^3*c^7 + 3*a^3*b^2*c^5*d^ 
2 + 3*a^4*b*c^3*d^4 + a^5*c*d^6 + (a*b^4*c^6*d + 3*a^2*b^3*c^4*d^3 + 3*a^3 
*b^2*c^2*d^5 + a^4*b*d^7)*x^3 + (a*b^4*c^7 + 3*a^2*b^3*c^5*d^2 + 3*a^3*b^2 
*c^3*d^4 + a^4*b*c*d^6)*x^2 + (a^2*b^3*c^6*d + 3*a^3*b^2*c^4*d^3 + 3*a^4*b 
*c^2*d^5 + a^5*d^7)*x), ((2*B*a^2*b*c^3*d - 3*A*a^2*b*c^2*d^2 - B*a^3*c*d^ 
3 + (2*B*a*b^2*c^2*d^2 - 3*A*a*b^2*c*d^3 - B*a^2*b*d^4)*x^3 + (2*B*a*b^2*c 
^3*d - 3*A*a*b^2*c^2*d^2 - B*a^2*b*c*d^3)*x^2 + (2*B*a^2*b*c^2*d^2 - 3*A*a 
^2*b*c*d^3 - B*a^3*d^4)*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*a*b^2*c^5 - 2*A*a*b^2*c^4*d - B*a^2*b*c^3*d^2 - A*a^2*b*c^2*d^3 
- 2*B*a^3*c*d^4 + A*a^3*d^5 - (A*b^3*c^4*d + 3*B*a*b^2*c^3*d^2 - A*a*b^...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (181) = 362\).

Time = 0.11 (sec) , antiderivative size = 669, normalized size of antiderivative = 3.52 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \, B b^{2} c^{3} x}{\sqrt {b x^{2} + a} a b^{2} c^{4} d + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d^{3} + \sqrt {b x^{2} + a} a^{3} d^{5}} + \frac {3 \, A b^{2} c^{2} x}{\sqrt {b x^{2} + a} a b^{2} c^{4} + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{3} d^{4}} - \frac {3 \, B b c^{2}}{\sqrt {b x^{2} + a} b^{2} c^{4} + 2 \, \sqrt {b x^{2} + a} a b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{2} d^{4}} + \frac {3 \, B b c x}{\sqrt {b x^{2} + a} a b c^{2} d + \sqrt {b x^{2} + a} a^{2} d^{3}} + \frac {3 \, A b c}{\frac {\sqrt {b x^{2} + a} b^{2} c^{4}}{d} + 2 \, \sqrt {b x^{2} + a} a b c^{2} d + \sqrt {b x^{2} + a} a^{2} d^{3}} - \frac {2 \, A b x}{\sqrt {b x^{2} + a} a b c^{2} + \sqrt {b x^{2} + a} a^{2} d^{2}} + \frac {B c}{\sqrt {b x^{2} + a} b c^{2} d x + \sqrt {b x^{2} + a} a d^{3} x + \sqrt {b x^{2} + a} b c^{3} + \sqrt {b x^{2} + a} a c d^{2}} - \frac {A}{\sqrt {b x^{2} + a} b c^{2} x + \sqrt {b x^{2} + a} a d^{2} x + \frac {\sqrt {b x^{2} + a} b c^{3}}{d} + \sqrt {b x^{2} + a} a c d} + \frac {B}{\sqrt {b x^{2} + a} b c^{2} + \sqrt {b x^{2} + a} a d^{2}} - \frac {3 \, B b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{4}} + \frac {3 \, A b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{3}} + \frac {B \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{2}} \] Input:

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

Output:

-3*B*b^2*c^3*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d + 2*sqrt(b*x^2 + a)*a^2*b*c^2* 
d^3 + sqrt(b*x^2 + a)*a^3*d^5) + 3*A*b^2*c^2*x/(sqrt(b*x^2 + a)*a*b^2*c^4 
+ 2*sqrt(b*x^2 + a)*a^2*b*c^2*d^2 + sqrt(b*x^2 + a)*a^3*d^4) - 3*B*b*c^2/( 
sqrt(b*x^2 + a)*b^2*c^4 + 2*sqrt(b*x^2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)* 
a^2*d^4) + 3*B*b*c*x/(sqrt(b*x^2 + a)*a*b*c^2*d + sqrt(b*x^2 + a)*a^2*d^3) 
 + 3*A*b*c/(sqrt(b*x^2 + a)*b^2*c^4/d + 2*sqrt(b*x^2 + a)*a*b*c^2*d + sqrt 
(b*x^2 + a)*a^2*d^3) - 2*A*b*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*d*x + sqrt(b*x^2 + a)*a*d^3*x + sqrt 
(b*x^2 + a)*b*c^3 + sqrt(b*x^2 + a)*a*c*d^2) - A/(sqrt(b*x^2 + a)*b*c^2*x 
+ sqrt(b*x^2 + a)*a*d^2*x + sqrt(b*x^2 + a)*b*c^3/d + sqrt(b*x^2 + a)*a*c* 
d) + B/(sqrt(b*x^2 + a)*b*c^2 + sqrt(b*x^2 + a)*a*d^2) - 3*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)^(5/2)*d^4) + 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)^(5/2)*d^3) + B*arcsinh(b*c*x/(sq 
rt(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)
                                                                                    
                                                                                    
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 1540, normalized size of antiderivative = 8.11 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*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**2*b*c**2*d**2 + 3*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)*a**2*b*c*d**3*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**2*b*c*d**3 
+ 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**2*b*d**4*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**3*d + 3*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)*a*b**2*c**2*d**2*x 
**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**2*d**2*x + 3*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**3 + 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*d**4*x**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)*b**3*c**3*d*x**2 - 2*sqr 
t(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**3*c**2*d**2*x**3 - 3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**2* 
d**2 - 3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**3*x - sqrt(a*d**2 
+ b*c**2)*log(c + d*x)*a**2*b*c*d**3 - sqrt(a*d**2 + b*c**2)*log(c + d*x)* 
a**2*b*d**4*x + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**3*d - 3*...