\(\int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^2} \, dx\) [153]

Optimal result

Integrand size = 24, antiderivative size = 150 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {(2 B c-A d+B d x) \sqrt {a+b x^2}}{d^2 (c+d x)}-\frac {\sqrt {b} (2 B c-A d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {\left (a B d^2+b c (2 B c-A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {\frac {d (2 B c-A d+B d x) \sqrt {a+b x^2}}{c+d x}-\frac {2 \left (a B d^2+b c (2 B c-A d)\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\sqrt {b} (2 B c-A d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {681, 27, 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.

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*B*c - A*d + B*d*x)*Sqrt[a + b*x^2])/(d^2*(c + d*x)) + (-((Sqrt[b]*(2*B 
*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d) - ((a*B*d^2 + b*c*(2*B* 
c - A*d))*ArcTanh[(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]))/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 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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(136)=272\).

Time = 1.36 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.94

method	result	size
risch	\(\frac {B \sqrt {b \,x^{2}+a}}{d^{2}}+\frac {\frac {\sqrt {b}\, \left (A d -2 B c \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d}+\frac {\left (2 A b c d -a B \,d^{2}-3 B b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\left (A \,d^{3} a +A b \,c^{2} d -a B c \,d^{2}-b B \,c^{3}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{d^{2}}\)	\(441\)
default	\(\frac {B \left (\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 {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}-\frac {\left (a \,d^{2}+b \,c^{2}\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}}}}\right )}{d^{2}}+\frac {\left (A d -B c \right ) \left (-\frac {d^{2} \left (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 )^{\frac {3}{2}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \left (\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 {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}-\frac {\left (a \,d^{2}+b \,c^{2}\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}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {2 b \,d^{2} \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b 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}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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 )}{8 b^{\frac {3}{2}}}\right )}{a \,d^{2}+b \,c^{2}}\right )}{d^{3}}\)	\(806\)

Input:

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

Output:

B/d^2*(b*x^2+a)^(1/2)+1/d^2*(b^(1/2)*(A*d-2*B*c)/d*ln(b^(1/2)*x+(b*x^2+a)^ 
(1/2))+1/d^2*(2*A*b*c*d-B*a*d^2-3*B*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))+1/d^3*(A*a*d^3+A*b*c^ 
2*d-B*a*c*d^2-B*b*c^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. 326 vs. \(2 (137) = 274\).

Time = 64.39 (sec) , antiderivative size = 1369, normalized size of antiderivative = 9.13 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {\sqrt {b x^{2} + a} B c}{d^{3} x + c d^{2}} - \frac {\sqrt {b x^{2} + a} A}{d^{2} x + c d} - \frac {2 \, B \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} + \frac {A \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{2}} + \frac {B b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} - \frac {A b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} + \frac {B \sqrt {a + \frac {b c^{2}}{d^{2}}} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} + \frac {\sqrt {b x^{2} + a} B}{d^{2}} \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1018, normalized size of antiderivative = 6.79 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(b*x^2+a)^(1/2)/(d*x+c)^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*c**2*d + 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)*a*b*c*d**2*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*c*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*d**3*x - 4*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**3 - 4*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**2*d*x - 
 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d - 2*sqrt(a*d**2 + b*c**2) 
*log(c + d*x)*a*b*c*d**2*x + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c*d* 
*2 + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*d**3*x + 4*sqrt(a*d**2 + b*c 
**2)*log(c + d*x)*b**2*c**3 + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c* 
*2*d*x - 2*sqrt(a + b*x**2)*a**2*d**4 - 2*sqrt(a + b*x**2)*a*b*c**2*d**2 + 
 4*sqrt(a + b*x**2)*a*b*c*d**3 + 2*sqrt(a + b*x**2)*a*b*d**4*x + 4*sqrt(a 
+ b*x**2)*b**2*c**3*d + 2*sqrt(a + b*x**2)*b**2*c**2*d**2*x - sqrt(b)*log( 
sqrt(a + b*x**2) - sqrt(b)*x)*a**2*c*d**3 - sqrt(b)*log(sqrt(a + b*x**2) - 
 sqrt(b)*x)*a**2*d**4*x - sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*b*c* 
*3*d - sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*b*c**2*d**2*x + 2*sqrt( 
b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*b*c**2*d**2 + 2*sqrt(b)*log(sqrt(a 
+ b*x**2) - sqrt(b)*x)*a*b*c*d**3*x + 2*sqrt(b)*log(sqrt(a + b*x**2) - ...