\(\int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^3} \, dx\) [154]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {(B c-A d) (a d-b c x) \sqrt {a+b x^2}}{2 d \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {B \sqrt {a+b x^2}}{d^2 (c+d x)}+\frac {\sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}+\frac {b \left (2 b B c^3+a d^2 (3 B c-A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^3 \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.54 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (-a d^2 (A d+B (c+2 d x))+b c \left (A d^2 x-B c (2 c+3 d x)\right )\right )}{\left (b c^2+a d^2\right ) (c+d x)^2}-\frac {2 b \left (2 b B c^3+a d^2 (3 B c-A d)\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}-2 \sqrt {b} B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {680, 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.

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

\(\Big \downarrow \) 680

\(\displaystyle -\frac {\int \frac {2 b \left (a d (B c-A d)-2 B \left (b c^2+a d^2\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{4 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (d x \left (2 a B d^2+b c (3 B c-A d)\right )+a d^2 (A d+B c)+2 b B c^3\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((2*b*B*c^3 + a*d^2*(B*c + A*d) + d*(2*a*B*d^2 + b*c*(3*B*c - A*d))*x 
)*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)*(c + d*x)^2) - (b*((-2*B*(b*c^2 + 
a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - ((2*b*B*c^3 + a 
*d^2*(3*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])))/(2*d^2*(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 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 680

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

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. \(1444\) vs. \(2(171)=342\).

Time = 1.48 (sec) , antiderivative size = 1445, normalized size of antiderivative = 7.57

method	result	size
default	\(\text {Expression too large to display}\)	\(1445\)

Input:

int((B*x+A)*(b*x^2+a)^(1/2)/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

B/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)^(3/2)-b*c*d/(a*d^2+b*c^2)*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2* 
b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(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)))+2*b/( 
a*d^2+b*c^2)*d^2*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2 
)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2) 
/d^2)^(1/2))))+(A*d-B*c)/d^4*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^ 
2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+1/2*b*c*d/(a*d^2+b*c^2)*(-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)^(3/ 
2)-b*c*d/(a*d^2+b*c^2)*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1 
/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(1/2))-(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)))+2*b/(a*d^2+b*c^2)* 
d^2*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2 
)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/d+ 
b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2...
 

Fricas [F(-1)]

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

integrate((B*x+A)*(b*x^2+a)^(1/2)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((B*x+A)*(b*x**2+a)**(1/2)/(d*x+c)**3,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (172) = 344\).

Time = 0.10 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.87 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {\sqrt {b x^{2} + a} B b c^{2}}{2 \, {\left (b c^{2} d^{3} x + a d^{5} x + b c^{3} d^{2} + a c d^{4}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B c}{2 \, {\left (b c^{2} d^{2} x^{2} + a d^{4} x^{2} + 2 \, b c^{3} d x + 2 \, a c d^{3} x + b c^{4} + a c^{2} d^{2}\right )}} - \frac {\sqrt {b x^{2} + a} A b c}{2 \, {\left (b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}\right )}} - \frac {\sqrt {b x^{2} + a} B b c}{2 \, {\left (b c^{2} d^{2} + a d^{4}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{2 \, {\left (b c^{2} d x^{2} + a d^{3} x^{2} + 2 \, b c^{3} x + 2 \, a c d^{2} x + \frac {b c^{4}}{d} + a c^{2} d\right )}} + \frac {\sqrt {b x^{2} + a} A b}{2 \, {\left (b c^{2} d + a d^{3}\right )}} - \frac {\sqrt {b x^{2} + a} B}{d^{3} x + c d^{2}} + \frac {B \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} + \frac {B b^{2} c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{6}} - \frac {A b^{2} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{5}} - \frac {3 \, B b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} + \frac {A b \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} \] Input:

integrate((B*x+A)*(b*x^2+a)^(1/2)/(d*x+c)^3,x, algorithm="maxima")
 

Output:

1/2*sqrt(b*x^2 + a)*B*b*c^2/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4) 
+ 1/2*(b*x^2 + a)^(3/2)*B*c/(b*c^2*d^2*x^2 + a*d^4*x^2 + 2*b*c^3*d*x + 2*a 
*c*d^3*x + b*c^4 + a*c^2*d^2) - 1/2*sqrt(b*x^2 + a)*A*b*c/(b*c^2*d^2*x + a 
*d^4*x + b*c^3*d + a*c*d^3) - 1/2*sqrt(b*x^2 + a)*B*b*c/(b*c^2*d^2 + a*d^4 
) - 1/2*(b*x^2 + a)^(3/2)*A/(b*c^2*d*x^2 + a*d^3*x^2 + 2*b*c^3*x + 2*a*c*d 
^2*x + b*c^4/d + a*c^2*d) + 1/2*sqrt(b*x^2 + a)*A*b/(b*c^2*d + a*d^3) - sq 
rt(b*x^2 + a)*B/(d^3*x + c*d^2) + B*sqrt(b)*arcsinh(b*x/sqrt(a*b))/d^3 + 1 
/2*B*b^2*c^3*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^6) - 1/2*A*b^2*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^5) - 3/2*B*b*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*a 
bs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^4) + 1/2*A*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^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (172) = 344\).

Time = 0.20 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.23 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^3} \, dx=-\frac {{\left (2 \, B b^{2} c^{3} + 3 \, B a b c d^{2} - A a b d^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} d^{3} + a d^{5}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {B \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{d^{3}} - \frac {4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B b^{2} c^{3} d - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b^{2} c^{2} d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B a b c d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a b d^{4} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B b^{\frac {5}{2}} c^{4} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {5}{2}} c^{3} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a b^{\frac {3}{2}} c^{2} d^{2} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} c d^{3} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} d^{4} - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a b^{2} c^{3} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b^{2} c^{2} d^{2} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b d^{4} + 3 \, B a^{2} b^{\frac {3}{2}} c^{2} d^{2} - A a^{2} b^{\frac {3}{2}} c d^{3} + 2 \, B a^{3} \sqrt {b} d^{4}}{{\left (b c^{2} d^{3} + a d^{5}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} \] Input:

integrate((B*x+A)*(b*x^2+a)^(1/2)/(d*x+c)^3,x, algorithm="giac")
 

Output:

-(2*B*b^2*c^3 + 3*B*a*b*c*d^2 - A*a*b*d^3)*arctan(-((sqrt(b)*x - sqrt(b*x^ 
2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^2*d^3 + a*d^5)*sqrt(-b* 
c^2 - a*d^2)) - B*sqrt(b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^3 - (4* 
(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*b^2*c^3*d - 2*(sqrt(b)*x - sqrt(b*x^2 + 
a))^3*A*b^2*c^2*d^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a*b*c*d^3 - (sqr 
t(b)*x - sqrt(b*x^2 + a))^3*A*a*b*d^4 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2* 
B*b^(5/2)*c^4 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*b^(5/2)*c^3*d + (sqrt( 
b)*x - sqrt(b*x^2 + a))^2*B*a*b^(3/2)*c^2*d^2 + (sqrt(b)*x - sqrt(b*x^2 + 
a))^2*A*a*b^(3/2)*c*d^3 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b)* 
d^4 - 8*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a*b^2*c^3*d + 2*(sqrt(b)*x - sqrt( 
b*x^2 + a))*A*a*b^2*c^2*d^2 - 5*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^2*b*c*d^ 
3 - (sqrt(b)*x - sqrt(b*x^2 + a))*A*a^2*b*d^4 + 3*B*a^2*b^(3/2)*c^2*d^2 - 
A*a^2*b^(3/2)*c*d^3 + 2*B*a^3*sqrt(b)*d^4)/((b*c^2*d^3 + a*d^5)*((sqrt(b)* 
x - sqrt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 + a))*sqrt(b)*c - a*d 
)^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1555, normalized size of antiderivative = 8.14 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(b*x^2+a)^(1/2)/(d*x+c)^3,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**2*b*c**2*d**3 + 2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqr 
t(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c*d**4*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*d**5*x**2 
 - 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**3*d**2 - 6*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**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**4*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)*b**3*c**5 - 4*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**4*d*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)*b**3*c** 
3*d**2*x**2 - sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**2*d**3 - 2*sqrt 
(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**4*x - sqrt(a*d**2 + b*c**2)*log 
(c + d*x)*a**2*b*d**5*x**2 + 3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c 
**3*d**2 + 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**2*d**3*x + 3*sqr 
t(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c*d**4*x**2 + 2*sqrt(a*d**2 + b*c** 
2)*log(c + d*x)*b**3*c**5 + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**4 
*d*x + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**3*d**2*x**2 - sqrt(a + 
 b*x**2)*a**3*d**6 - sqrt(a + b*x**2)*a**2*b*c**2*d**4 + sqrt(a + b*x**...